![Preface
In this book, we continue the investigations of global attractors and invariant sets
for dynamical systems by means of Lyapunov functions and adapted metrics. The
effectiveness of such approaches for the approximation and localization of attractors
for different classes of dynamical systems was already shown in Abramovich et al.
[1, 2] and in [30, 32]. In particular, Lyapunov functions and adapted metrics were
constructed for global stability problems and the existence of homoclinic orbits in
the Lorenz system using frequency-domain methods and reduction principles.
In 1980, investigators of differential equations and general dynamical systems
were greatly impressed by a paper about upper estimates of the Hausdorff dimen-
sion of flow and map invariant sets written by Douady and Oesterlé [12]. The
Douady–Oesterlé approach, the significance of which can be compared with that of
Liouville’s theorem, has been developed and modificated in many papers for var-
ious types of dimension characteristics of attractors generated by dynamical sys-
tems: Ledrappier [22], Constantin et al. [11], Smith [42], Eden et al. [13], Chen [9],
Hunt [17], Boichenko and Leonov [4].
After Ya. B. Pesin had worked out [39] a general scheme of introducing metric
dimension characteristics, this method made it possible to define from a unique
point of view various types of outer measures and dimensions, such as the
Hausdorff dimension, the fractal dimension, the information dimension as well as
the topological and metric entropies. The Pesin scheme naturally led to the char-
acterization of a class of Carathéodory measures [25, 27], which are adapted to the
specific character of attractors of autonomous differential equations, i.e. to the fact
that these attractors consist wholly of trajectories. The neighborhoods of pieces
of these trajectories form a covering of the attractor. It serves as the base for
introducing the special outer Carathéodory measures. A number of effective tools
for estimating these measures were developed within the theory of differential
equations in Euclidean space and well-known results by Borg [8], Hartman and
Olech [15], when analysing the orbital stability of solutions. The most important
property, used in dimension theory, is the fact that the Carathéodory measures are
majorants for the associated Hausdorff measures.
v](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-10-320.jpg)
![Early in the nineties of the last century, G. A. Leonov and his co-workers
observed deep inner connections between the mentioned direct method of
Lyapunov in stability theory and estimation technics for outer measures in
dimension theory. Introducing the Lyapunov functions and varying Riemannian
metrices (Leonov [24], Noack and Reitmann [38]) into upper estimates of dimen-
sion characteristics of invariant sets made it possible to generalize and improve
[4, 6, 28, 29] some well-known results of R. A. Smith, P. Constantin, C. Foias,
A. Eden, and R. Temam.
On the other hand, Pugh’s closing lemma [40] and theorems about the spanning
of two-dimensional surfaces on a given closed curve gave the opportunity to apply
some theorems about the contraction of Hausdorff measures to global stability
investigations of time-continuous dynamical systems (Smith [42], Leonov [24], Li
and Muldowney [35]). In this book, the effectiveness of introducing the Lyapunov
functions into dimensional characteristics is shown for a number of concrete
dynamical systems: the Hénon map, the systems of Lorenz and Rössler as well as
their generalizations for various physical systems and models (rotation of a rigid
body in a resisting medium, convection of liquid in a rotating ellipsoid, interaction
between waves in plasma, etc.).
Additionally to the derivation of upper Hausdorff dimension estimates, exact
formulas for the Lyapunov dimensions for Lorenz type systems were shown
[26, 28]. Many of these results were presented in [7].
In the following decade, the modified Douady–Oesterlé approach was also used
for new classes of attractors [33].
It was also possible to get different versions of the Douady–Oesterlé theorem for
piecewise continuous maps and differential equations [37, 41]. For cocycles gen-
erated by non-autonomous systems, the upper Hausdorff dimension estimates are
derived in [31, 34]. Some of these results are included in the present book which
provides a systematic presentation of research activities in the dimension theory of
dynamical systems in finite-dimensional Euclidean spaces and manifolds. Let us
briefly sketch the contents of the book.
In Part I, we consider the basic facts from attractor theory, exterior products and
dimension theory. Chapter 1 is devoted to the investigation of various types of
global attractors of dynamical systems in general metric spaces (global B-attractors,
minimal global B-attractors and others). The theoretical results are applied to the
generalized Lorenz system and dynamical systems on the flat cylinder. One section
is concerned with the existence proof of a homoclinic orbit in the Lorenz system
(Leonov [23], Hastings and Troy [16], Chen [10]).
In Chap. 2, some facts on singular values of matrices, the exterior calculus for
spaces and matrices and the Lozinskii matrix norm, necessary for estimation
techniques of outer measures, are presented. In addition to this, the Yakubovich–
Kalman frequency theorem and the Kalman–Szegö theorem about the solvability of
certain matrix inequalities are formulated and used for the estimation of singular
values.
vi Preface](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-11-320.jpg)
![Chapter 3 is an introduction to dimension theory. It starts with the definition and
the basic properties of the topological dimension in the spirit of Hurewicz and
Wallman [18]. Next, the notions of Hausdorff measure, Hausdorff dimension and
fractal dimension are introduced. After this, the topological entropy of a continuous
map is discussed. The last part of the chapter deals with Pesin’s scheme of intro-
ducing the Carathéodory dimension characteristics.
In Part II, we investigate dimension properties of dynamical systems in
Euclidean spaces. It includes estimates of topological dimension of the Hausdorff
and fractal dimensions for invariant sets of concrete physical systems and estimates
of the Lyapunov dimension.
Chapter 4 is concerned with the investigation of dimension properties of almost
periodic flows [3]. We thank M. M. Anikushin for helping us to prepare the first
version of Chap. 4.
Chapter 5 begins with the so-called limit theorem about the evolution of
Hausdorff measures under the action of smooth maps in Euclidean spaces. This
theorem gives the opportunity to include into Hausdorff dimension and Hausdorff
measure estimates, certain varying functions which turn over into Lyapunov
functions when the theorem is applied to differential equations. The method of
varying functions is used in estimations of fractal dimension and topological
entropy. Simultaneously, with the introduction of Lyapunov functions in estimates
for the Hausdorff measure, the logarithmic norms were used by Muldowney [36],
for estimating the two-dimensional Riemannian volumes of compact sets shifted
along the orbits of differential equations. One of the main goals of Chap. 5 is to
combine the Lyapunov function and logarithmic norm approaches (Boichenko and
Leonov [5]), in order to solve a number of problems in the qualitative theory of
ordinary differential equations, such as the generalization of the Liouville formula
and the Bendixson criterion.
Chapter 6 is devoted to finite-dimensional dynamical systems in Euclidean space
and its aim is to explain, in a simple but rigorous way, the connection between the
key works in the area: Kaplan and Yorke (the concept of Lyapunov dimension [19]
1979), Douady and Oesterlé (estimation of Hausdorff dimension via the Lyapunov
dimension of maps [12]), Constantin, Eden, Foias, and Temam (estimation of
Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of
dynamical systems [13]), Leonov (estimation of the Lyapunov dimension via the
direct Lyapunov method [20, 24]), and numerical methods for the computation of
Lyapunov exponents and Lyapunov dimension [21]. We also concentrate in this
chapter on the Kaplan–Yorke formula and the Lyapunov dimension formulas for
the Lorenz and Hénon attractors (Leonov [26]).
In Part III, we consider dimension properties for dynamical systems on mani-
folds. Chapter 7 gives a presentation of the exterior calculus in general linear
spaces. It contains also some results about orbital stability for vector fields on
manifolds.
Chapter 8 is devoted to dimension estimates of invariant sets and attractors of
dynamical systems on Riemannian manifolds. The Douady–Oesterlé theorem for
the upper Hausdorff dimension estimates for invariant sets of smooth dynamical
Preface vii](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-12-320.jpg)
![systems on Riemannian manifolds is proved. Chapter 8 contains also an important
result on the estimation of the fractal dimension of an invariant set on an arbitrary
finite-dimensional smooth manifold by the upper Lyapunov dimension, which goes
back to Hunt [17], Gelfert [14]. Then we discuss the construction of the special
Carathéodory measures for the estimation of Hausdorff measures connected with
flow invariant sets on Riemannian manifolds.
In Chap. 9, we derive dimension and entropy estimates for invariant sets and
global B-attractors of cocycles. A version of the Douady–Oesterlé theorem (Leonov
et al. [31]) is proved for local cocycles in a Euclidean space and for cocycles on
Riemannian manifolds. As examples, we consider cocycles, generated by the
Rössler system with variable coefficients. We also introduce time-discrete cocycles
on fibered spaces and define the topological entropy of such cocycles. We thank
A. O. Romanov for helping us to prepare this chapter.
In Chap. 10, we derive some versions of the Douady–Oesterlé theorem for
systems with singularities. In the first part of this chapter, we consider a special
class of non-injective maps, for which we introduce a factor describing the “degree
of non-injectivity” (Boichenko et al. [6]). This factor can be included in the
dimension estimates of Chap. 8 in order to weaken the condition to the singular
value function. In the second part of Chap. 10, we derive the upper Hausdorff
dimension estimates for invariant sets of a class of not necessarily invertible and
piecewise smooth maps on manifolds with controllable preimages of the
non-differentiability sets in terms of the singular values of the derivative of the
smoothly extended map (Reitmann and Schnabel [41], Neunhäuserer [37]) These
estimates generalize some Douady–Oesterlé type results for differentiable maps in a
Euclidean space, derived in Chaps. 5 and 8.
In the last section of Chap. 10, we discuss some classes of functionals which are
useful for the estimation of topological and metric dimensions.
St. Petersburg, Russia Nikolay Kuznetsov
Volker Reitmann
References
1. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for
oscillations in discrete systems. I., Oscillations in the sense of Yakubovich in discrete sys-
tems. Wiss. Zeitschr. Techn. Univ. Dresden. 25(5/6), 1153–1163 (1977) (German)
2. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for
oscillations in discrete systems. II., Oscillations in discrete phase systems. Wiss. Zeitschr.
Techn. Univ. Dresden. 26(1), 115–122 (1977) (German)
3. Anikushin, M.M.: Dimension theory approach to the complexity of almost periodic trajec-
tories. Intern. J. Evol. Equ. 10(3–4), 215–232 (2017)
4. Boichenko, V.A., Leonov, G.A.: Lyapunov’s direct method in the estimation of the Hausdorff
dimension of attractors. Acta Appl. Math. 26, 1–60 (1992)
viii Preface](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-13-320.jpg)
![5. Boichenko, V.A., Leonov, G.A.: Lyapunov functions, Lozinskii norms, and the Hausdorff
measure in the qualitative theory of differential equations. Amer. Math. Soc. Transl. 193(2),
1–26 (1999)
6. Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann,V.: Hausdorff and fractal dimension
estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre
Anwendungen (ZAA). 17(1), 207–223 (1998)
7. Boichenko, V.A., Leonov, G.A., Reitmann, V.: Dimension Theory for Ordinary Differential
Equations. Teubner, Stuttgart (2005)
8. Borg, G.: A condition for existence of orbitally stable solutions of dynamical systems. Kungl.
Tekn. Högsk. Handl. Stockholm. 153, 3–12 (1960)
9. Chen, Zhi-Min.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic
attractors. Chaos, Solitons & Fractals 3, 575–582 (1993)
10. Chen, X.: Lorenz equations, part I: existence and nonexistence of homoclinic orbits.
SIAM J. Math. Anal. 27(4), 1057–1069 (1996)
11. Constantin, P., Foias, C., Temam, R.: Attractors representing turbulent flows. Amer. Math.
Soc. Memoirs., Providence, Rhode Island. 53(314), (1985)
12. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser.
A. 290, 1135–1138 (1980)
13. Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dynam. Diff. Equ.
3, 133–177 (1991) [Preprint No. 8804, The Institute for Applied Mathematics and Scientific
Computing, Indiana University, 1988]
14. Gelfert, K.: Maximum local Lyapunov dimension bounds the box dimension. Direct proof for
invariant sets on Riemannian manifolds. Zeitschrift für Analysis und ihre Anwendungen
(ZAA). 22(3), 553–568 (2003)
15. Hartman, P., Olech, C.: On global asymptotic stability of solutions of ordinary differential
equations. Trans. Amer. Math. Soc. 104, 154–178 (1962)
16. Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Diff.
Equ. 127(1), 41–53 (1996)
17. Hunt, B.: Maximum local Lyapunov dimension bounds the box dimension of chaotic
attractors. Nonlinearity. 9, 845–852 (1996)
18. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton Univ. Press, Princeton (1948)
19. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In:
Functional Differential Equations and Approximations of Fixed Points, 204–227, Springer,
Berlin (1979)
20. Kuznetsov, N.V.: The Lyapunov dimension and its estimation via the Leonov method.
Physics Letters A, 380(25–26), 2142–2149 (2016)
21. Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Prasad, A., Shrimali, M.D.: Finite-time
Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dyn. 92 (2),
267–285 (2018)
22. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math.
Phys. 81, 229–238 (1981)
23. Leonov, G.A.: On the estimation of the bifurcation parameter values of the Lorenz system.
Uspekhi Mat. Nauk. 43(3), 189–200 (1988) (Russian); English transl. Russian Math. Surveys.
43(3), 216–217 (1988)
24. Leonov, G.A.: Estimation of the Hausdorff dimension of attractors of dynamical systems.
Diff. Urav. 27(5), 767–771 (1991) (Russian); English transl. Diff. Equations, 27, 520–524
(1991)
25. Leonov, G.A.: Construction of a special outer Carathéodory measure for the estimation of the
Hausdorff dimension of attractors. Vestn. S. Peterburg Gos. Univ. 1(22), 24–31 (1995)
(Russian); English transl. Vestn. St. Petersburg Univ. Math. Ser. 1, 28(4), 24–30 (1995)
26. Leonov, G.A.: Lyapunov dimensions formulas for Hénon and Lorenz attractors. Alg. & Anal.
13, 155–170 (2001) (Russian); English transl. St. Petersburg Math. J. 13(3), 453–464 (2002)
Preface ix](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-14-320.jpg)
![Chapter 1
Attractors and Lyapunov Functions
Abstract The main tool in estimating dimensions of invariant sets and entropies
of dynamical systems developed in this book is based on Lyapunov functions. In
this chapter we introduce the basic concept of global attractors. The existence of a
global attractor for a dynamical system follows from the dissipativity of the system.
In order to show the last property we use Lyapunov functions. In this chapter we
also consider some applications of Lyapunov functions to stability problems of the
Lorenz system. A central result is the existence of homoclinic orbits in the Lorenz
system for certain parameters.
1.1 Dynamical Systems, Limit Sets and Attractors
1.1.1 Dynamical Systems in Metric Spaces
Suppose that (M, ρ) is a complete metric space. Let T be one of the sets R, R+
, Z or
Z+
. A map ϕ(·)
(·) : T × M → M resp. a triple ({ϕt
}t∈T, M, ρ) is called a dynamical
system on (M, ρ) if the following conditions are satisfied ([2, 11]):
(1) ϕ0
(u) = u , ∀ u ∈ M ;
(2) ϕt+s
(u) = ϕt
(ϕs
(u)) , ∀ t, s ∈ T, ∀ u ∈ M ;
(3) If T ∈ {R, R+} the map (t, u) ∈ T × M → ϕt
(u) is continuous;
if T ∈ {Z, Z+} the map u ∈ M → ϕt
(u) is continuous on M for any t ∈ T.
If the metric space (M, ρ) is fixed we denote the dynamical system shortly by
{ϕt
}t∈T. The sets T and M are called time sets and phase space, respectively. The
dynamical system ({ϕt
}t∈T, M, ρ) forms a group, if T ∈ {R, Z}, and a semi-group if
T ∈ {R+, Z+}. If T ∈ {R, R+} we say that the dynamical system is with continuous
time, if T ∈ {Z, Z+} we say that the system is with discrete time. A dynamical system
({ϕt
}t∈T, M, ρ) is called flow if T = R, semi-flow if T = R+, and cascade if T = Z.
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical
Systems: Theory and Computation, Emergence, Complexity and Computation 38,
https://doi.org/10.1007/978-3-030-50987-3_1
3](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-25-320.jpg)
![4 1 Attractors and Lyapunov Functions
Example 1.1 Consider the autonomous differential equation
ϕ̇ = f (ϕ) , (1.1)
where f : Rn
→ Rn
is assumed to be locally Lipschitz. The Euclidean norm in Rn
is denoted by | · |. Suppose also that any maximal solution ϕ(·, u) of (1.1) starting
in u at t = 0 exists for any t ∈ R. Let now ϕt
(·) ≡ ϕ(t, ·) : Rn
→ Rn
be the time
t-map of (1.1). Clearly, that by the solution properties of (1.1) (uniqueness theo-
rem and theorem of continuous dependence on initial condition, [12, 34]) the triple
({ϕt
}t∈R, Rn
, | · |) defines a dynamical system with the additive group T = R, i.e. a
flow.
Example 1.2 Assume that
ϕ̇ = f (t, ϕ) (1.2)
is a non-autonomous differential equation with f : R × Rn
→ Rn
. Let us suppose
that f is continuously differentiable, T -periodic in the first argument and that any
solution exists on R. Denote the solution of (1.2) starting in u at t = t0 by ϕ(·, t0, u).
Again by the uniqueness theorem and the theorem of continuous dependence of
solutions on initial conditions for ODE’s it follows that the family of maps ϕm
(·) ≡
ϕ(m T, 0, ·), m ∈ Z, defines a dynamical system ({ϕm
}m∈Z, Rn
, | · |) which is a cas-
cade. Let us demonstrate this. Clearly, ϕ0
(u) = ϕ(0, 0, u) = u, ∀ u ∈ Rn
. In order to
show the property (2) of a dynamical system we consider arbitrary m, k ∈ Z. Define
the two functions c1(t) := ϕ(t, 0, ϕ(mT, 0, u)) and c2(t) := ϕ(t + mT, 0, u). From
the T -periodicity of f in the first variable it follows that c2 is also a solution of (1.2)
on R, i.e.
ċ2(t) = f (t + mT, ϕ(t + mT, 0, u)) = f (t, c2(t)).
Since c2(0) = ϕ(mT, 0, u), it follows by the uniqueness theorems that
c1(t) = c2(t), ∀ t ∈ R.
If we put t = kT , the last property results in c1(kT ) ≡ ϕk
(ϕm
(u)) = c2(kT ) ≡
ϕk+m
(u).
Example 1.3 Suppose that
ϕ : M → M (1.3)
is a continuous invertible map on the complete metric space (M, ρ). Let us define
the family of maps
ϕm
:=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
ϕ ◦ ϕ ◦ · · · ◦ ϕ
m-times
for m = 1, 2, . . . ,
idM for m = 0 ,
ϕ−1
◦ ϕ−1
◦ · · · ◦ ϕ−1
-m-times
for m = −1, −2, . . . .
(1.4)](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-26-320.jpg)
![6 1 Attractors and Lyapunov Functions
is the orbit through u. If T ∈ {R, Z} we consider also the positive and the negative
semi-orbit through u defined by
γ +
(u) :=
t∈T∩R+
ϕt
(u) resp. γ −
(u) :=
t∈T∩R−
ϕt
(u) .
An orbit γ (u) is called stationary, critical or an equilibrium if γ (u) = {u}. The orbit
γ (u)ofadynamicalsystemiscalled T -periodicwithperiod T if T 0 isthesmallest
positive number in T such that ϕt
(u) = ϕt+T
(u), ∀ t ∈ T. A set Z ⊂ M is said to be
positively invariant if ϕt
(Z) ⊂ Z, ∀ t ∈ T ∩ R+, invariant if ϕt
(Z) = Z, ∀ t ∈ T,
and negatively invariant, if ϕt
(Z) ⊃ Z, ∀ t ∈ T ∩ R+. The positively invariant set
Z of the dynamical system ({ϕt
}t∈T, M, ρ) is said to be stable if in any neighborhood
U of Z there exists a neighborhood U such that ϕt
(U ) ⊂ U, ∀t ∈ T+. Z is called
asymptotically stable if it is stable and ϕt
(u) → Z as t → +∞ for each u ∈ U .
The set Z is said to be globally asymptotically stable if Z is stable and ϕt
(u) → Z
as t → +∞ for each u ∈ M. The set Z is called uniformly asymptotically stable if
it is stable and limt→+∞{dist(ϕt
(u), Z) | u ∈ U } = 0.
For any u ∈ M the ω-limit set of u under {ϕt
}t∈T is the set
ω(u) := {υ ∈ M | ∃{tn}n∈N , tn ∈ T , tn → +∞ , ϕtn
(u) → υ for n → +∞} .
For a subset Z ⊂ M we define its ω-limit set ω(Z) under {ϕt
}t∈T as the set of the
limits of all converging sequences of the form ϕtn
(un), where un ∈ Z, tn ∈ T, and
tn → +∞.
Example 1.6 Consider a class of modified horseshoe maps ϕ which are defined
on an open neighborhood U = (−δ, 1 + δ) × (−δ, 1 + δ) ⊂ R2
of the unit square
C = [0, 1] × [0, 1], where δ 0 is a sufficiently small number. The map is defined
for points (x, y) ∈ C in such a way that it first contracts C horizontally with a factor
α 1
2
and stretches it vertically with a function f , then it is folded along a horizontal
line such that the vertical edge of the resulting rectangle is greater than 2, and finally
it is formed into an horseshoe (see Fig.1.1) in such a way that the map can be con-
tinuously extended to U and is continuously differentiable on an open neighborhood
U of K =
∞
i=−∞ ϕi
(C) , where ϕi
(·) for negative numbers i means the preimage
under the map ϕ−i
.
For example we take α = 1
3
and let the function f stretch C with factor β1 = 3,
if y ≤ 51
65
=: h, with a factor β2 between 3 and 5, if 51
65
y 4
5
and with factor 5,
if y ≥ 4
5
, the resulting rectangle is folded on the image of the line y = 51
65
and then
it is formed to an horseshoe in such a way, that the map satisfies
ϕ(x, y) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
3
x, 3y − 1
13
if 0 ≤ y ≤ 14
39
,
1 − 1
3
x, 148
65
− 3y
if 83
195
≤ y ≤ 148
195
,
1 − 1
3
x, 5y − 4
if 4
5
≤ y ≤ 1 .](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-28-320.jpg)
![1.1 Dynamical Systems, Limit Sets and Attractors 7
Fig. 1.1 Modified horseshoe map
The set K =
∞
i=−∞ ϕi
(C) is invariant under the map. Therefore if we take
K = K,
then ϕ j
(K) ⊂
K is satisfied for any j = 1, 2, . . . . The set K can be constructed step
by step starting with K0
= C. At every step i 1 we get Ki
:= Ki−1
∩ ϕ(Ki−1
) ∩
ϕ−1
(Ki−1
) and in the limit the invariant set K as K =
∞
i=0 Ki
. The set Ki
consists
of 6i
rectangles where the lengths of the edges are horizontally 1
3i and vertically 1
3i
and 1
5i , respectively (see Fig.1.1).
It is easy to verify that the following proposition is true ([4, 7, 38]).
Proposition 1.1 For any subset Z ⊂ M the ω-limit set under {ϕt
}t∈T is given by
ω(Z) =
s≥0
s∈T
t≥s
t∈T
ϕt (Z) .
Here for a set Z ⊂ M we denote by Z its closure in the topology of the metric
space (M, ρ).
If T ∈ {R, Z} we also consider the α-limit set of a point u ∈ M under {ϕt
}t∈T
defined by
α(u) := {υ ∈ M | ∃{tn}n∈N , tn ∈ T, tn → −∞, ϕtn
(u) → υ for n → +∞}
and the α-limit set ω(Z) of a set Z ⊂ M under {ϕt
}t∈T given as the set of the limits of
all converging sequences of the form ϕtn
(un), where pn ∈ Z, tn ∈ T, and tn → −∞.
A set Zmin ⊂ M is called minimal for ({ϕt
}t∈T, M, ρ) if it is closed, invariant, and
does not have any proper subset with the same properties. The following proposition
is taken from [33, 36].
Proposition 1.2 Suppose that Z ⊂ M is non-empty, compact and invariant for
({ϕt
}t∈T, M, ρ). Then Z contains a minimal set Zmin.
Proof (For the case that T is a semi-group). If Z has no proper subset, which is
closed and invariant, then Z is minimal, and the proposition is proved.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-29-320.jpg)
![8 1 Attractors and Lyapunov Functions
Suppose that there exists Z1 ⊂ Z, Z1 = Z, such that Z1 is closed and invariant.
If Z1 contains no proper subset being closed and invariant, then it is minimal.
Suppose again that there exists a closed invariant set Z2 ⊂ Z1 with Z2 = Z1. If
we can continue this process and at any step we obtain a new minimal set Zi , we get
the sequence of closed invariant sets
Z0 := Z ⊃ Z1 ⊃ Z2 ⊃ · · · .
The intersection of these sets Zω :=
∞
i=0 Zi is non-empty and compact. Let us show
that Zω is invariant. Suppose that u ∈ Zω is arbitrary. Then for any integer k ≥ 0
we have u ∈ Zk. It follows that ϕt
(u) ∈ Zk , t ≥ 0, for any k. But this means that
ϕt
(u) ∈ Zω and, consequently, Zω ⊂ ϕt
(Zω). The inverse inclusion can be proved
analogously. Thus the set Zω is invariant.
If the set Zω is not minimal then there exists a closed and invariant set Zω+1 ⊂ Zω
with Zω+1 = Zω. If we can continue this process and if β is the transfinite limit
number for which the sets Zα are constructed for all α β, we put
Zβ :=
αβ
Zα .
Clearly, the set Zβ is closed and invariant. Thus we get the transfinite sequence of
sets
Z ⊃ Z1 ⊃ · · · ⊃ Zk ⊃ · · · ⊃ Zω ⊃ · · · ⊃ Zβ ⊃ · · · .
According to the Baire-Hausdorff theorem ([16], Theorem A.14.1) there exists a
transfinite number β of the second class such that Zβ = Zβ+1, i.e. the set Zβ has no
proper closed and invariant subset. Consequently, Zβ is a minimal set.
The next result follows immediately from Proposition 1.2.
Proposition 1.3 Suppose that the positive semi-orbit of ({ϕt
}t∈T, M, ρ) , starting
in p, is relatively compact. Then ω(p) contains a minimal set.
Some important properties of ω-limit sets are proved in the next proposition ([18]).
Proposition 1.4 Let ({ϕt
}t∈T, M, ρ) be a dynamical system with a semi-group as
time set and let Z ⊂ M be a non-empty set such that for some t0 0, t0 ∈ T, the
set
t≥t0,t∈T
ϕt
(Z) is relatively compact.
Then ω(Z) is non-empty, compact and invariant. Furthermore, ω(Z) is a minimal
closed set which attracts Z.
Proof Recall that relative compactness of a set means that the closure of this set
is compact. Since Z = ∅ the sets Bs =
t≥s,t∈T
ϕt
(Z) are non-empty for all s ≥ 0,
s ∈ T. Consequently, the sets Bs are non-empty compact sets for s ≥ t0 and Bs1
⊂ Bs2](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-30-320.jpg)
![1.1 Dynamical Systems, Limit Sets and Attractors 9
for all s1 ≥ s2 in T. Therefore ω(Z) =
s≥0,s∈T
Bs is a non-empty compact set and
attracts Z.
If u ∈ ϕt
(ω(Z)) then for a certain υ ∈ ω(Z) we have u = ϕt
(υ). Hence there
exists a sequence um ∈ Z and a sequence {tm}, tm ∈ T, tm → +∞ as m → +∞,
such that limm→+∞ ϕt
(ϕtm
(u)) = limm→+∞ ϕt+tm
(um) = ϕt
(υ) = u. But this means
that u ∈ ω(Z).
Let us prove the reverse inclusion. If υ ∈ ω(Z) a sequence um ∈ Z and a sequence
{tm}fromTexistsuchthattm → +∞asm → +∞, 1 + t0 + t ≤ t1 t2 · · · ,and
ϕtm
(um) → υ for m → +∞ . (1.7)
For tm ≥ t the sequence ϕtm −t
(um) belongs to the relatively compact set
t≥t0+1,
t∈T
ϕt
(Z). Consequently, passing if necessary to a subsequence, we may sup-
pose that there exists a point u ∈ M such that
ϕtm −t
(um) → u as m → +∞ .
But this means that u ∈ ω(Z). Since
ϕtm
(um) = ϕt
(ϕtm −t
(um)) → ϕt
(u) as m → +∞ ,
by (1.7) we have υ = ϕt
(u). Thus, υ ∈ ϕt
(ω(Z)).
It remains to show that ω(Z) is a minimal closed set which attracts Z. Let us
argue as in [18]. Suppose the contrary and let C be a proper closed subset of ω(Z)
which attracts Z. As ω(Z) is compact so is C. Choose any υ ∈ ω(Z) C. For ε
0 small enough the ε-neighborhoods Uε(υ) and Uε(C) do not intersect. Since C
attracts Z there is a t = t(ε) ≥ 0 such that ϕt
(Z) ⊂ Uε(C), ∀t ≥ t(ε). On the other
hand since υ ∈ ω(Z), υ = limk→∞ ϕtk
(uk) for some uk ∈ Z and tk → +∞ is a
sequence. Consequently, ϕtk
(Z) ∩ Uε(υ) = ∅ for sufficiently large tk. Hence Uε(υ)
∩ Uε(C) = ∅, a contradiction.
Assume that ({ϕt
}t∈T, M, ρ) is a dynamical system with a group as time set and
u ∈ M is an arbitrary point. The sets
Ws
(u) := {υ ∈ M| lim
t→+∞
ϕt
(υ) = u} and
Wu
(u) := {υ ∈ M| lim
t→−∞
ϕt
(υ) = u}
are called stable and unstable manifold, respectively, in u. Since the orbits are invari-
ant, the sets Ws
(u) and Wu
(u) are also invariant. Suppose that u and υ are equilibria
of the dynamical system. Then any orbit which is contained in Ws
(u) ∩ Wu
(υ) is
called heteroclinic if u = υ, and homoclinic if u = υ.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-31-320.jpg)
![1.1 Dynamical Systems, Limit Sets and Attractors 11
Proposition 1.5 Let A be a global B-attractor and ε 0 an arbitrary number. Then
the ε-neighborhood of A is B-absorbing for the dynamical system.
Remark 1.1 Minimal global attractors and B-attractors where introduced by
O.A. Ladyzhenskaya in [18]. Our Definition 1.1 follows the representation given
in [18]. Important properties of minimal global attractors are also derived in [6, 7,
15, 35, 37].
The existence of a global B-attractor is shown in the next proposition ([17]).
Note that if a global B-attractor exists, then it contains a minimal global B-
attractor.
Proposition 1.6 Suppose that the dynamical system ({ϕt
}t∈T, M, ρ) is
B-dissipative according to the bounded B-absorbing set B0 and there exists a t0 0
such that the set
t≥t0,t∈T
ϕt
(B0) is relatively compact. Then
AM,min := {ω(B) | B ⊂ M, B bounded}
is a minimal global B-attractor and
AM,min :=
u∈M
ω(u)
is a minimal global attractor of the dynamical system.
Proof Since by Proposition 1.4 every bounded set B ⊂ M is attracted to its ω-
limit set ω(B) and to AM,min, it is attracted to ω(B) ∩ AM,min. Since ω(B) is
minimal, it lies in AM,min. The set AM,min, is invariant and minimal. It follows that
ω(AM,min) = AM,min and the representation for AM,min, is shown.
The fact that
u∈M ω(u) is a minimal global attractor follows from the properties
of an ω-limit set.
Remark 1.2 In contrast to the minimal global attractor given by Proposition 1.6 a
minimal global attractor can be unbounded. The dynamical system generated by the
ODE
ẋ = 0 , ẏ = − ay , a 0
has as a minimal global B-attractor AR2,min the x-axis (see Sect. 2.1, Chap. 2)
Other examples of minimal global attractors and global B-attractors will be con-
sidered in the sequel.
A dynamical system ({ϕt
}t∈T, M, ρ) is called locally completely continuous if
for any u ∈ M there exists a δ = δ(u) 0 and an l = l(u) 0,l(u) ∈ T+, such
that ϕl
(Bδ(u)) is relatively compact. It is clear that a dynamical system given in a
locally compact space is locally completely continuous.
The next proposition is a result of [4].](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-33-320.jpg)
![12 1 Attractors and Lyapunov Functions
Proposition 1.7 For a locally completely continuous dynamical system pointwise
dissipativity and B-dissipativity are equivalent.
Proof We have to show that a dynamical system which is pointwise dissipative is
also B-dissipative. From the pointwise dissipativity it follows that there exists a
non-empty compact set
K ⊂ M such that for each ε 0 and u ∈ M there exists a
δ(u) 0 and a τ(ε, u) such that
dist (ϕt
(υ),
K) ε (1.8)
for all t ≥ τ(ε, u), t ∈ T, and all υ ∈ Bδ(u)(u). Suppose B is an arbitrary bounded set
whichis containedinacompact setK. Thenfor anyu ∈ K thereexists aδ = δ(u) 0
and τ = τ(ε, u) 0 such that (1.8) is satisfied. Consider an open cover {Bδ(u)(u)}u∈K
of K. Since K is compact and M is a complete metric space there is a finite subcover
{Bδ(ui )(ui )}m
i=1 of K.
Define ˜
l(ε, K) := max{τ(ε, ui )|i = 1, . . . , m}. Then it follows from (1.8) that
dist (ϕt
(K), K) ε, ∀t ˜
l(ε, K).
Since for dynamical systems in locally compact metric spaces pointwise dissipa-
tivity and B-dissipativity are equivalent we call these properties shortly dissipativity.
The next proposition is proved in [7]. It shows that Lyapunov functions can give
a good inside in the structure of an attractor.
In this book the term Lyapunov function for a dynamical system ({ϕt
}t∈T, M, ρ)
means a scalar valued continuous function V which is considered along the orbits
and whose properties allow some conclusions about the qualitative behaviour of the
dynamical system. If M is a manifold and V is differentiable the properties of V
depend on the Lie derivative of V w.r.t. the dynamical system (see Subsect. 1.2.3).
Proposition 1.8 Suppose for the dynamical system ({ϕt
}t∈T, M, ρ) with a group as
time set there exists a compact global B-attractor AM and a continuous function
V : M → R with the following properties:
(1)Foranyu ∈ Mthefunction V (ϕt
(u))isnon-increasingwithrespecttot ∈ T+;
(2) If for some t0 0, t0 ∈ T+, the equation V (u) = V (ϕt0
(u)) holds, then u is
an equilibrium of the dynamical system.
Then: (a) AM = Wu
(C), where C is the set of equilibrium points of the dynamical
system.
(b) The global minimal attractor
AM,min of the dynamical system is C.
Remark 1.3 Suppose ({ϕt
}t∈T, M, ρ) is a dynamical system with a group as time
set, which has a bounded minimal global B-attractor A. Then Wu
(C) ⊂ A, where C
is the set of equilibria of the dynamical system. For a proof see [7].
For dynamical systems ({ϕt
}t∈T, M, ρ) given on a Riemannian smooth n-dimen-
sional manifold (M, g) we define a Milnor attractor as a closed invariant set
A
having the property limt→+∞ dist(ϕt
(u), A) = 0 for each u ∈ S, where S is a set of
positive Lebesgue measure.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-34-320.jpg)
![16 1 Attractors and Lyapunov Functions
Proposition 1.9 The dynamical system ({ϕt
}t∈T, Rn
, | · |) is dissipative in the sense
of Levinson if and only if there exists a bounded set D ⊂ Rn
that attracts any point
in Rn
.
Proof Let the dynamical system be dissipative in the sense of Levinson. Choose as
the set D a ball of radius R, where R is from Definition 1.2, and with center in the
origin. It is obvious that such a D attracts every point in Rn
.
Conversely, let D be a set for the dynamical system which attracts every point of
Rn
, and let ε 0 be an arbitrary number. Choose R so large that the ball of radius R
with center in the origin contains the ε-neighborhood Dε of D. It is clear that such
an R satisfies Definition 1.2.
It is easy to see that if the dynamical system is dissipative in the sense of Levinson
withD asregionofdissipativity,thenanyattractorAofthedynamicalsystemsatisfies
the inclusion A ⊂ D.
1.2.2 Dissipativity and Completeness of The Lorenz System
Consider the Lorenz system ([28, 30, 39])
ẋ = −σ x + σ y , ẏ = rx − y − xz , ż = −bz + xy (1.12)
where σ,r and b are positive parameters. Let us show that equation (1.12) is dissi-
pative. Introduce the auxilary function V : R3
→ R+ given by
V (x, y, z) :=
1
2
x2
+ y2
+ (z − σ − r)2
. (1.13)
Direct computation along an arbitrary solution u = (x, y, z) of (1.12) shows that the
derivative of V along
V̇ (x, y, z) = − σ x2
− y2
−
b
2
(z − σ − r)2
+
b
2
(σ + r)2
.
Thus in R3
we have
V̇ ≤
b
2
(σ + r)2
. (1.14)
On the set
E1 := (x, y, z) | σ x2
+ y2
+
b
2
(z − σ − r)2
≤
b
2
(σ + r)2](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-38-320.jpg)
![1.2 Dissipativity 17
the inequality V̇ ≥ 0 is true and on the set R3
E1 we have V̇ 0. For large R the ball
BR = {(x, y, z) | V (x, y, z) R} contains the ellipsoid E1. On the boundary of BR,
i.e. on the set SR = {(x, y, z) | V (x, y, z) = R} the inequality V̇ 0 is satisfied. It
follows that BR is a bounded absorbing set. If we put κ = min{σ, 1, b
2
} we conclude
that along the solution of (1.12)
V̇ ≤ −2κV +
b
2
(σ + r)2
.
This means that any solution of (1.12) enters the ellipsoid
E2 := (x, y, z) |
1
2
x2
+ y2
+ (z − σ − r)2
≤
b
4κ
(σ + r)2
and remains there during the positive existence interval. From (1.14) we have
V (x(t), y(t), z(t)) ≤ V (x(0), y(0), z(0)) +
b
2
(σ + r)2
t
for t ≥ 0. Since V cannot go to infinity in a finite positive time, each of |x(t)|, |y(t)|,
and |z(t)| cannot go to infinity in a finite positive time. Thus the Lorenz system is
complete in positive time. Thus the Lorenz system defines a semi-flow in R3
. From
(1.13) it follows that for a sufficiently large κ1 0 we have
V̇ + κ1V ≥
b
2
(σ + r)2
=: c1 (1.15)
From (1.15) we get
d
dt
(eκ1 t
V ) ≥ c1eκ1t
.
Thus for t ≤ 0 we have
V (x(0), y(0), z(0)) − eκ1 t
V (x(t), y(t), z(t)) ≥
c1
κ1
[1 − eκ1 t
]
or V (x(t), y(t), z(t)) ≤ e−κ1 t
V (x(0), y(0), z(0)) +
c1
κ1
[1 − e−κ1 t
] .
Thus V (x(t), y(t), z(t)) cannot go to infinity in finite negative time. Hence each
of |x(t)|, |y(t)| and |z(t)| cannot got to infinity in finite negative time and the system
is complete in negative time ([8]).
Let us obtain other estimates for the region of dissipativity for (1.12). Consider
next the function
V1(x, y, z) :=
1
2
x2
+
1
2
y2
+
1
2
z2
− (σ + r)z .](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-39-320.jpg)
![18 1 Attractors and Lyapunov Functions
Let us show that for an arbitrary solution u = (x, y, z) of (1.12) with b 1 we have
lim sup
t→+∞
V1(x(t), y(t), z(t)) ≤ c2 , (1.16)
where c2 := (σ+r)2
(b−2)2
8(b−1)
. Indeed, a calculation shows that
V̇1 + 2V1 = −(σ − 1)x2
− (b − 1)z2
+ (σ + r)(b − z)z
≤ −(b − 1)z2
+ (σ + r)(b − 2)z ≤ 2 c2 .
Therefore we have
d
dt
(V1 − c2) + 2 (V1 − c2) ≤ 0 .
Multiplying the last inequality by e2t
we get for t ≥ 0
d
dt
[(V1 − c2)e2t
] ≤ 0 . (1.17)
Integrating (1.17) on [0, t], we obtain
V1(x(t), y(t), z(t)) − c2 ≤ [V1(x(0), y(0), z(0)) − c2] e−2t
,
from which the inequality (1.16) results. From (1.16) it follows that the ellipsoid
{(x, y, z) ∈ R3
| x2
+
1
2
y2
+
1
2
z2
− (σ + r)z ≤ c2}
is a region of dissipativity for (1.12).
Let us now show that for an arbitrary solution u = (x, y, z) of (1.12)
lim sup
t→+∞
[y2
(t) + (z(t) − r)2
] ≤ l2
r2
(1.18)
and, if 2σ − b ≥ 0,
lim inf
t→+∞
[2 σz(t) − x2
(t)] ≥ 0 . (1.19)
The parameter l in (1.18) is defined by
l :=
1 , if b ≤ 2 ,
b
2
√
b−1
, if b ≥ 2 .
(1.20)
In order to prove (1.18) we put for (x, y, z) ∈ R3
V2(y, z) :=
1
2
y2
+ (z − r)2
.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-40-320.jpg)
![1.2 Dissipativity 19
Suppose that κ0 := min{1, b}. Then for any κ ∈ (0, κ0) we have
V̇2 + 2κV2 = (κ − 1)y2
+ (κ − b)z2
− 2 r
κ −
b
2
z + κr2
≤ (κ − b) z −
r(κ − b/2)
κ − b
!2
−
r2
(κ − b/2)2
κ − b
+ κr2
≤ κ −
(κ − b/2)2
κ − b
!
r2
=
b2
r2
4 (b − κ)
.
It follows that
lim sup
t→+∞
V2(y(t), z(t)) ≤
b2
r2
8 (b − κ)κ
. (1.21)
Minimizing the right-hand side of (1.21) over κ ∈ (0, κ0) we obtain (1.18). To prove
(1.19) we put
V3(x, z) := σz −
1
2
x2
.
The direct computation shows that
V̇3 = −b σz −
2σ
b
1
2
x2
!
≥ −b V3 .
The last relation implies (1.19).
From (1.18) it follows that the region of dissipativity D satisfies the inclusion
D ⊂ D1 ,
where D1 := {(x, y, z) | y2
+ (z − r)2
l2
r2
} is a cylinder in R3
.
Under the condition 2σ − b ≥ 0 it follows from (1.18) and (1.19) that
D ⊂ D1 ∩ {(x, y, z) | z ≥ 0} .
Remark 1.4 Using the Lyapunov function (1.13) and Proposition 1.7 one sees that
the Lorenz system has a compact attracting set which attracts bounded sets. It follows
that the Lorenz system is B-dissipative and has a minimal B-attractor AR3,min which
satisfies AR3,min ⊂ D. Note that any other attractor of (1.12) also belongs to D.
Remark 1.5 Let us consider the system
ẋ = −σ x + σ y , ẏ = rx − y + xz , ż = −bz + xy (1.22)
withpositiveparametersσ,r andb.ThissystemdiffersfromtheLorenzsystem(1.12)
only in the sign of the nonlinearity xz in the second equation and the divergence of the
right-hand side of (1.22) is −(σ + 1 + b) 0, i.e. the same as in the Lorenz system.
However, it was shown in [14] that system (1.22) for any positive parameters has](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-41-320.jpg)
![20 1 Attractors and Lyapunov Functions
solutions converging to infinity for t → +∞. But this means that system (1.22) is
not dissipative. We need certain extra conditions on the right-hand side in order to
guarantee dissipativity.
1.2.3 Lyapunov-Type Results for Dissipativity
Let us consider the dynamical system ({ϕt
}t∈T, M, ρ) on the Riemannian n-
dimensional Ck
-manifold (M, g) which is, for continuous time, given by (1.5)
and for discrete time by (1.6). Suppose that there exists a scalar valued function
V : M → R which is C1
in the continuous-time case and C0
in the discrete-time
case. Define the Lie derivative V̇ (u) w.r.t. the dynamical system in the continuous-
time case by
V̇ (u) :=
d
dt
V (ϕt
(u))|t=0 = (F(u), grad V (u)) (1.23)
and in the discrete-time case by
V̇ (u) := V (ϕ(u)) − V (u). (1.24)
Let us establish the following theorem which is a generalization of a result from
[40, 41], obtained for differential equations in Rn
.
Proposition 1.10 Suppose that there exists a function as introduced above and such
that the following conditions are satisfied:
(1) V is proper for M, i.e. for any compact set K ⊂ R the set V −1
(K) ⊂ M is
compact and V is bounded from below on M;
(2) There exists an r 0 such that V̇ (u) ≤ 0 for u /
∈ Br (0) ;
(3) The dynamical system does not have a motion ϕ(·)
(υ) with ϕt
(υ) /
∈ Br (0) and
V̇ (ϕt
(υ)) ≡ 0 for t ≥ t0 .
Then the dynamical system ({ϕt
}t∈T, M, ρ) is dissipative.
Proof Let us put η:= max
u∈Br (0)
V (u) and consider the setD:= {u ∈M|V (x)≤η}. In
the discrete-time case we choose η so large that additionally D ⊃ ϕ1
(Br (0)). By
assumption (1) we can write D = {u ∈ M | θ ≤ V (x) ≤ η}, where θ := infu∈M
V (u) −∞. Since K := [θ, η] is compact again by assumption (1) we conclude
that D is bounded. It follows from the definition of D that ϕt
(D) ⊂ D for all t ∈ T+,
proposed that u ∈ Br (0).
Let us show this. Assume to the contrary that there is a u ∈ D and a time t1 ∈ T+
such ϕt1
(u) /
∈ D.
Consider at first the continuous-time case. Here exists a maximum time t such
that 0 t t1 and ρ(ϕt
(u), 0) = r or put t := 0.
It follows that V (u) ≤ η and on the interval (t , t1) we have ρ(ϕt
(u), 0) r.
Now we conclude by continuity that V (ϕt1
(u)) ≤ V (u) ≤ η, a contradiction. In the](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-42-320.jpg)
![22 1 Attractors and Lyapunov Functions
Fig. 1.3 Minimal global attractor of (1.26)
Fig. 1.4 Deformation of a small ball under the flow of (1.26)
Fig. 1.5 Minimal global B-attractor of (1.26)
In order to apply Proposition 1.10 one has to construct a Lyapunov-type function V
satisfying the assumptions (1)–(3). Very often this is a sufficiently difficult problem.
In some cases one can avoid this as the next proposition ([32]) shows.
Consider the dynamical system ({ϕt
}t∈T, Rn
, | · |) which is given for continuous
time by the ODE
ϕ̇ = Aϕ + g(ϕ) , (1.27)](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-44-320.jpg)
![1.2 Dissipativity 23
and for discrete time by the map
u → Au + g(u) , u ∈ Rn
. (1.28)
In both cases A is an n × n matrix and g : Rn
→ Rn
is continuous. The matrix
A is assumed to be stable, i.e. all eigenvalues of A have negative real part in the
continuous-time case, and all eigenvalues have moduli smaller one in the discrete-
time case.
Proposition 1.11 Suppose that the dynamical system is given by (1.29) resp. (1.30)
and g is a bounded map. Then the dynamical system is dissipative.
Proof Suppose that |g(u)| ≤ c0 in Rn
with some constant c0 0. Any motion ϕ(·)
(u)
of the dynamical system can be written as
ϕt
(u) = eAt
u +
t
0
eA(t−τ)
g(ϕτ
(u)) dτ , t ≥ 0 , (1.29)
in the continuous-time case, and as
ϕt
(u) = At
u +
t−1
τ=0
At−τ−1
g(ϕτ
(u)) , t = 1, 2, . . . , (1.30)
in the discrete-time case.
From (1.29), and the stability of A it follows that there exist constants γ 0 and
c1 0 such that
|ϕt
(u)| ≤ c1(e−γ t
|u| + c0
∞
0
eγ (t−τ)
dτ), t ≥ 0 . (1.31)
From (1.30) and the stability of A we get with some constants δ ∈ (0, 1) and c2 0
the representation
|ϕt
(u)| ≤ c2
δt
|u| + c0
∞
τ=0
δ t+τ+1
, t = 1, 2, . . . . (1.32)
From (1.31) and (1.32) the assertion follows immediately.
Definition 1.3 The equilibrium p of the dynamical system ({ϕt
}t∈T, M, ρ) is said
to be globally asymptotically stable if p is asymptotically Lyapunov stable and for
any q ∈ M we have ϕt
(q) → p for t → +∞.
The next theorem was proved by E. A. Barbashin and N. N. Krasovskii in [1] for
continuous time. For discrete time it was shown in [29].](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-45-320.jpg)
![26 1 Attractors and Lyapunov Functions
It was shown in Sect.1.2 that the surfaces
S1 := {(r − z)2
+ y2
= l2
+ ς} and S2 := {z − x2
/(2σ) = −ς},
where ς 0 and l is given by (1.20), are transversal (“contact-free”) for the solutions
of system (1.35). Hence the following inequalities hold on a global attractor of system
(1.35):
(r − z)2
+ y2
≤ l2
, z ≥ x2
/(2σ) . (1.37)
Hence it follows that on a global attractor of system (1.36)
−
l
√
2 (r − 1)
−
√
σ ξ
√
r − 1
≤ η ≤
l
√
2 (r − 1)
−
√
σ ξ
√
r − 1
, (1.38)
ζ −Bξ2
/2, ∀ ξ = 0 . (1.39)
Using estimate (1.37), we introduce the comparison system ([19, 20])
ξ̇ = η, η̇ = −μη + ξ − ξ3
, (1.40)
which is equivalent to the first-order equation
P
d P
dξ
+ μP − ξ + ξ3
= 0. (1.41)
Let us consider positive solutions P1(ξ) of (1.41) on the set [0, ξ0) with initial condi-
tion P1(ξ0) = 0. They define for system (1.36) in the half-space {ξ ≥ 0} the contact-
free surfaces
η = P1(ξ), η 0, ξ ∈ [0, ξ0] , {η 0, ξ = ξ0}. (1.42)
Negative solutions P2(ξ) of (1.41) on (−ξ0, 0] with the initial condition
P2(−ξ0) = 0 define for system (1.36) in the half-space {ξ ≤ 0} the contact-free
surfaces
η = P2(ξ), η 0, ξ ∈ [−ξ0, 0] , {η 0, ξ = −ξ0}. (1.43)
From this and from estimate (1.38) it follows that if the graph of the function
η = P1(ξ) intersects the graph of the straight line
η =
l
√
2 (r − 1)
−
√
σ
√
r − 1
ξ](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-48-320.jpg)
![1.3 Existence of a Homoclinic Orbit in the Lorenz System 27
in a certain point ξ1 on the interval (0, ξ0), then the inequalities
ξ ξ0, η P1(ξ) for ξ ∈ [ξ1, ξ0] (1.44)
hold on a global attractor
A of system (1.36). Similarly, if the graph of the function
η = P2(ξ) intersects the graph of the straight line
η = −
l
√
2 (r − 1)
−
√
σ ξ
√
r − 1
in a certain point ξ2 on the interval (−ξ0, 0) then the inequalities
ξ −ξ0, η P2(ξ) for [−ξ0, ξ2] (1.45)
are true on the global attractor
A of system (1.36). Note that the surfaces
{ζ = C − Bξ2
/2, C Bξ2
0 /2} are contact-free for system (1.36) in the strip
{|ξ| ≤ ξ0}. Hence the estimate
ζ ≤ B(ξ2
0 − ξ2
)/2 (1.46)
holds on a global attractor of system (1.36). We have thus proved the following
result ([24]).
Theorem 1.2 Estimates (1.37)–(1.38), (1.44)–(1.46) hold on a global attractor of
system (1.35).
Let us give now a simple estimate of ξ0. To do this we note that the inequali-
ties (1.44) hold if the graph of η = P1(ξ) intersects the graph of the straight line
η = l/(
√
2 (r − 1)). From the positiveness of μ in equation (1.41) it follows that
P1(ξ)2
(ξ2
− ξ2
0 ) −
1
2
(ξ4
− ξ4
0 ).
Therefore, a sufficient condition for the above intersection to take place is that
(1 − ξ2
0 ) −
1
2
(1 − ξ4
0 ) =
l2
2(r − 1)2
.
This inequality implies that
ξ0 =
1 +
l
r − 1
. (1.47)
Similar reasoning may also be applied to estimate (1.45). It follows from relation
(1.47) that any global attractor of (1.36) lies in a domain which is bounded uniformly
with respect to the parameterr ∈ (1, +∞). For a global B-attractor in the case b ≤ 2,](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-49-320.jpg)
![28 1 Attractors and Lyapunov Functions
the estimates (1.37)–(1.38) are asymptotically the best possible as r → +∞. Indeed,
in this case, as r → +∞ the following inequalities hold on the global B-attractor:
|η| ≤ 1/
√
2, |ξ| ≤
√
2.
We recall that a part of a global B-attractor consists of the unstable manifold of
the zero equilibrium, which may be represented approximately (for small ε) by the
formulae
ζ = −Bξ2
/2, η2
= ξ2
− ξ4
/2 .
So for large r the global B-attractor has points close to the planes {|ξ| =
√
2},
{|η| = 1/
√
2}.
1.3.3 The Existence of Homoclinic Orbits
Let ξ+
, η+
, ζ+
denote a solution of (1.36) associated with the positive branch of the
unstable manifold of the saddle point (0, 0, 0), that goes into the half-plane {ξ 0},
that is, a solution of (1.36) such that
lim
t→−∞
(ξ+
(t), η+
(t), ζ+
(t)) = (0, 0, 0)
and ξ+
(t) 0 for t ∈ (−∞, T ). Here T is a certain number or +∞. It is well-known
([20, 25, 27]) that if the values of σ and b and the value of r are close enough to 1,
then T = +∞.
Let us consider a smooth path s ∈ [0, 1] → (b(s) , σ(s) , r(s)) in the parameter
space {b, σ,r}. The main result of this section is the following theorem ([24]).
Theorem 1.3 Suppose that for system (1.36) with parameters b(0), σ(0), r(0) there
exist numbers T τ such that the relations
ξ+
(T ) = η+
(τ) = 0, ξ+
(t) 0, ∀ t T, (1.48)
η+
(t) = 0, ∀ t T, t = τ (1.49)
hold. Suppose also that for system (1.36) with parameters b(1), σ(1), r(1) the
inequality
ξ+
(t) 0, ∀ t ∈ R (1.50)
is true. Then there exists a number s0 ∈ [0, 1] such that system (1.36) with parameters
b(s0), σ(s0), r(s0) has a solution (ξ+
, η+
, ζ+
) corresponding to a homoclinic orbit.
In order to prove this assertion we shall need the following lemmas.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-50-320.jpg)
![1.3 Existence of a Homoclinic Orbit in the Lorenz System 29
Lemma 1.1 If the conditions
η+
(τ) = 0, η+
(t) 0, ∀ t ∈ (−∞, τ),
hold for system (1.36), then η̇+
(τ) 0.
Proof Suppose the contrary, i.e. η̇+
(τ) = 0. Then we derive from the two last equa-
tions of system (1.36) that
η̈+
(τ) = Aζ+
(τ) ξ+
(τ) . (1.51)
It follows from the relations η+
(t) 0, ξ+
(t) 0, ∀ t ∈ (−∞, τ) and from the last
equation of (1.36) that ζ+
(t) 0 , ∀ t ∈ (−∞, τ]. This inequality and (1.51) imply
the inequality η̈+
(τ) 0 follows. But this contradicts the assumption η̇+
(τ) = 0 and
the conditions of the lemma. This contradiction proves Lemma 1.1.
Lemma 1.2 Consider system (1.36). Suppose that the relations (1.48), (1.49) and
the inequalities
η+
(t) 0, ∀ t ∈ (−∞, τ), η+
(t) ≤ 0, ∀ t ∈ (τ, T ) (1.52)
are true. Then inequality (1.49) also holds.
Proof Suppose the contrary. Then we conclude that a number ς ∈ (τ, T ), exists such
that
η+
(ς) = η̇+
(ς) = 0, η̈+
(ς) = Aξ+
(ς)ζ+
(ς) 0, η+
(t) 0, ∀ t ∈ (ς, T )
are valid. Note that the orbit corresponding to the solution (ξ(t), η(t), ζ(t)) =
(0, 0, ζ(0) exp(−At)) belongs to the stable manifold of the saddle point (0, 0, 0).
Hence, from the conditions (1.48), (1.49) and from the above relations it follows
that the positive branch of the unstable manifold corresponding to the solution
(ξ+
, η+
, ζ+
) and the stable manifold intersect. Then the positive branch of the unsta-
ble manifold belongs completely to the stable manifold of the saddle, and the relation
ξ+
(t) 0, ∀ t ≥ ς is valid. The latter relation contradicts the hypothesis (1.48). This
contradiction proves Lemma 1.2.
Remark 1.6 It is possible to give the following geometrical interpretation of this
proof in the phase space with the coordinates ξ, η, ζ. A piece of the stable manifold of
the saddle ξ = η = ζ = 0 is situated “under” the set {ξ 0, η = 0, ζ ≤ 1 − γ ξ2
}.
This property does not allow the trajectory with the initial data from the set to reach
the plane ξ = 0 if it remains in the quadrant {ξ ≥ 0, η ≤ 0}.
Let us consider the polynomial
λ3
+ aλ2
+ bλ + c, (1.53)
where a, b, c are positive numbers.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-51-320.jpg)
![30 1 Attractors and Lyapunov Functions
Lemma 1.3 Either all zeros of the polynomial (1.53) have negative real parts, or
two of them have non-zero imaginary parts.
Proof It is well-known ([9]) that all the zeros of (1.53) have negative real parts if
and only if ab c. If ab = c the polynomial (1.53) has two pure imaginary zeros.
Suppose now that for certain a, b, c with ab c the polynomial (1.53) has only real
zeros. Since the coefficients are positive it follows that these zeros are negative. This
leads to the inequality ab c which contradicts our assumption.
Proof of Theorem 1.3 It is well-known ([12]) that the semi-orbit of system (1.36)
{(ξ+
(t), η+
(t), ζ+
(t)) | t ∈ (−∞, t0)} depends continuously on parameter s. Here
t0 is an arbitrary fixed number. It follows from this and from Lemma 1.1 that, if
conditions (1.48)–(1.49) hold for system (1.36) with parameters b(s1), σ(s1), r(s1)
then these conditions also hold for b(s), σ(s), r(s) provided that s ∈ (s1 − δ, s1 + δ).
Here δ is a certain sufficiently small number and the numbers τ and T depend on
parameter s. It follows from the above reasoning that the relations (1.48)–(1.49) are
valid for a certain interval (0, s0). Further we shall assume that (0, s0) is the maximal
interval where the relations (1.48)–(1.49) are valid. Let us demonstrate that there
exists a certain homoclinic orbit which corresponds to the values b(s0), σ(s0), r(s0).
We first note that for these parameters and some value τ
η+
(t) 0, ∀ t τ, η+
(t) ≤ 0, ∀ t ≥ τ,
ξ+
(t) 0, ∀ t ∈ (−∞, +∞).
(1.54)
Indeed, if there exist numbers T2 T1 τ, for which
ξ+
(t) 0, ∀ t ∈ (−∞, T2), ξ+
(T2) = 0, η+
(T1) 0,
η+
(t) 0, ∀ t τ, η+
(τ) = 0, η̇+
(τ) 0
are true then for s sufficiently close to s0 and such that s s0 the inequality
η+
(T1) 0 still holds. This contradicts the definition of the number s0. If there exist
numbers T1 τ,forwhichη+
(T1) 0, η+
(t) 0,∀ t τ, η+
(τ) = 0, η̇+
(τ) 0,
and ξ+
(t) 0, ∀ t ∈ (−∞, +∞), then again for s sufficiently close to s0 and such
that s s0 the inequality η+
(T1) 0 remains true and again we have a contradic-
tion with the definition of the number s0. If there exist numbers T τ, for which
ξ+
(t) 0, ∀ t T , ξ+
(T ) = 0, η+
(t) 0, ∀ t τ, η+
(t) ≤ 0, ∀ t ∈ [τ, T ], then
the inequality (1.49) is true according Lemma 1.2. Consequently for s = s0 relations
(1.48)–(1.49) are fulfilled and then (0, s0) is not the maximal interval, for which these
relations are true. By these contradictions inequalities (1.54) are proven. It follows
from (1.54) that for s = s0 only an equilibrium can be the ω-limit set of the orbit of
(ξ+
, η+
, ζ+
). Let us demonstrate that the equilibrium (ξ, η, ζ) = (1/
√
γ , 0, 0) can
not be an ω-limit point of this orbit. The linearization in the neighborhood of this
equilibrium gives the characteristic polynomial
λ3
+ (A + μ)λ2
+ (Aμ + 2/γ )λ + 2A.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-52-320.jpg)
![1.3 Existence of a Homoclinic Orbit in the Lorenz System 31
Suppose that for s = s0 the ω-limit set of the positive branch of the unstable manifold
correspondingtothesolutionξ+
, η+
, ζ+
includesthepoint(ξ, η, ζ) = (1/
√
γ , 0, 0).
By Lemma 1.3 and by the fact that the semi-orbits {ξ+
(t), η+
(t), ζ+
(t) | t ∈
(−∞, t0)} depend continuously on the parameter s we obtain the following asser-
tion. For the values s which are sufficiently close to s0 the positive branch of the
unstable manifold corresponding to (ξ+
, η+
, ζ+
) either tend to an equilibrium state
(ξ, η, ζ) = (1/
√
γ , 0, 0) as t → +∞, or oscillate in some time-interval with chang-
ing sign of the coordinate η. Both of these possibilities contradict properties (1.48)–
(1.49). Hence, for system (1.36) with parameters b(s0), σ(s0), r(s0) the orbit of
(ξ+
, η+
, ζ+
) tends to the trivial equilibrium state as t → +∞.
Remark 1.7 Note that the proof of Theorem 1.3 actually yields a stronger result,
which may be formulated as follows. If relations (1.48)–(1.49) hold for s ∈ [0, s0),
but not for s = s0, then system (1.36) with parameters b(s0), σ(s0), r(s0) has a
homoclinic orbit.
Let us apply Theorem 1.3 in various specific cases. Fix the numbers b and σ. It is
well-known ([20, 27]) that inequality (1.50) is true for r sufficiently close to 1. We
will show that if
3σ − 2b 1 (1.55)
and r is sufficiently large, then relations (1.48)–(1.49) will hold. Indeed, consider
the system
Q
dQ
dξ
= −μQ − Pξ − ϕ(ξ), Q
d P
dξ
= −AP − BQξ, (1.56)
which is equivalent to system (1.36) in the sets {ξ ≥ 0, η 0} and {ξ ≥ 0, η 0},
where P and Q are solutions of (1.56) which are functions of ξ. Since Theorem 1.2
implies that the quantities (ξ+
(t), η+
(t), ζ+
(t)) are bounded uniformly with respect
to the parameter r, we can carry out an asymptotic integration of the solutions of
system (1.56) with a small parameter ε that correspond to the branch of the unstable
manifold under consideration. In the first approximation these solutions may be
written in the form
Q1(ξ)2
= ξ2
−
ξ4
2
− 2μ
ξ
0
ξ
#
1 − ξ2/2 dξ − 2AB
ξ
0
ξ
1 −
#
1 − ξ2/2
dξ,
Q1(ξ) ≥ 0, P1(ξ) = −
$
β
2
%
ξ2
+ AB
1 −
#
1 − ξ2/2
,](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-53-320.jpg)
![32 1 Attractors and Lyapunov Functions
Q2(ξ)2 =
ξ2 −
ξ4
2
− 2μ
√
2
ξ
ξ
1 − ξ2/2 dξ −
4
3
μ + 2AB
√
2
ξ
ξ
$
1 +
1 − ξ2/2
%
dξ −
2
3
AB,
Q2(ξ) ≤ 0, P2(ξ) = −
$
B
2
%
ξ2 + AB
$
1 +
1 − ξ2/2
%
.
It follows from these formulae that if inequality (1.56) holds, then for some T τ
relations (1.48)–(1.49) will also hold and at the same time
ζ+
(T ) = P2(0) = 2AB,
η+
(T ) = Q2(0) = −
#
8(AB − μ)/3 = −
8ε(3σ − 2b − 1)/(3
√
σ).
Thus, all the conditions of Theorem 1.3 hold for the path s → (b(s), σ(s),r(s))
with b(s) ≡ b, σ(s) = σ,r(0) = r1,r(1) = r2, where r1 is sufficiently large and r2
is sufficiently close to 1. We may therefore formulate the following result.
Corollary 1.1 For any positive numbers b and σ satisfying the inequality (1.55) a
number r ∈ (1, +∞) exists, such that system (1.36) with these parameters b, σ and
r has a solution (ξ+
, η+
, ζ+
) corresponding to a homoclinic orbit.
Remark 1.8 Corollary 1.1 was first obtained in [21, 22] and discussed later in [5,
13, 23].
Now fix σ = 10 and r = 28, and consider the parameter b ∈ (0, +∞). It is well-
known ([5]) that for
b
3σ − 1
2
condition (1.50) is fulfilled. To analyse system (1.36) for small b, we reduce it to the
form
ξ̇ = η, η̇ = −μη − uξ + ξ − ξ3
, u̇ = −Au +
ε(2σ − b)
√
σ
ξ2
, (1.57)
where u = ζ + Bξ2
/2. Since the semi-orbit {(ξ+
(t), η+
(t), ζ+
(t))|t ∈ (−∞, t0]}
depends continuously on the parameter b, it follows that, when b is small, the system
(1.57) may be replaced by the system
ξ̇ = η, η̇ = −
ε(σ + 1)
√
σ
η − uξ + ξ − ξ3
, u̇ = 2ε
√
σ ξ2
. (1.58)
Numerical integration of the solution (ξ+
, η+
, ζ+
) of system (1.58) for σ = 10,
r = 28showsthatconditions(1.48)–(1.49)aresatisfied.Hence,theabovearguments,
using Theorem 1.3, yield the following.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-54-320.jpg)
![E
VON DEM MANNE, DER AUSZOG,
ERDBEEREN ZU SUCHEN UND
PFIFFERLINGE MIT HEIMBRACHTE
INE sehr schöne Geschichte.
Von mir.
Und außerdem eine sehr kurze Geschichte.
Aber auch kurze Geschichten können schön sein.
Ich liebe die kurzen Geschichten, die schön sind.
Dies ist eine.
Wenigstens meiner Meinung nach.
Also: ein Mann ging in den Wald, um Erdbeeren zu suchen.
Sogenannte Walderdbeeren.
(Weil sie im Walde wachsen!)
Aber er fand keine.
Aber Pfifferlinge fand er.
Einen ganzen Sack voll.
Er ging heim mit seinem Sack voller Pfifferlinge oder Pfefferlinge.
In Sachsen sagt man «Gehlchen».
Die Sachsen müssen immer eine Extrawurst haben.
Na, und die schmorte er sich.[1]
Und aß sie.
Und die schmeckten sehr gut.
[1] Die Pilze, meine Verehrten!
In Sachsen sagt man «schmeckten sehr s c h ö n ».
Die schmeckten also sehr schön.
Und da freute sich der Mann schrecklich und vergaß völlig, daß er
in den Wald gegangen war, um Erdbeeren zu suchen.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-59-320.jpg)
![mich. Sein Spießgeselle peitscht auf mich ein. Ich weiß nicht, was
gehauen und gestochen ist.
(Einst war ich eine Blume.)
Man führt mich hinweg von meiner Wiese. Ade, du Wiese, ade!
— — —
In der Abendstunde erreichen wir ein Gehöft.
Einst war ich eine Blume, ich denke dran.
Blume bin ich nimmer; bin eine armselige, wehrlose Kuh, muh.
(Hilft mir der Stoizismus etwas?)
Rasch tritt der Tod die Kühe an: Eine Ledermaske mit einem bösen
Stirnbolzen wird mir aufgestülpt — — — ein Schlag, und ich stürze
hin. Da hilft kein Muhen.
Mit einem Rohrstock pfählt man mir das arme Hirn. Das macht
mich traurig. Oder lustig? Ich weiß nicht, ich glaube, ich bin tot.
Kuh bin ich gewesen.
Blume bin ich gewesen.
Ich entsinne mich wirr . . . es ist mir, ja . . . vor langer, langer Zeit
— war ich ein Falter. Aber ich weiß es nicht.
Daß ich Blume war, weiß ich mit Sicherheit. Ich lege meinen Huf
dafür ins Feuer.
Es ist vorbei.
Bin weder Kuh noch Blume mehr.
— — — — —
Bin Wurst. Salamiwurst. Ich koste das Pfund 1.80 M.[2] Ich bin
erstklassige Ware, elektrisch hergestellt.
Den Stoizismus habe ich behalten. Dennoch stimmt es trübe,
Wurst sein zu müssen, wenn man Blume hat sein dürfen.
Ich bin mir Wurst. Ich nehme es hin. Muh. (Eigentlich dürfte ich
als Wurst nimmer muhen. Ich nehme das Muh als anachronistisch
zurück.)
Ich habe keine Freude mehr auf der Welt.
Ich bin eine kalte Wurst. Nichts tangiert mich.
Wenn ich mein Leben überdenke, so muß ich frank gestehen:
Wurst sein, das ist das Schlimmste nicht. Mensch sein ist weitaus
schlimmer!](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-79-320.jpg)
![Doch Kuh sein, das ist schöner als Wurst sein.
Das Allerallerschönste freilich war: Blume sein, Blume gewesen
sein, Blume sein gedurft zu haben.
Mir war’s verstattet.
[2] Wer’s glaubt.
Ich war Blume, ich war Blume!
O Blumen, ihr seid glücklicher als Kuh und Wurst!
O Blumen, nichts auf Erden ist glücklicher denn ihr.
O Blumen — —
* *
*
Vom wurstigen Standpunkt gesehen, ist es vielleicht das Vorteilhafteste, Kuh zu
sein.
Die Kuh ist besser dran als die Blume.
Denn während eine Kuh sehr wohl Blumen fressen kann, kann eine Blume nichts
fressen.
Und eine Wurst kann auch nichts fressen: nicht Kuh, nicht Blume.
Kuh gewesen sein gedurft zu haben ist also — mit Vorbehalt — noch erhebender
als Blume gewesen sein gedurft zu haben.
Ich wünsch’ euch eine gute Nacht und mir, wieder Kuh werden zu dürfen.](https://image.slidesharecdn.com/11251997-250518084000-b58726b0/85/Attractor-Dimension-Estimates-For-Dynamical-Systems-Theory-And-Computation-Dedicated-To-Gennady-Leonov-1st-Ed-Nikolay-Kuznetsov-80-320.jpg)
Attractor Dimension Estimates For Dynamical Systems Theory And Computation Dedicated To Gennady Leonov 1st Ed Nikolay Kuznetsov
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- 5. Emergence, Complexity and Computation ECC Nikolay Kuznetsov Volker Reitmann Attractor Dimension Estimates for Dynamical Systems: Theory and Computation Dedicated to Gennady Leonov
- 6. Emergence, Complexity and Computation Volume 38 Series Editors Ivan Zelinka, Technical University of Ostrava, Ostrava, Czech Republic Andrew Adamatzky, University of the West of England, Bristol, UK Guanrong Chen, City University of Hong Kong, Hong Kong, China Editorial Board Ajith Abraham, MirLabs, USA Ana Lucia, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil Juan C. Burguillo, University of Vigo, Spain Sergej Čelikovský, Academy of Sciences of the Czech Republic, Czech Republic Mohammed Chadli, University of Jules Verne, France Emilio Corchado, University of Salamanca, Spain Donald Davendra, Technical University of Ostrava, Czech Republic Andrew Ilachinski, Center for Naval Analyses, USA Jouni Lampinen, University of Vaasa, Finland Martin Middendorf, University of Leipzig, Germany Edward Ott, University of Maryland, USA Linqiang Pan, Huazhong University of Science and Technology, Wuhan, China Gheorghe Păun, Romanian Academy, Bucharest, Romania Hendrik Richter, HTWK Leipzig University of Applied Sciences, Germany Juan A. Rodriguez-Aguilar , IIIA-CSIC, Spain Otto Rössler, Institute of Physical and Theoretical Chemistry, Tübingen, Germany Vaclav Snasel, Technical University of Ostrava, Czech Republic Ivo Vondrák, Technical University of Ostrava, Czech Republic Hector Zenil, Karolinska Institute, Sweden
- 7. The Emergence, Complexity and Computation (ECC) series publishes new developments, advancements and selected topics in the fields of complexity, computation and emergence. The series focuses on all aspects of reality-based computation approaches from an interdisciplinary point of view especially from applied sciences, biology, physics, or chemistry. It presents new ideas and interdisciplinary insight on the mutual intersection of subareas of computation, complexity and emergence and its impact and limits to any computing based on physical limits (thermodynamic and quantum limits, Bremermann’s limit, Seth Lloyd limits…) as well as algorithmic limits (Gödel’s proof and its impact on calculation, algorithmic complexity, the Chaitin’s Omega number and Kolmogorov complexity, non-traditional calculations like Turing machine process and its consequences,…) and limitations arising in artificial intelligence. The topics are (but not limited to) membrane computing, DNA computing, immune computing, quantum computing, swarm computing, analogic computing, chaos computing and computing on the edge of chaos, computational aspects of dynamics of complex systems (systems with self-organization, multiagent systems, cellular automata, artificial life,…), emergence of complex systems and its computational aspects, and agent based computation. The main aim of this series is to discuss the above mentioned topics from an interdisciplinary point of view and present new ideas coming from mutual intersection of classical as well as modern methods of computation. Within the scope of the series are monographs, lecture notes, selected contributions from specialized conferences and workshops, special contribution from international experts. More information about this series at http://www.springer.com/series/10624
- 8. Nikolay Kuznetsov • Volker Reitmann Attractor Dimension Estimates for Dynamical Systems: Theory and Computation Dedicated to Gennady Leonov 123
- 9. Nikolay Kuznetsov Department of Applied Cybernetics Faculty of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia Volker Reitmann Department of Applied Cybernetics Faculty of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia Faculty of Information Technology University of Jyväskylä Jyväskylä, Finland ISSN 2194-7287 ISSN 2194-7295 (electronic) Emergence, Complexity and Computation ISBN 978-3-030-50986-6 ISBN 978-3-030-50987-3 (eBook) https://doi.org/10.1007/978-3-030-50987-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 10. Preface In this book, we continue the investigations of global attractors and invariant sets for dynamical systems by means of Lyapunov functions and adapted metrics. The effectiveness of such approaches for the approximation and localization of attractors for different classes of dynamical systems was already shown in Abramovich et al. [1, 2] and in [30, 32]. In particular, Lyapunov functions and adapted metrics were constructed for global stability problems and the existence of homoclinic orbits in the Lorenz system using frequency-domain methods and reduction principles. In 1980, investigators of differential equations and general dynamical systems were greatly impressed by a paper about upper estimates of the Hausdorff dimen- sion of flow and map invariant sets written by Douady and Oesterlé [12]. The Douady–Oesterlé approach, the significance of which can be compared with that of Liouville’s theorem, has been developed and modificated in many papers for var- ious types of dimension characteristics of attractors generated by dynamical sys- tems: Ledrappier [22], Constantin et al. [11], Smith [42], Eden et al. [13], Chen [9], Hunt [17], Boichenko and Leonov [4]. After Ya. B. Pesin had worked out [39] a general scheme of introducing metric dimension characteristics, this method made it possible to define from a unique point of view various types of outer measures and dimensions, such as the Hausdorff dimension, the fractal dimension, the information dimension as well as the topological and metric entropies. The Pesin scheme naturally led to the char- acterization of a class of Carathéodory measures [25, 27], which are adapted to the specific character of attractors of autonomous differential equations, i.e. to the fact that these attractors consist wholly of trajectories. The neighborhoods of pieces of these trajectories form a covering of the attractor. It serves as the base for introducing the special outer Carathéodory measures. A number of effective tools for estimating these measures were developed within the theory of differential equations in Euclidean space and well-known results by Borg [8], Hartman and Olech [15], when analysing the orbital stability of solutions. The most important property, used in dimension theory, is the fact that the Carathéodory measures are majorants for the associated Hausdorff measures. v
- 11. Early in the nineties of the last century, G. A. Leonov and his co-workers observed deep inner connections between the mentioned direct method of Lyapunov in stability theory and estimation technics for outer measures in dimension theory. Introducing the Lyapunov functions and varying Riemannian metrices (Leonov [24], Noack and Reitmann [38]) into upper estimates of dimen- sion characteristics of invariant sets made it possible to generalize and improve [4, 6, 28, 29] some well-known results of R. A. Smith, P. Constantin, C. Foias, A. Eden, and R. Temam. On the other hand, Pugh’s closing lemma [40] and theorems about the spanning of two-dimensional surfaces on a given closed curve gave the opportunity to apply some theorems about the contraction of Hausdorff measures to global stability investigations of time-continuous dynamical systems (Smith [42], Leonov [24], Li and Muldowney [35]). In this book, the effectiveness of introducing the Lyapunov functions into dimensional characteristics is shown for a number of concrete dynamical systems: the Hénon map, the systems of Lorenz and Rössler as well as their generalizations for various physical systems and models (rotation of a rigid body in a resisting medium, convection of liquid in a rotating ellipsoid, interaction between waves in plasma, etc.). Additionally to the derivation of upper Hausdorff dimension estimates, exact formulas for the Lyapunov dimensions for Lorenz type systems were shown [26, 28]. Many of these results were presented in [7]. In the following decade, the modified Douady–Oesterlé approach was also used for new classes of attractors [33]. It was also possible to get different versions of the Douady–Oesterlé theorem for piecewise continuous maps and differential equations [37, 41]. For cocycles gen- erated by non-autonomous systems, the upper Hausdorff dimension estimates are derived in [31, 34]. Some of these results are included in the present book which provides a systematic presentation of research activities in the dimension theory of dynamical systems in finite-dimensional Euclidean spaces and manifolds. Let us briefly sketch the contents of the book. In Part I, we consider the basic facts from attractor theory, exterior products and dimension theory. Chapter 1 is devoted to the investigation of various types of global attractors of dynamical systems in general metric spaces (global B-attractors, minimal global B-attractors and others). The theoretical results are applied to the generalized Lorenz system and dynamical systems on the flat cylinder. One section is concerned with the existence proof of a homoclinic orbit in the Lorenz system (Leonov [23], Hastings and Troy [16], Chen [10]). In Chap. 2, some facts on singular values of matrices, the exterior calculus for spaces and matrices and the Lozinskii matrix norm, necessary for estimation techniques of outer measures, are presented. In addition to this, the Yakubovich– Kalman frequency theorem and the Kalman–Szegö theorem about the solvability of certain matrix inequalities are formulated and used for the estimation of singular values. vi Preface
- 12. Chapter 3 is an introduction to dimension theory. It starts with the definition and the basic properties of the topological dimension in the spirit of Hurewicz and Wallman [18]. Next, the notions of Hausdorff measure, Hausdorff dimension and fractal dimension are introduced. After this, the topological entropy of a continuous map is discussed. The last part of the chapter deals with Pesin’s scheme of intro- ducing the Carathéodory dimension characteristics. In Part II, we investigate dimension properties of dynamical systems in Euclidean spaces. It includes estimates of topological dimension of the Hausdorff and fractal dimensions for invariant sets of concrete physical systems and estimates of the Lyapunov dimension. Chapter 4 is concerned with the investigation of dimension properties of almost periodic flows [3]. We thank M. M. Anikushin for helping us to prepare the first version of Chap. 4. Chapter 5 begins with the so-called limit theorem about the evolution of Hausdorff measures under the action of smooth maps in Euclidean spaces. This theorem gives the opportunity to include into Hausdorff dimension and Hausdorff measure estimates, certain varying functions which turn over into Lyapunov functions when the theorem is applied to differential equations. The method of varying functions is used in estimations of fractal dimension and topological entropy. Simultaneously, with the introduction of Lyapunov functions in estimates for the Hausdorff measure, the logarithmic norms were used by Muldowney [36], for estimating the two-dimensional Riemannian volumes of compact sets shifted along the orbits of differential equations. One of the main goals of Chap. 5 is to combine the Lyapunov function and logarithmic norm approaches (Boichenko and Leonov [5]), in order to solve a number of problems in the qualitative theory of ordinary differential equations, such as the generalization of the Liouville formula and the Bendixson criterion. Chapter 6 is devoted to finite-dimensional dynamical systems in Euclidean space and its aim is to explain, in a simple but rigorous way, the connection between the key works in the area: Kaplan and Yorke (the concept of Lyapunov dimension [19] 1979), Douady and Oesterlé (estimation of Hausdorff dimension via the Lyapunov dimension of maps [12]), Constantin, Eden, Foias, and Temam (estimation of Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems [13]), Leonov (estimation of the Lyapunov dimension via the direct Lyapunov method [20, 24]), and numerical methods for the computation of Lyapunov exponents and Lyapunov dimension [21]. We also concentrate in this chapter on the Kaplan–Yorke formula and the Lyapunov dimension formulas for the Lorenz and Hénon attractors (Leonov [26]). In Part III, we consider dimension properties for dynamical systems on mani- folds. Chapter 7 gives a presentation of the exterior calculus in general linear spaces. It contains also some results about orbital stability for vector fields on manifolds. Chapter 8 is devoted to dimension estimates of invariant sets and attractors of dynamical systems on Riemannian manifolds. The Douady–Oesterlé theorem for the upper Hausdorff dimension estimates for invariant sets of smooth dynamical Preface vii
- 13. systems on Riemannian manifolds is proved. Chapter 8 contains also an important result on the estimation of the fractal dimension of an invariant set on an arbitrary finite-dimensional smooth manifold by the upper Lyapunov dimension, which goes back to Hunt [17], Gelfert [14]. Then we discuss the construction of the special Carathéodory measures for the estimation of Hausdorff measures connected with flow invariant sets on Riemannian manifolds. In Chap. 9, we derive dimension and entropy estimates for invariant sets and global B-attractors of cocycles. A version of the Douady–Oesterlé theorem (Leonov et al. [31]) is proved for local cocycles in a Euclidean space and for cocycles on Riemannian manifolds. As examples, we consider cocycles, generated by the Rössler system with variable coefficients. We also introduce time-discrete cocycles on fibered spaces and define the topological entropy of such cocycles. We thank A. O. Romanov for helping us to prepare this chapter. In Chap. 10, we derive some versions of the Douady–Oesterlé theorem for systems with singularities. In the first part of this chapter, we consider a special class of non-injective maps, for which we introduce a factor describing the “degree of non-injectivity” (Boichenko et al. [6]). This factor can be included in the dimension estimates of Chap. 8 in order to weaken the condition to the singular value function. In the second part of Chap. 10, we derive the upper Hausdorff dimension estimates for invariant sets of a class of not necessarily invertible and piecewise smooth maps on manifolds with controllable preimages of the non-differentiability sets in terms of the singular values of the derivative of the smoothly extended map (Reitmann and Schnabel [41], Neunhäuserer [37]) These estimates generalize some Douady–Oesterlé type results for differentiable maps in a Euclidean space, derived in Chaps. 5 and 8. In the last section of Chap. 10, we discuss some classes of functionals which are useful for the estimation of topological and metric dimensions. St. Petersburg, Russia Nikolay Kuznetsov Volker Reitmann References 1. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. I., Oscillations in the sense of Yakubovich in discrete sys- tems. Wiss. Zeitschr. Techn. Univ. Dresden. 25(5/6), 1153–1163 (1977) (German) 2. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. II., Oscillations in discrete phase systems. Wiss. Zeitschr. Techn. Univ. Dresden. 26(1), 115–122 (1977) (German) 3. Anikushin, M.M.: Dimension theory approach to the complexity of almost periodic trajec- tories. Intern. J. Evol. Equ. 10(3–4), 215–232 (2017) 4. Boichenko, V.A., Leonov, G.A.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992) viii Preface
- 14. 5. Boichenko, V.A., Leonov, G.A.: Lyapunov functions, Lozinskii norms, and the Hausdorff measure in the qualitative theory of differential equations. Amer. Math. Soc. Transl. 193(2), 1–26 (1999) 6. Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann,V.: Hausdorff and fractal dimension estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 17(1), 207–223 (1998) 7. Boichenko, V.A., Leonov, G.A., Reitmann, V.: Dimension Theory for Ordinary Differential Equations. Teubner, Stuttgart (2005) 8. Borg, G.: A condition for existence of orbitally stable solutions of dynamical systems. Kungl. Tekn. Högsk. Handl. Stockholm. 153, 3–12 (1960) 9. Chen, Zhi-Min.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors. Chaos, Solitons & Fractals 3, 575–582 (1993) 10. Chen, X.: Lorenz equations, part I: existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27(4), 1057–1069 (1996) 11. Constantin, P., Foias, C., Temam, R.: Attractors representing turbulent flows. Amer. Math. Soc. Memoirs., Providence, Rhode Island. 53(314), (1985) 12. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A. 290, 1135–1138 (1980) 13. Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dynam. Diff. Equ. 3, 133–177 (1991) [Preprint No. 8804, The Institute for Applied Mathematics and Scientific Computing, Indiana University, 1988] 14. Gelfert, K.: Maximum local Lyapunov dimension bounds the box dimension. Direct proof for invariant sets on Riemannian manifolds. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 22(3), 553–568 (2003) 15. Hartman, P., Olech, C.: On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104, 154–178 (1962) 16. Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Diff. Equ. 127(1), 41–53 (1996) 17. Hunt, B.: Maximum local Lyapunov dimension bounds the box dimension of chaotic attractors. Nonlinearity. 9, 845–852 (1996) 18. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton Univ. Press, Princeton (1948) 19. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In: Functional Differential Equations and Approximations of Fixed Points, 204–227, Springer, Berlin (1979) 20. Kuznetsov, N.V.: The Lyapunov dimension and its estimation via the Leonov method. Physics Letters A, 380(25–26), 2142–2149 (2016) 21. Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Prasad, A., Shrimali, M.D.: Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dyn. 92 (2), 267–285 (2018) 22. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981) 23. Leonov, G.A.: On the estimation of the bifurcation parameter values of the Lorenz system. Uspekhi Mat. Nauk. 43(3), 189–200 (1988) (Russian); English transl. Russian Math. Surveys. 43(3), 216–217 (1988) 24. Leonov, G.A.: Estimation of the Hausdorff dimension of attractors of dynamical systems. Diff. Urav. 27(5), 767–771 (1991) (Russian); English transl. Diff. Equations, 27, 520–524 (1991) 25. Leonov, G.A.: Construction of a special outer Carathéodory measure for the estimation of the Hausdorff dimension of attractors. Vestn. S. Peterburg Gos. Univ. 1(22), 24–31 (1995) (Russian); English transl. Vestn. St. Petersburg Univ. Math. Ser. 1, 28(4), 24–30 (1995) 26. Leonov, G.A.: Lyapunov dimensions formulas for Hénon and Lorenz attractors. Alg. & Anal. 13, 155–170 (2001) (Russian); English transl. St. Petersburg Math. J. 13(3), 453–464 (2002) Preface ix
- 15. 27. Leonov, G.A., Gelfert, K., Reitmann, V.: Hausdorff dimension estimates by use of a tubular Carathéodory structure and their application to stability theory. Nonlinear Dyn. Syst. Theory, 1(2), 169–192 (2001) 28. Leonov, G.A., Lyashko, S.: Eden’s hypothesis for a Lorenz system. Vestn. S. Peterburg Gos. Univ., Matematika. 26(3), 15–18 (1993) (Russian); English transl. Vestn. St. Petersburg Univ. Math. Ser. 1, 26(3), 14–16 (1993) 29. Leonov, G.A., Ponomarenko, D.V., Smirnova, V.B.: Frequency-Domain Methods for Nonlinear Analysis. World Scientific, Singapore-New Jersey-London-Hong Kong (1996) 30. Leonov, G. A., Reitmann, V.: Localization of Attractors for Nonlinear Systems. Teubner-Texte zur Mathematik, Bd. 97, B. G. Teubner Verlagsgesellschaft, Leipzig, (1987) (German) 31. Leonov, G.A., Reitmann, V., Slepuchin, A.S.: Upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Dokl. Akad. Nauk, T. 439, No. 6 (2011) (Russian); English transl. Dokl. Mathematics. 84(1), 551–554 (2011) 32. Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local Methods for Pendulum-like Feedback Systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart- Leipzig (1992) 33. Leonov, G.A., Kuznetsov, N.V., Mokaev T.N.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Special Topics. 224(8), 1421–1458 (2015) 34. Maltseva, A.A., Reitmann, V.: Existence and dimension properties of a global B-pullback attractor for a cocycle generated by a discrete control system. J. Diff. Equ. 53(13), 1703–1714 (2017) 35. Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Diff. Equ. 106(1), 27–39 (1993) 36. Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mountain J. Math. 20, 857–871 (1990) 37. Neunhäuserer, J.: A Douady-Oesterlé type estimate for the Hausdorff dimension of invariant sets of piecewise smooth maps. Preprint, University of Technology Dresden (2000) 38. Noack, A., Reitmann, V.: Hausdorff dimension estimates for invariant sets of time-dependent vector fields. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 15(2), 457–473 (1996) 39. Pesin, Ya. B.: Dimension type characteristics for invariant sets of dynamical systems. Uspekhi Mat. Nauk. 43(4), 95–128 (1988) (Russian); English transl. Russian Math. Surveys. 43(4), 111–151 (1988) 40. Pugh, C.C.: An improved closing lemma and a general density theorem. Amer. J. Math. 89, 1010–1021 (1967) 41. Reitmann, V., Schnabel, U.: Hausdorff dimension estimates for invariant sets of piecewise smooth maps. ZAMM 80(9), 623–632 (2000) 42. Smith, R.A.: Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proc. Roy. Soc. Edinburgh. 104A, 235–259 (1986) x Preface
- 16. Acknowledgements While still working on this book, our coauthor Prof. G. A. Leonov, corresponding member of the Russian Academy of Science, died in 2018. This work is dedicated to his memory, with our deepest and most sincere admiration, gratitude, and love. He was an excellent mathematician with a sharp view on problems and a wonderful colleague and friend. He will stay forever in our mind. The preparation of this book was carried out in 2017–2019 at the St. Petersburg State University, at the Institute for Problems in Mechanical Engineering of the Russian Academy of Science, and at the University of Jyväskylä within the framework of the Russian Science Foundation projects 14-21-00041 and 19-41-02002. One of the authors (V.R.) was supported in 2017–2018 by the Johann Gottfried Herder Programme of the German Academic Exchange Service (DAAD). The authors of the book are greatly indebted to Margitta Reitmann for her accurate typing of the manuscript in L A TEX. St. Petersburg, Russia Nikolay Kuznetsov February 2020 Volker Reitmann xi
- 17. Contents Part I Basic Elements of Attractor and Dimension Theories 1 Attractors and Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Dynamical Systems, Limit Sets and Attractors . . . . . . . . . . . . . 3 1.1.1 Dynamical Systems in Metric Spaces . . . . . . . . . . . . . 3 1.1.2 Minimal Global Attractors . . . . . . . . . . . . . . . . . . . . . 10 1.2 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Dissipativity in the Sense of Levinson . . . . . . . . . . . . . 15 1.2.2 Dissipativity and Completeness of The Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.3 Lyapunov-Type Results for Dissipativity . . . . . . . . . . . 20 1.3 Existence of a Homoclinic Orbit in the Lorenz System . . . . . . . 25 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.2 Estimates for the Shape of Global Attractors . . . . . . . . 25 1.3.3 The Existence of Homoclinic Orbits . . . . . . . . . . . . . . 28 1.4 The Generalized Lorenz System . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.1 Definition of the System . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.2 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4.3 Global Asymptotic Stability . . . . . . . . . . . . . . . . . . . . 35 1.4.4 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Singular Values, Exterior Calculus and Logarithmic Norms . . . . . 41 2.1 Singular Values and Covering of Ellipsoids . . . . . . . . . . . . . . . 41 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.1.2 Definition of Singular Values . . . . . . . . . . . . . . . . . . . 43 2.1.3 Lemmas on Covering of Ellipsoids . . . . . . . . . . . . . . . 45 xiii
- 18. 2.2 Singular Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.1 The Fischer-Courant Theorem . . . . . . . . . . . . . . . . . . . 47 2.2.2 The Binet–Cauchy Theorem . . . . . . . . . . . . . . . . . . . . 48 2.2.3 The Inequalities of Horn, Weyl and Fan . . . . . . . . . . . 50 2.3 Compound Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.3.1 Multiplicative Compound Matrices . . . . . . . . . . . . . . . 52 2.3.2 Additive Compound Matrices . . . . . . . . . . . . . . . . . . . 56 2.3.3 Applications to Stability Theory . . . . . . . . . . . . . . . . . 58 2.4 Logarithmic Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.1 Lozinskii’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.2 Generalization of the Liouville Equation . . . . . . . . . . . 66 2.4.3 Applications to Orbital Stability . . . . . . . . . . . . . . . . . 69 2.5 The Yakubovich-Kalman Frequency Theorem . . . . . . . . . . . . . 73 2.5.1 The Frequency Theorem for ODE’s. . . . . . . . . . . . . . . 73 2.5.2 The Frequency Theorem for Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.6 Frequency-Domain Estimation of Singular Values . . . . . . . . . . 77 2.6.1 Linear Differential Equations. . . . . . . . . . . . . . . . . . . . 77 2.6.2 Linear Difference Equations . . . . . . . . . . . . . . . . . . . . 81 2.7 Convergence in Systems with Several Equilibrium States . . . . . 84 2.7.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.7.2 Convergence in the Lorenz System . . . . . . . . . . . . . . . 88 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3 Introduction to Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1 Topological Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.1 The Inductive Topological Dimension . . . . . . . . . . . . . 96 3.1.2 The Covering Dimension . . . . . . . . . . . . . . . . . . . . . . 101 3.2 Hausdorff and Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . 105 3.2.1 The Hausdorff Measure and the Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.2 Fractal Dimension and Lower Box Dimension . . . . . . . 114 3.2.3 Self-similar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.2.4 Dimension of Cartesian Products . . . . . . . . . . . . . . . . . 122 3.3 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.3.1 The Bowen-Dinaburg Definition . . . . . . . . . . . . . . . . . 125 3.3.2 The Characterization by Open Covers . . . . . . . . . . . . . 128 3.3.3 Some Properties of the Topological Entropy . . . . . . . . 131 3.4 Dimension-Like Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 136 3.4.1 Carathéodory Measure, Dimension and Capacity . . . . . 136 3.4.2 Properties of the Carathéodory Dimension and Carathéodory Capacity . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 xiv Contents
- 19. Part II Dimension Estimates for Almost Periodic Flows and Dynamical Systems in Euclidean Spaces 4 Dimensional Aspects of Almost Periodic Dynamics . . . . . . . . . . . . . 149 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2 Topological Dimension of Compact Groups . . . . . . . . . . . . . . . 150 4.3 Frequency Module and Cartwright’s Theorem on Almost Periodic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.4 Minimal Sets in Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . 157 4.5 Almost Periodic Solutions of Almost Periodic ODEs . . . . . . . . 158 4.6 Frequency Spectrum of Almost Periodic Solutions for DDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.7 Fractal Dimensions of Almost Periodic Trajectories and The Liouville Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 171 4.8 Fractal Dimensions of Forced Almost Periodic Regimes in Chua’s Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5 Dimension and Entropy Estimates for Dynamical Systems . . . . . . . 191 5.1 Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.1.1 The Limit Theorem for Hausdorff Measures. . . . . . . . . 191 5.1.2 Corollaries of the Limit Theorem for Hausdorff Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.1.3 Application of the Limit Theorem to the Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2 The Application of the Limit Theorem to ODE’s . . . . . . . . . . . 206 5.2.1 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.2.2 Estimates of the Hausdorff Measure and of Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.2.3 The Generalized Bendixson Criterion . . . . . . . . . . . . . 212 5.2.4 On the Finiteness of the Number of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.2.5 Convergence Theorems. . . . . . . . . . . . . . . . . . . . . . . . 214 5.3 Convergence in Third-Order Nonlinear Systems Arising from Physical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.3.1 The Generalized Lorenz System . . . . . . . . . . . . . . . . . 215 5.3.2 Euler’s Equations Describing the Rotation of a Rigid Body in a Resisting Medium . . . . . . . . . . . 220 5.3.3 A Nonlinear System Arising from Fluid Convection in a Rotating Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . 221 5.3.4 A System Describing the Interaction of Three Waves in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Contents xv
- 20. 5.4 Estimates of Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . 225 5.4.1 Maps with a Constant Jacobian . . . . . . . . . . . . . . . . . . 225 5.4.2 Autonomous Differential Equations Which are Conservative on the Invariant Set . . . . . . . . . . . . . 229 5.5 Fractal Dimension Estimates for Invariant Sets and Attractors of Concrete Systems . . . . . . . . . . . . . . . . . . . . . 230 5.5.1 The Rössler System . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.5.2 Lorenz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.5.3 Equations of the Third Order . . . . . . . . . . . . . . . . . . . 239 5.5.4 Equations Describing the Interaction Between Waves in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 5.6 Estimates of the Topological Entropy . . . . . . . . . . . . . . . . . . . . 247 5.6.1 Ito’s Generalized Entropy Estimate for Maps . . . . . . . . 247 5.6.2 Application to Differential Equations . . . . . . . . . . . . . . 252 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6 Lyapunov Dimension for Dynamical Systems in Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.1 Singular Value Function and Invariant Sets of Maps of Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.2 Lyapunov Dimension of Maps. . . . . . . . . . . . . . . . . . . . . . . . . 263 6.3 Lyapunov Dimensions of a Dynamical System . . . . . . . . . . . . . 265 6.3.1 Lyapunov Exponents: Various Definitions . . . . . . . . . . 269 6.3.2 Kaplan-Yorke Formula of the Lyapunov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 6.4 Analytical Estimates of the Lyapunov Dimension and its Invariance with Respect to Diffeomorphisms . . . . . . . . . 279 6.5 Analytical Formulas of Exact Lyapunov Dimension for Well-Known Dynamical Systems . . . . . . . . . . . . . . . . . . . . 286 6.5.1 Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 6.5.2 Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.5.3 Glukhovsky-Dolzhansky System . . . . . . . . . . . . . . . . . 291 6.5.4 Yang and Tigan Systems . . . . . . . . . . . . . . . . . . . . . . 292 6.5.5 Shimizu-Morioka System . . . . . . . . . . . . . . . . . . . . . . 293 6.6 Attractors of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 294 6.6.1 Computation of Attractors and Lyapunov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.7 Computation of the Finite-Time Lyapunov Exponents and Dimension in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . 298 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 xvi Contents
- 21. Part III Dimension Estimates on Riemannian Manifolds 7 Basic Concepts for Dimension Estimation on Manifolds . . . . . . . . . 309 7.1 Exterior Calculus in Linear Spaces, Singular Values of an Operator and Covering Lemmas . . . . . . . . . . . . . . . . . . . 309 7.1.1 Multiplicative and Additive Compounds of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7.1.2 Singular Values of an Operator Acting Between Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 7.1.3 Lemmas on Covering of Ellipsoids in an Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 7.1.4 Singular Value Inequalities for Operators . . . . . . . . . . . 323 7.2 Orbital Stability for Flows on Manifolds . . . . . . . . . . . . . . . . . 325 7.2.1 The Andronov-Vitt Theorem . . . . . . . . . . . . . . . . . . . . 325 7.2.2 Various Types of Variational Equations . . . . . . . . . . . . 326 7.2.3 Asymptotic Orbital Stability Conditions . . . . . . . . . . . . 329 7.2.4 Characteristic Exponents . . . . . . . . . . . . . . . . . . . . . . . 343 7.2.5 Orbital Stability Conditions in Terms of Exponents . . . 347 7.2.6 Estimating the Singular Values and Orbital Stability. . . 348 7.2.7 Frequency-Domain Conditions for Orbital Stability in Feedback Control Equations on the Cylinder . . . . . . 354 7.2.8 Dynamical Systems with a Local Contraction Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8 Dimension Estimates on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1 Hausdorff Dimension Estimates for Invariant Sets of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1.2 Hausdorff Dimension Bounds for Invariant Sets of Maps on Manifolds . . . . . . . . . . . . . . . . . . . . . 366 8.1.3 Time-Dependent Vector Fields on Manifolds . . . . . . . . 371 8.1.4 Convergence for Autonomous Vector Fields . . . . . . . . 376 8.2 The Lyapunov Dimension as Upper Bound of the Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 8.2.1 Statement of the Results . . . . . . . . . . . . . . . . . . . . . . . 379 8.2.2 Proof of Theorem 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . 380 8.2.3 Global Lyapunov Exponents and Upper Lyapunov Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 8.2.4 Application to the Lorenz System . . . . . . . . . . . . . . . . 388 8.3 Hausdorff Dimension Estimates by Use of a Tubular Carathéodory Structure and their Application to Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Contents xvii
- 22. 8.3.1 The System in Normal Variation . . . . . . . . . . . . . . . . . 390 8.3.2 Tubular Carathéodory Structure . . . . . . . . . . . . . . . . . . 394 8.3.3 Dimension Estimates for Sets Which are Negatively Invariant for a Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 396 8.3.4 Flow Invariant Sets with an Equivariant Tangent Bundle Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 8.3.5 Generalizations of the Theorems of Hartman-Olech and Borg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 9 Dimension and Entropy Estimates for Global Attractors of Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 9.1 Basic Facts from Cocycle Theory in Non-fibered Spaces. . . . . . 411 9.1.1 Definition of a Cocycle. . . . . . . . . . . . . . . . . . . . . . . . 411 9.1.2 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 9.1.3 Global B-Attractors of Cocycles . . . . . . . . . . . . . . . . . 413 9.1.4 Extension System Over the Bebutov Flow on a Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 9.2 Local Cocycles Over the Base Flow in Non-fibered Spaces . . . . 418 9.2.1 Definition of a Local Cocycle . . . . . . . . . . . . . . . . . . . 418 9.2.2 Upper Bounds of the Hausdorff Dimension for Local Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 9.2.3 Upper Estimates for the Hausdorff Dimension of Local Cocycles Generated by Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 9.2.4 Upper Estimates for the Hausdorff Dimension of a Negatively Invariant Set of the Non-autonomous Rössler System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 9.3 Dimension Estimates for Cocycles on Manifolds (Non-fibered Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 9.3.1 The Douady-Oesterlé Theorem for Cocycles on a Finite Dimensional Riemannian Manifold. . . . . . . 429 9.3.2 Upper Bounds for the Haussdorff Dimension of Negatively Invariant Sets of Discrete-Time Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 9.3.3 Frequency Conditions for Dimension Estimates for Discrete Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . 438 9.3.4 Upper Bound for Hausdorff Dimension of Invariant Sets and B-attractors of Cocycles Generated by Ordinary Differential Equations on Manifolds . . . . . . . 440 9.3.5 Upper Bounds for the Hausdorff Dimension of Attractors of Cocycles Generated by Differential Equations on the Cylinder. . . . . . . . . . . . . . . . . . . . . . 444 xviii Contents
- 23. 9.4 Discrete-Time Cocycles on Fibered Spaces. . . . . . . . . . . . . . . . 450 9.4.1 Definition of Cocycles on Fibered Spaces . . . . . . . . . . 450 9.4.2 Global Pullback Attractors . . . . . . . . . . . . . . . . . . . . . 451 9.4.3 The Topological Entropy of Fibered Cocycles . . . . . . . 451 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 10 Dimension Estimates for Dynamical Systems with Some Degree of Non-injectivity and Nonsmoothness. . . . . . . . . . . . . . . . . 457 10.1 Dimension Estimates for Non-injective Smooth Maps . . . . . . . . 457 10.1.1 Hausdorff Dimension Estimates . . . . . . . . . . . . . . . . . . 457 10.1.2 Fractal Dimension Estimates . . . . . . . . . . . . . . . . . . . . 465 10.2 Dimension Estimates for Piecewise C1 -Maps . . . . . . . . . . . . . . 471 10.2.1 Decomposition of Invariant Sets of Piecewise Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 10.2.2 A Class of Piecewise C1 -Maps . . . . . . . . . . . . . . . . . . 472 10.2.3 Douady-Oesterlé-Type Estimates . . . . . . . . . . . . . . . . . 476 10.2.4 Consideration of the Degree of Non-injectivity . . . . . . 479 10.2.5 Introduction of Long Time Behavior Information . . . . . 481 10.2.6 Estimation of the Hausdorff Dimension for Invariant Sets of Piecewise Smooth Vector Fields . . . . . . . . . . . 484 10.3 Dimension Estimates for Maps with Special Singularity Sets . . . 491 10.3.1 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . 492 10.3.2 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . 493 10.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 10.4 Lower Dimension Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 499 10.4.1 Frequency-Domain Conditions for Lower Topological Dimension Bounds of Global B-Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 10.4.2 Lower Estimates of the Hausdorff Dimension of Global B-Attractors . . . . . . . . . . . . . . . . . . . . . . . . 503 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 Appendix A: Basic Facts from Manifold Theory . . . . . . . . . . . . . . . . . . . 509 Appendix B: Miscellaneous Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Contents xix
- 24. Part I Basic Elements of Attractor and Dimension Theories
- 25. Chapter 1 Attractors and Lyapunov Functions Abstract The main tool in estimating dimensions of invariant sets and entropies of dynamical systems developed in this book is based on Lyapunov functions. In this chapter we introduce the basic concept of global attractors. The existence of a global attractor for a dynamical system follows from the dissipativity of the system. In order to show the last property we use Lyapunov functions. In this chapter we also consider some applications of Lyapunov functions to stability problems of the Lorenz system. A central result is the existence of homoclinic orbits in the Lorenz system for certain parameters. 1.1 Dynamical Systems, Limit Sets and Attractors 1.1.1 Dynamical Systems in Metric Spaces Suppose that (M, ρ) is a complete metric space. Let T be one of the sets R, R+ , Z or Z+ . A map ϕ(·) (·) : T × M → M resp. a triple ({ϕt }t∈T, M, ρ) is called a dynamical system on (M, ρ) if the following conditions are satisfied ([2, 11]): (1) ϕ0 (u) = u , ∀ u ∈ M ; (2) ϕt+s (u) = ϕt (ϕs (u)) , ∀ t, s ∈ T, ∀ u ∈ M ; (3) If T ∈ {R, R+} the map (t, u) ∈ T × M → ϕt (u) is continuous; if T ∈ {Z, Z+} the map u ∈ M → ϕt (u) is continuous on M for any t ∈ T. If the metric space (M, ρ) is fixed we denote the dynamical system shortly by {ϕt }t∈T. The sets T and M are called time sets and phase space, respectively. The dynamical system ({ϕt }t∈T, M, ρ) forms a group, if T ∈ {R, Z}, and a semi-group if T ∈ {R+, Z+}. If T ∈ {R, R+} we say that the dynamical system is with continuous time, if T ∈ {Z, Z+} we say that the system is with discrete time. A dynamical system ({ϕt }t∈T, M, ρ) is called flow if T = R, semi-flow if T = R+, and cascade if T = Z. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Emergence, Complexity and Computation 38, https://doi.org/10.1007/978-3-030-50987-3_1 3
- 26. 4 1 Attractors and Lyapunov Functions Example 1.1 Consider the autonomous differential equation ϕ̇ = f (ϕ) , (1.1) where f : Rn → Rn is assumed to be locally Lipschitz. The Euclidean norm in Rn is denoted by | · |. Suppose also that any maximal solution ϕ(·, u) of (1.1) starting in u at t = 0 exists for any t ∈ R. Let now ϕt (·) ≡ ϕ(t, ·) : Rn → Rn be the time t-map of (1.1). Clearly, that by the solution properties of (1.1) (uniqueness theo- rem and theorem of continuous dependence on initial condition, [12, 34]) the triple ({ϕt }t∈R, Rn , | · |) defines a dynamical system with the additive group T = R, i.e. a flow. Example 1.2 Assume that ϕ̇ = f (t, ϕ) (1.2) is a non-autonomous differential equation with f : R × Rn → Rn . Let us suppose that f is continuously differentiable, T -periodic in the first argument and that any solution exists on R. Denote the solution of (1.2) starting in u at t = t0 by ϕ(·, t0, u). Again by the uniqueness theorem and the theorem of continuous dependence of solutions on initial conditions for ODE’s it follows that the family of maps ϕm (·) ≡ ϕ(m T, 0, ·), m ∈ Z, defines a dynamical system ({ϕm }m∈Z, Rn , | · |) which is a cas- cade. Let us demonstrate this. Clearly, ϕ0 (u) = ϕ(0, 0, u) = u, ∀ u ∈ Rn . In order to show the property (2) of a dynamical system we consider arbitrary m, k ∈ Z. Define the two functions c1(t) := ϕ(t, 0, ϕ(mT, 0, u)) and c2(t) := ϕ(t + mT, 0, u). From the T -periodicity of f in the first variable it follows that c2 is also a solution of (1.2) on R, i.e. ċ2(t) = f (t + mT, ϕ(t + mT, 0, u)) = f (t, c2(t)). Since c2(0) = ϕ(mT, 0, u), it follows by the uniqueness theorems that c1(t) = c2(t), ∀ t ∈ R. If we put t = kT , the last property results in c1(kT ) ≡ ϕk (ϕm (u)) = c2(kT ) ≡ ϕk+m (u). Example 1.3 Suppose that ϕ : M → M (1.3) is a continuous invertible map on the complete metric space (M, ρ). Let us define the family of maps ϕm := ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ ◦ ϕ ◦ · · · ◦ ϕ m-times for m = 1, 2, . . . , idM for m = 0 , ϕ−1 ◦ ϕ−1 ◦ · · · ◦ ϕ−1 -m-times for m = −1, −2, . . . . (1.4)
- 27. 1.1 Dynamical Systems, Limit Sets and Attractors 5 Using well-known properties of the composition of continuous invertible maps, i.e. of homeomorphisms, it is easy to show that the properties (1)–(3) of a dynamical system with the additive group T = Z are satisfied. This means that (1.4) defines a cascade. Example 1.4 Let (M, g) be a Riemannian n-dimensional Ck -manifold (k ≥ 2), F : M → T M a C2 -vector field (see Sect. A.6, Appendix A). Consider the corresponding differential equation ϕ̇ = F(ϕ) . (1.5) Assume that any maximal integral curve ϕ(·, u) of (1.5) satisfying ϕ(0, u) = u exists on R. Define ϕ(·) (u) := ϕ(·, u) and denote with ρ the metric generated by the metric tensor g. Then ({ϕt }t∈R, M, ρ) is a flow defined by the vector field (1.5). Instead of (1.5) we can consider a C1 -diffeomorphism ϕ : M → M . (1.6) It is clear that (1.6) generates the dynamical system ({ϕm }m∈Z, M, ρ) . Example 1.5 Suppose that Ω+ 2 := ω = (ω0, ω1, . . .) | ωi ∈ {0, 1} is the set of all one-sided infinite sequences of the symbols 0 and 1. The metric on Ω+ 2 is given by ρ(ω, ω ) := ∞ i=0 2−i | ωi − ωi | , where ω = (ω0, ω1, . . .) and ω = (ω0, ω1, . . .) are from Ω+ 2 . It is easy to see that ρ is really a metric and (Ω+ 2 , ρ) is a complete metric space. Define the (left) shift map ϑ : Ω+ 2 → Ω+ 2 by ϑ(ω) = (ω1, ω2, . . .) for ω = (ω0, ω1, . . .) ∈ Ω+ 2 . The map ϑ is continuous since for arbitrary ω, ω ∈ Ω+ 2 we have ρ(ϑ(ω), ϑ(ω )) = ∞ i=0 2−i | ωi+1 − ωi+1 | = 2 ∞ i=0 1 2i+1 | ωi+1 − ωi+1 | ≤ 2 ∞ i=−1 1 2i+1 | ωi+1 − ωi+1 | = 2 ρ(ω, ω ) . It follows that {ϑm }m∈Z+ , Ω+ 2 , ρ is a dynamical system with discrete time. Let us define now some properties of a dynamical system ({ϕt }t∈T, M, ρ). For an arbitrary fixed u ∈ M, the map t → ϕt (u), t ∈ T defines a motion of the dynamical system starting from u at time t = 0. For any u ∈ M the set γ (u) := t∈T ϕt (u)
- 28. 6 1 Attractors and Lyapunov Functions is the orbit through u. If T ∈ {R, Z} we consider also the positive and the negative semi-orbit through u defined by γ + (u) := t∈T∩R+ ϕt (u) resp. γ − (u) := t∈T∩R− ϕt (u) . An orbit γ (u) is called stationary, critical or an equilibrium if γ (u) = {u}. The orbit γ (u)ofadynamicalsystemiscalled T -periodicwithperiod T if T 0 isthesmallest positive number in T such that ϕt (u) = ϕt+T (u), ∀ t ∈ T. A set Z ⊂ M is said to be positively invariant if ϕt (Z) ⊂ Z, ∀ t ∈ T ∩ R+, invariant if ϕt (Z) = Z, ∀ t ∈ T, and negatively invariant, if ϕt (Z) ⊃ Z, ∀ t ∈ T ∩ R+. The positively invariant set Z of the dynamical system ({ϕt }t∈T, M, ρ) is said to be stable if in any neighborhood U of Z there exists a neighborhood U such that ϕt (U ) ⊂ U, ∀t ∈ T+. Z is called asymptotically stable if it is stable and ϕt (u) → Z as t → +∞ for each u ∈ U . The set Z is said to be globally asymptotically stable if Z is stable and ϕt (u) → Z as t → +∞ for each u ∈ M. The set Z is called uniformly asymptotically stable if it is stable and limt→+∞{dist(ϕt (u), Z) | u ∈ U } = 0. For any u ∈ M the ω-limit set of u under {ϕt }t∈T is the set ω(u) := {υ ∈ M | ∃{tn}n∈N , tn ∈ T , tn → +∞ , ϕtn (u) → υ for n → +∞} . For a subset Z ⊂ M we define its ω-limit set ω(Z) under {ϕt }t∈T as the set of the limits of all converging sequences of the form ϕtn (un), where un ∈ Z, tn ∈ T, and tn → +∞. Example 1.6 Consider a class of modified horseshoe maps ϕ which are defined on an open neighborhood U = (−δ, 1 + δ) × (−δ, 1 + δ) ⊂ R2 of the unit square C = [0, 1] × [0, 1], where δ 0 is a sufficiently small number. The map is defined for points (x, y) ∈ C in such a way that it first contracts C horizontally with a factor α 1 2 and stretches it vertically with a function f , then it is folded along a horizontal line such that the vertical edge of the resulting rectangle is greater than 2, and finally it is formed into an horseshoe (see Fig.1.1) in such a way that the map can be con- tinuously extended to U and is continuously differentiable on an open neighborhood U of K = ∞ i=−∞ ϕi (C) , where ϕi (·) for negative numbers i means the preimage under the map ϕ−i . For example we take α = 1 3 and let the function f stretch C with factor β1 = 3, if y ≤ 51 65 =: h, with a factor β2 between 3 and 5, if 51 65 y 4 5 and with factor 5, if y ≥ 4 5 , the resulting rectangle is folded on the image of the line y = 51 65 and then it is formed to an horseshoe in such a way, that the map satisfies ϕ(x, y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 3 x, 3y − 1 13 if 0 ≤ y ≤ 14 39 , 1 − 1 3 x, 148 65 − 3y if 83 195 ≤ y ≤ 148 195 , 1 − 1 3 x, 5y − 4 if 4 5 ≤ y ≤ 1 .
- 29. 1.1 Dynamical Systems, Limit Sets and Attractors 7 Fig. 1.1 Modified horseshoe map The set K = ∞ i=−∞ ϕi (C) is invariant under the map. Therefore if we take K = K, then ϕ j (K) ⊂ K is satisfied for any j = 1, 2, . . . . The set K can be constructed step by step starting with K0 = C. At every step i 1 we get Ki := Ki−1 ∩ ϕ(Ki−1 ) ∩ ϕ−1 (Ki−1 ) and in the limit the invariant set K as K = ∞ i=0 Ki . The set Ki consists of 6i rectangles where the lengths of the edges are horizontally 1 3i and vertically 1 3i and 1 5i , respectively (see Fig.1.1). It is easy to verify that the following proposition is true ([4, 7, 38]). Proposition 1.1 For any subset Z ⊂ M the ω-limit set under {ϕt }t∈T is given by ω(Z) = s≥0 s∈T t≥s t∈T ϕt (Z) . Here for a set Z ⊂ M we denote by Z its closure in the topology of the metric space (M, ρ). If T ∈ {R, Z} we also consider the α-limit set of a point u ∈ M under {ϕt }t∈T defined by α(u) := {υ ∈ M | ∃{tn}n∈N , tn ∈ T, tn → −∞, ϕtn (u) → υ for n → +∞} and the α-limit set ω(Z) of a set Z ⊂ M under {ϕt }t∈T given as the set of the limits of all converging sequences of the form ϕtn (un), where pn ∈ Z, tn ∈ T, and tn → −∞. A set Zmin ⊂ M is called minimal for ({ϕt }t∈T, M, ρ) if it is closed, invariant, and does not have any proper subset with the same properties. The following proposition is taken from [33, 36]. Proposition 1.2 Suppose that Z ⊂ M is non-empty, compact and invariant for ({ϕt }t∈T, M, ρ). Then Z contains a minimal set Zmin. Proof (For the case that T is a semi-group). If Z has no proper subset, which is closed and invariant, then Z is minimal, and the proposition is proved.
- 30. 8 1 Attractors and Lyapunov Functions Suppose that there exists Z1 ⊂ Z, Z1 = Z, such that Z1 is closed and invariant. If Z1 contains no proper subset being closed and invariant, then it is minimal. Suppose again that there exists a closed invariant set Z2 ⊂ Z1 with Z2 = Z1. If we can continue this process and at any step we obtain a new minimal set Zi , we get the sequence of closed invariant sets Z0 := Z ⊃ Z1 ⊃ Z2 ⊃ · · · . The intersection of these sets Zω := ∞ i=0 Zi is non-empty and compact. Let us show that Zω is invariant. Suppose that u ∈ Zω is arbitrary. Then for any integer k ≥ 0 we have u ∈ Zk. It follows that ϕt (u) ∈ Zk , t ≥ 0, for any k. But this means that ϕt (u) ∈ Zω and, consequently, Zω ⊂ ϕt (Zω). The inverse inclusion can be proved analogously. Thus the set Zω is invariant. If the set Zω is not minimal then there exists a closed and invariant set Zω+1 ⊂ Zω with Zω+1 = Zω. If we can continue this process and if β is the transfinite limit number for which the sets Zα are constructed for all α β, we put Zβ := αβ Zα . Clearly, the set Zβ is closed and invariant. Thus we get the transfinite sequence of sets Z ⊃ Z1 ⊃ · · · ⊃ Zk ⊃ · · · ⊃ Zω ⊃ · · · ⊃ Zβ ⊃ · · · . According to the Baire-Hausdorff theorem ([16], Theorem A.14.1) there exists a transfinite number β of the second class such that Zβ = Zβ+1, i.e. the set Zβ has no proper closed and invariant subset. Consequently, Zβ is a minimal set. The next result follows immediately from Proposition 1.2. Proposition 1.3 Suppose that the positive semi-orbit of ({ϕt }t∈T, M, ρ) , starting in p, is relatively compact. Then ω(p) contains a minimal set. Some important properties of ω-limit sets are proved in the next proposition ([18]). Proposition 1.4 Let ({ϕt }t∈T, M, ρ) be a dynamical system with a semi-group as time set and let Z ⊂ M be a non-empty set such that for some t0 0, t0 ∈ T, the set t≥t0,t∈T ϕt (Z) is relatively compact. Then ω(Z) is non-empty, compact and invariant. Furthermore, ω(Z) is a minimal closed set which attracts Z. Proof Recall that relative compactness of a set means that the closure of this set is compact. Since Z = ∅ the sets Bs = t≥s,t∈T ϕt (Z) are non-empty for all s ≥ 0, s ∈ T. Consequently, the sets Bs are non-empty compact sets for s ≥ t0 and Bs1 ⊂ Bs2
- 31. 1.1 Dynamical Systems, Limit Sets and Attractors 9 for all s1 ≥ s2 in T. Therefore ω(Z) = s≥0,s∈T Bs is a non-empty compact set and attracts Z. If u ∈ ϕt (ω(Z)) then for a certain υ ∈ ω(Z) we have u = ϕt (υ). Hence there exists a sequence um ∈ Z and a sequence {tm}, tm ∈ T, tm → +∞ as m → +∞, such that limm→+∞ ϕt (ϕtm (u)) = limm→+∞ ϕt+tm (um) = ϕt (υ) = u. But this means that u ∈ ω(Z). Let us prove the reverse inclusion. If υ ∈ ω(Z) a sequence um ∈ Z and a sequence {tm}fromTexistsuchthattm → +∞asm → +∞, 1 + t0 + t ≤ t1 t2 · · · ,and ϕtm (um) → υ for m → +∞ . (1.7) For tm ≥ t the sequence ϕtm −t (um) belongs to the relatively compact set t≥t0+1, t∈T ϕt (Z). Consequently, passing if necessary to a subsequence, we may sup- pose that there exists a point u ∈ M such that ϕtm −t (um) → u as m → +∞ . But this means that u ∈ ω(Z). Since ϕtm (um) = ϕt (ϕtm −t (um)) → ϕt (u) as m → +∞ , by (1.7) we have υ = ϕt (u). Thus, υ ∈ ϕt (ω(Z)). It remains to show that ω(Z) is a minimal closed set which attracts Z. Let us argue as in [18]. Suppose the contrary and let C be a proper closed subset of ω(Z) which attracts Z. As ω(Z) is compact so is C. Choose any υ ∈ ω(Z) C. For ε 0 small enough the ε-neighborhoods Uε(υ) and Uε(C) do not intersect. Since C attracts Z there is a t = t(ε) ≥ 0 such that ϕt (Z) ⊂ Uε(C), ∀t ≥ t(ε). On the other hand since υ ∈ ω(Z), υ = limk→∞ ϕtk (uk) for some uk ∈ Z and tk → +∞ is a sequence. Consequently, ϕtk (Z) ∩ Uε(υ) = ∅ for sufficiently large tk. Hence Uε(υ) ∩ Uε(C) = ∅, a contradiction. Assume that ({ϕt }t∈T, M, ρ) is a dynamical system with a group as time set and u ∈ M is an arbitrary point. The sets Ws (u) := {υ ∈ M| lim t→+∞ ϕt (υ) = u} and Wu (u) := {υ ∈ M| lim t→−∞ ϕt (υ) = u} are called stable and unstable manifold, respectively, in u. Since the orbits are invari- ant, the sets Ws (u) and Wu (u) are also invariant. Suppose that u and υ are equilibria of the dynamical system. Then any orbit which is contained in Ws (u) ∩ Wu (υ) is called heteroclinic if u = υ, and homoclinic if u = υ.
- 32. 10 1 Attractors and Lyapunov Functions LetZ ⊂ Mbeanarbitraryinvariantsubset.Thenthestableandunstablemanifold of Z are the sets Ws (Z) := {υ ∈ M| lim t→+∞ dist(ϕt (υ), Z) = 0} and Wu (Z) := {υ ∈ M| lim t→−∞ dist(ϕt (υ), Z) = 0}, respectively. 1.1.2 Minimal Global Attractors Now we come to some of the basic definitions in our book. For arbitrary nonempty sets Z1, Z2 ⊂ M we define dist(Z1, Z2) := sup u∈Z1 inf υ∈Z2 ρ(u, υ). By Uε(Z) we denote the ε-neighborhood of a set Z, i.e. Uε(Z) := {υ ∈ M | dist (υ, Z) ε}. Definition 1.1 Suppose that ({ϕt }t∈T, M, ρ) is a dynamical system. (1) We say that a set Z0 ⊂ M attracts the set Z ⊂ M if for any ε 0 there exists a t0 = t0(ε, Z) such that for all t ≥ t0, t ∈ T, we have ϕt (Z) ⊂ Uε(Z0). (2) An attractor A for ({ϕt }t∈T, M, ρ) is a non-empty closed and invariant set which attracts all points from some set Z with a non-empty interior. The largest set with non-empty interior which is attracted by A is called the domain of attraction. (3) A global attractor for ({ϕt }t∈T, M, ρ) is a non-empty, closed and invariant set which attracts all points of M. (4) A global B-attractor is a non-empty, closed and invariant set which attracts any bounded set B of M. (5) A minimal global attractor (minimal global B-attractor) is a global attractor (global B-attractor) which is a minimal set among all global attractors (global B-attractors). (6) A set Z0 ⊂ M is said to be B-absorbing for ({ϕt }t∈T, M, ρ) if for any bounded set B in M there exists a t0 = t0(B) such that ϕt (B) ⊂ Z0 for any t ≥ t0, t ∈ T. (7) A dynamical system is said to be pointwise dissipative (B-dissipative) if it pos- sesses a pointwise absorbing (B-absorbing set) B0. The set B0 is called region of pointwise dissipativity (of B-dissipativity). (8) A set Z0 ⊂ M is said to be pointwise absorbing for ({ϕt }t∈T, M, ρ) if for any u ∈ M there exists a t0 = t0(u) such that ϕt (u) ⊂ Z0 for any t ≥ t0, t ∈ T. Let us use the following abbreviations for the attractors of a dynamical system ({ϕt }t∈T, M, ρ): A—an arbitrary attractor, AM—a global B-attractor, AM,min— a minimal global B-attractor, AM—a global attractor, AM,min—a minimal global attractor. A direct consequence of Definition 1.1 is the following proposition.
- 33. 1.1 Dynamical Systems, Limit Sets and Attractors 11 Proposition 1.5 Let A be a global B-attractor and ε 0 an arbitrary number. Then the ε-neighborhood of A is B-absorbing for the dynamical system. Remark 1.1 Minimal global attractors and B-attractors where introduced by O.A. Ladyzhenskaya in [18]. Our Definition 1.1 follows the representation given in [18]. Important properties of minimal global attractors are also derived in [6, 7, 15, 35, 37]. The existence of a global B-attractor is shown in the next proposition ([17]). Note that if a global B-attractor exists, then it contains a minimal global B- attractor. Proposition 1.6 Suppose that the dynamical system ({ϕt }t∈T, M, ρ) is B-dissipative according to the bounded B-absorbing set B0 and there exists a t0 0 such that the set t≥t0,t∈T ϕt (B0) is relatively compact. Then AM,min := {ω(B) | B ⊂ M, B bounded} is a minimal global B-attractor and AM,min := u∈M ω(u) is a minimal global attractor of the dynamical system. Proof Since by Proposition 1.4 every bounded set B ⊂ M is attracted to its ω- limit set ω(B) and to AM,min, it is attracted to ω(B) ∩ AM,min. Since ω(B) is minimal, it lies in AM,min. The set AM,min, is invariant and minimal. It follows that ω(AM,min) = AM,min and the representation for AM,min, is shown. The fact that u∈M ω(u) is a minimal global attractor follows from the properties of an ω-limit set. Remark 1.2 In contrast to the minimal global attractor given by Proposition 1.6 a minimal global attractor can be unbounded. The dynamical system generated by the ODE ẋ = 0 , ẏ = − ay , a 0 has as a minimal global B-attractor AR2,min the x-axis (see Sect. 2.1, Chap. 2) Other examples of minimal global attractors and global B-attractors will be con- sidered in the sequel. A dynamical system ({ϕt }t∈T, M, ρ) is called locally completely continuous if for any u ∈ M there exists a δ = δ(u) 0 and an l = l(u) 0,l(u) ∈ T+, such that ϕl (Bδ(u)) is relatively compact. It is clear that a dynamical system given in a locally compact space is locally completely continuous. The next proposition is a result of [4].
- 34. 12 1 Attractors and Lyapunov Functions Proposition 1.7 For a locally completely continuous dynamical system pointwise dissipativity and B-dissipativity are equivalent. Proof We have to show that a dynamical system which is pointwise dissipative is also B-dissipative. From the pointwise dissipativity it follows that there exists a non-empty compact set K ⊂ M such that for each ε 0 and u ∈ M there exists a δ(u) 0 and a τ(ε, u) such that dist (ϕt (υ), K) ε (1.8) for all t ≥ τ(ε, u), t ∈ T, and all υ ∈ Bδ(u)(u). Suppose B is an arbitrary bounded set whichis containedinacompact setK. Thenfor anyu ∈ K thereexists aδ = δ(u) 0 and τ = τ(ε, u) 0 such that (1.8) is satisfied. Consider an open cover {Bδ(u)(u)}u∈K of K. Since K is compact and M is a complete metric space there is a finite subcover {Bδ(ui )(ui )}m i=1 of K. Define ˜ l(ε, K) := max{τ(ε, ui )|i = 1, . . . , m}. Then it follows from (1.8) that dist (ϕt (K), K) ε, ∀t ˜ l(ε, K). Since for dynamical systems in locally compact metric spaces pointwise dissipa- tivity and B-dissipativity are equivalent we call these properties shortly dissipativity. The next proposition is proved in [7]. It shows that Lyapunov functions can give a good inside in the structure of an attractor. In this book the term Lyapunov function for a dynamical system ({ϕt }t∈T, M, ρ) means a scalar valued continuous function V which is considered along the orbits and whose properties allow some conclusions about the qualitative behaviour of the dynamical system. If M is a manifold and V is differentiable the properties of V depend on the Lie derivative of V w.r.t. the dynamical system (see Subsect. 1.2.3). Proposition 1.8 Suppose for the dynamical system ({ϕt }t∈T, M, ρ) with a group as time set there exists a compact global B-attractor AM and a continuous function V : M → R with the following properties: (1)Foranyu ∈ Mthefunction V (ϕt (u))isnon-increasingwithrespecttot ∈ T+; (2) If for some t0 0, t0 ∈ T+, the equation V (u) = V (ϕt0 (u)) holds, then u is an equilibrium of the dynamical system. Then: (a) AM = Wu (C), where C is the set of equilibrium points of the dynamical system. (b) The global minimal attractor AM,min of the dynamical system is C. Remark 1.3 Suppose ({ϕt }t∈T, M, ρ) is a dynamical system with a group as time set, which has a bounded minimal global B-attractor A. Then Wu (C) ⊂ A, where C is the set of equilibria of the dynamical system. For a proof see [7]. For dynamical systems ({ϕt }t∈T, M, ρ) given on a Riemannian smooth n-dimen- sional manifold (M, g) we define a Milnor attractor as a closed invariant set A having the property limt→+∞ dist(ϕt (u), A) = 0 for each u ∈ S, where S is a set of positive Lebesgue measure.
- 35. 1.1 Dynamical Systems, Limit Sets and Attractors 13 If the manifold is compact we define the minimal global Milnor attractor as a minimal closed invariant set A having the property limt→∞ dist(ϕt (u), A) = 0 for any u ∈ S, where S is a Lebesgue measurable set with the full Lebesgue measure, i.e. μL (S) = μL(M). In the following we denote a Milnor attractor by A and a minimal global Milnor attractor by Amin. We will consider the minimal global Milnor attractor also for dynamical systems given in Rn and possessing a bounded open positively invariant absorbing set B0. In this case we can restrict our system on the positive semi-group T+ on B0, considering B0 with relative topology as compact manifold. Example 1.7 Let us consider Van der Pol’s equation ẍ + ε(x2 − 1)ẋ + x = 0 where ε 0 is a parameter. This equation can be written as planar system ẋ = y , ẏ = −ε(x2 − 1)y − x . (1.9) It is well-known that (1.9) generates a semi-flow ({ϕt }t≥0, R2 , | · |) and the origin (0, 0) is an unstable equilibrium of this semi-flow. Furthermore, there is a single orbitally stable periodic orbit (Fig.1.2). Any orbit of the semi-flow, different from the equilibrium (0, 0), tends for t → +∞ to this periodic orbit. It is easy to see that the minimal global B-attractor is AR2,min is the closed disk around the origin and bounded by the unit circle S1 = {(x, y) | x2 + y2 = 1}, the minimalglobalattractoris AR2,min = S1 ∪ {(0, 0)},theminimalglobalMilnorattrac- tor is Amin = S1 and a non-global attractor is given by A = S1 . Fig. 1.2 Attractors of Van der Pol’s system
- 36. 14 1 Attractors and Lyapunov Functions Consider the dynamical system ({ϕt }t∈T, (M, ρ)) on the metric space (M, ρ). Suppose B = B(M) is the σ-algebra of Borel sets on M and μ is a finite Borel measure on B, i.e., μ(M) +∞. The bounded set Aμ(M) ⊂ M is called global Milnor attractor w.r.t. the dynamical system ({ϕt }t∈T, (M, ρ)) and the measure μ if Aμ(M) is a minimal, closed and invariant set having the property limt→∞ dist(ϕt (u), Aμ(M)) for μ-a.e. point u ∈ M. Sometimes the global Milnor attractor is called stochastic attractor. If the metric space (M, ρ) is compact we define the minimal global Mil- nor attractor as a minimal closed invariant set Amin,μ(M) having the property limt→∞ dist(ϕt (u), Amin,μ(M)) = 0 for μ-a.e. point u ∈ M. Suppose that M = E is a linear metric space and E∗ is the dual to E, i.e., the linear space of linear bounded functionals on E. The sequence {un}∞ n=1 from E is called weakly convergent to u ∈ E if (un) → (u) , ∀ ∈ E∗ . We denote this by un u for n → ∞. The set Z ⊂ E is called weakly closed if it contains the weak limit u of arbitrary weakly convergent sequences {un} ⊂ Z. In the weak topology the open sets are given by arbitrary unions of sets O(υ; 1, 2, . . . , n; ε1, ε2, . . . , εn) := u ∈ E | 1(u − υ) | ε1 , 2(u − υ) ε2, . . . , |n(u − υ) | εn where υ ∈ E , i ∈ E∗ , εi 0 (i = 1, 2, . . . , n) are numbers. By definition the empty set ∅ is open. A non-empty set O which is open in the weak topology and which contains the set Z ⊂ E is called weak neighborhood of Z. Suppose that M = E is a linear metric space. The set Aw(M) ⊂ M is called weak global B-attractor w.r.t. ({ϕt }t∈T, (E, ρ)) if Aw(M) is a bounded and weakly closed invariant set such that for any weak neighborhood O of the set Aw(M) and any bounded set B ⊂ M there exists a t0 = t0(O, B) such that ϕt (B) ⊂ O for all t ≥ t0. Let us note that if the linear space M = E has finite dimension and for the dynamical system the global attractor AM exists and is weakly closed then also exists Aw(M) and Aw(M) = AM. In Table1.1 we present the various types of attractors and their symbols. Example 1.8 Let us consider as complete metric space M the Hilbert space L2 (a, b) of quadratically integrable functions on (a, b). It follows from the Riesz the- orem that the any linear bounded functional on L2 (a, b) is given by (u) = Ω uυdx, whereυ ∈ L2 (a, b)andΩ = (a, b).Thuswehavethepropertiesun u forn → ∞ in L2 (a, b) ⇔ Ω unυdx → Ω uυdx for n → ∞ and any υ ∈ L2 (a, b). Assume that {ei }∞ i=1 is an orthonormal basis of L2 (a, b). Then any function υ ∈ L2 (a, b) can be represented as υ = ∞ i1 ci ei , where ci = Ω υei dx, i = 1, 2, . . . , are the Fourier coefficients satisfying υ2 L2(a,b) = ∞ i1 c2 i ∞. Consider the functions un = en , n = 1, 2, . . . . Then for any υ ∈ L2 (a, b) we have
- 37. 1.1 Dynamical Systems, Limit Sets and Attractors 15 Table 1.1 Types of attractors and their symbols Symbol Type of attractor Sections A Arbitrary attractor 1.1.2 AM Global B-attractor 1.1.2 AM,min Minimal global B-attractor 1.1.2 AM Global attractor 1.1.2 AM,min Minimal global attractor 1.1.2 A Milnor attractor 1.1.2 Amin Minimal global Milnor attractor 1.1.2 Aw(M) Weak global B-attractor 1.1.2 lim n→∞ Ω enυdx = lim n→∞ Ω cne2 ndx = lim n→∞ cn = 0, i.e., en 0 for n → ∞ in L2(a, b). From the other side we have en 0 as n → ∞ in L2 (a, b) since en2 = 1, n = 1, 2, . . . . 1.2 Dissipativity 1.2.1 Dissipativity in the Sense of Levinson This section is devoted to the concepts of dissipativity, region of dissipativity, and its estimation for autonomous differential equations. These notions arose for the first time in stability theory. Later they turned out to be very useful in the study of attractors since they give the possibility to localize attractors in the phase space. Consider the dynamical system ({ϕt }t∈T, Rn , | · |) which in the continuous-time case is given by the autonomous ODE ϕ̇ = f (ϕ) , (1.10) where f : Rn → Rn is continuously differentiable, and in the discrete-time case is given by the continuous map ϕ : Rn → Rn . (1.11) Definition 1.2 The dynamical system ({ϕt }t∈T, Rn , | · |) is called dissipative in the sense of Levinson, if there exists an R 0 such that for any u ∈ Rn lim sup t→+∞ |ϕt (u)| R .
- 38. 16 1 Attractors and Lyapunov Functions Proposition 1.9 The dynamical system ({ϕt }t∈T, Rn , | · |) is dissipative in the sense of Levinson if and only if there exists a bounded set D ⊂ Rn that attracts any point in Rn . Proof Let the dynamical system be dissipative in the sense of Levinson. Choose as the set D a ball of radius R, where R is from Definition 1.2, and with center in the origin. It is obvious that such a D attracts every point in Rn . Conversely, let D be a set for the dynamical system which attracts every point of Rn , and let ε 0 be an arbitrary number. Choose R so large that the ball of radius R with center in the origin contains the ε-neighborhood Dε of D. It is clear that such an R satisfies Definition 1.2. It is easy to see that if the dynamical system is dissipative in the sense of Levinson withD asregionofdissipativity,thenanyattractorAofthedynamicalsystemsatisfies the inclusion A ⊂ D. 1.2.2 Dissipativity and Completeness of The Lorenz System Consider the Lorenz system ([28, 30, 39]) ẋ = −σ x + σ y , ẏ = rx − y − xz , ż = −bz + xy (1.12) where σ,r and b are positive parameters. Let us show that equation (1.12) is dissi- pative. Introduce the auxilary function V : R3 → R+ given by V (x, y, z) := 1 2 x2 + y2 + (z − σ − r)2 . (1.13) Direct computation along an arbitrary solution u = (x, y, z) of (1.12) shows that the derivative of V along V̇ (x, y, z) = − σ x2 − y2 − b 2 (z − σ − r)2 + b 2 (σ + r)2 . Thus in R3 we have V̇ ≤ b 2 (σ + r)2 . (1.14) On the set E1 := (x, y, z) | σ x2 + y2 + b 2 (z − σ − r)2 ≤ b 2 (σ + r)2
- 39. 1.2 Dissipativity 17 the inequality V̇ ≥ 0 is true and on the set R3 E1 we have V̇ 0. For large R the ball BR = {(x, y, z) | V (x, y, z) R} contains the ellipsoid E1. On the boundary of BR, i.e. on the set SR = {(x, y, z) | V (x, y, z) = R} the inequality V̇ 0 is satisfied. It follows that BR is a bounded absorbing set. If we put κ = min{σ, 1, b 2 } we conclude that along the solution of (1.12) V̇ ≤ −2κV + b 2 (σ + r)2 . This means that any solution of (1.12) enters the ellipsoid E2 := (x, y, z) | 1 2 x2 + y2 + (z − σ − r)2 ≤ b 4κ (σ + r)2 and remains there during the positive existence interval. From (1.14) we have V (x(t), y(t), z(t)) ≤ V (x(0), y(0), z(0)) + b 2 (σ + r)2 t for t ≥ 0. Since V cannot go to infinity in a finite positive time, each of |x(t)|, |y(t)|, and |z(t)| cannot go to infinity in a finite positive time. Thus the Lorenz system is complete in positive time. Thus the Lorenz system defines a semi-flow in R3 . From (1.13) it follows that for a sufficiently large κ1 0 we have V̇ + κ1V ≥ b 2 (σ + r)2 =: c1 (1.15) From (1.15) we get d dt (eκ1 t V ) ≥ c1eκ1t . Thus for t ≤ 0 we have V (x(0), y(0), z(0)) − eκ1 t V (x(t), y(t), z(t)) ≥ c1 κ1 [1 − eκ1 t ] or V (x(t), y(t), z(t)) ≤ e−κ1 t V (x(0), y(0), z(0)) + c1 κ1 [1 − e−κ1 t ] . Thus V (x(t), y(t), z(t)) cannot go to infinity in finite negative time. Hence each of |x(t)|, |y(t)| and |z(t)| cannot got to infinity in finite negative time and the system is complete in negative time ([8]). Let us obtain other estimates for the region of dissipativity for (1.12). Consider next the function V1(x, y, z) := 1 2 x2 + 1 2 y2 + 1 2 z2 − (σ + r)z .
- 40. 18 1 Attractors and Lyapunov Functions Let us show that for an arbitrary solution u = (x, y, z) of (1.12) with b 1 we have lim sup t→+∞ V1(x(t), y(t), z(t)) ≤ c2 , (1.16) where c2 := (σ+r)2 (b−2)2 8(b−1) . Indeed, a calculation shows that V̇1 + 2V1 = −(σ − 1)x2 − (b − 1)z2 + (σ + r)(b − z)z ≤ −(b − 1)z2 + (σ + r)(b − 2)z ≤ 2 c2 . Therefore we have d dt (V1 − c2) + 2 (V1 − c2) ≤ 0 . Multiplying the last inequality by e2t we get for t ≥ 0 d dt [(V1 − c2)e2t ] ≤ 0 . (1.17) Integrating (1.17) on [0, t], we obtain V1(x(t), y(t), z(t)) − c2 ≤ [V1(x(0), y(0), z(0)) − c2] e−2t , from which the inequality (1.16) results. From (1.16) it follows that the ellipsoid {(x, y, z) ∈ R3 | x2 + 1 2 y2 + 1 2 z2 − (σ + r)z ≤ c2} is a region of dissipativity for (1.12). Let us now show that for an arbitrary solution u = (x, y, z) of (1.12) lim sup t→+∞ [y2 (t) + (z(t) − r)2 ] ≤ l2 r2 (1.18) and, if 2σ − b ≥ 0, lim inf t→+∞ [2 σz(t) − x2 (t)] ≥ 0 . (1.19) The parameter l in (1.18) is defined by l := 1 , if b ≤ 2 , b 2 √ b−1 , if b ≥ 2 . (1.20) In order to prove (1.18) we put for (x, y, z) ∈ R3 V2(y, z) := 1 2 y2 + (z − r)2 .
- 41. 1.2 Dissipativity 19 Suppose that κ0 := min{1, b}. Then for any κ ∈ (0, κ0) we have V̇2 + 2κV2 = (κ − 1)y2 + (κ − b)z2 − 2 r κ − b 2 z + κr2 ≤ (κ − b) z − r(κ − b/2) κ − b !2 − r2 (κ − b/2)2 κ − b + κr2 ≤ κ − (κ − b/2)2 κ − b ! r2 = b2 r2 4 (b − κ) . It follows that lim sup t→+∞ V2(y(t), z(t)) ≤ b2 r2 8 (b − κ)κ . (1.21) Minimizing the right-hand side of (1.21) over κ ∈ (0, κ0) we obtain (1.18). To prove (1.19) we put V3(x, z) := σz − 1 2 x2 . The direct computation shows that V̇3 = −b σz − 2σ b 1 2 x2 ! ≥ −b V3 . The last relation implies (1.19). From (1.18) it follows that the region of dissipativity D satisfies the inclusion D ⊂ D1 , where D1 := {(x, y, z) | y2 + (z − r)2 l2 r2 } is a cylinder in R3 . Under the condition 2σ − b ≥ 0 it follows from (1.18) and (1.19) that D ⊂ D1 ∩ {(x, y, z) | z ≥ 0} . Remark 1.4 Using the Lyapunov function (1.13) and Proposition 1.7 one sees that the Lorenz system has a compact attracting set which attracts bounded sets. It follows that the Lorenz system is B-dissipative and has a minimal B-attractor AR3,min which satisfies AR3,min ⊂ D. Note that any other attractor of (1.12) also belongs to D. Remark 1.5 Let us consider the system ẋ = −σ x + σ y , ẏ = rx − y + xz , ż = −bz + xy (1.22) withpositiveparametersσ,r andb.ThissystemdiffersfromtheLorenzsystem(1.12) only in the sign of the nonlinearity xz in the second equation and the divergence of the right-hand side of (1.22) is −(σ + 1 + b) 0, i.e. the same as in the Lorenz system. However, it was shown in [14] that system (1.22) for any positive parameters has
- 42. 20 1 Attractors and Lyapunov Functions solutions converging to infinity for t → +∞. But this means that system (1.22) is not dissipative. We need certain extra conditions on the right-hand side in order to guarantee dissipativity. 1.2.3 Lyapunov-Type Results for Dissipativity Let us consider the dynamical system ({ϕt }t∈T, M, ρ) on the Riemannian n- dimensional Ck -manifold (M, g) which is, for continuous time, given by (1.5) and for discrete time by (1.6). Suppose that there exists a scalar valued function V : M → R which is C1 in the continuous-time case and C0 in the discrete-time case. Define the Lie derivative V̇ (u) w.r.t. the dynamical system in the continuous- time case by V̇ (u) := d dt V (ϕt (u))|t=0 = (F(u), grad V (u)) (1.23) and in the discrete-time case by V̇ (u) := V (ϕ(u)) − V (u). (1.24) Let us establish the following theorem which is a generalization of a result from [40, 41], obtained for differential equations in Rn . Proposition 1.10 Suppose that there exists a function as introduced above and such that the following conditions are satisfied: (1) V is proper for M, i.e. for any compact set K ⊂ R the set V −1 (K) ⊂ M is compact and V is bounded from below on M; (2) There exists an r 0 such that V̇ (u) ≤ 0 for u / ∈ Br (0) ; (3) The dynamical system does not have a motion ϕ(·) (υ) with ϕt (υ) / ∈ Br (0) and V̇ (ϕt (υ)) ≡ 0 for t ≥ t0 . Then the dynamical system ({ϕt }t∈T, M, ρ) is dissipative. Proof Let us put η:= max u∈Br (0) V (u) and consider the setD:= {u ∈M|V (x)≤η}. In the discrete-time case we choose η so large that additionally D ⊃ ϕ1 (Br (0)). By assumption (1) we can write D = {u ∈ M | θ ≤ V (x) ≤ η}, where θ := infu∈M V (u) −∞. Since K := [θ, η] is compact again by assumption (1) we conclude that D is bounded. It follows from the definition of D that ϕt (D) ⊂ D for all t ∈ T+, proposed that u ∈ Br (0). Let us show this. Assume to the contrary that there is a u ∈ D and a time t1 ∈ T+ such ϕt1 (u) / ∈ D. Consider at first the continuous-time case. Here exists a maximum time t such that 0 t t1 and ρ(ϕt (u), 0) = r or put t := 0. It follows that V (u) ≤ η and on the interval (t , t1) we have ρ(ϕt (u), 0) r. Now we conclude by continuity that V (ϕt1 (u)) ≤ V (u) ≤ η, a contradiction. In the
- 43. 1.2 Dissipativity 21 discrete-time case there must exist a time t2 ∈ (0, t1) ∩ Z+ such that ϕt2 (u) ∈ D ∩ Br (0), but ϕt2+1 (u) / ∈ D. But this is impossible by the choice of D in the discrete-time case. Let us show now that for any point u ∈ M with u / ∈ Br (0) there exists a time t1 0 such that ϕt1 (u) ∈ Br (0). Suppose the opposite, i.e. ϕt (u) / ∈ Br (0) for all t ∈ T+. Then the positive semi-orbit of the motion ϕ(·) (u) is bounded. Indeed, by condition (2) we have V (ϕt (u)) ≤ V (u) for all t ∈ T+. From this and assumption (1) we obtain the boundedness of the semi-orbit. In virtue of this boundedness according to Proposition 1.4, the ω-limit set of the semi-orbit of ϕ(·) (u) is non-empty. Let υ ∈ ω(u) be an arbitrary point. According to our assumption we have υ / ∈ Br (0). The function V (ϕt (u)) is bounded on T+ and does not increase. Therefore, there exists the limit lim t→+∞ V (ϕt (u)) = V (υ) . (1.25) Consider the motion ϕ(·) (υ). By Proposition 1.4 we have ϕt (υ) ∈ ω(u) for all t ∈ T+. It follows that for any t ∈ T+ there exists a sequence tm → +∞ as m → +∞ such that ϕtm (u) → ϕt (υ) as m → +∞. Hence by (1.25) we have V (ϕt (υ)) ≡ V (υ) , which contradicts the assumption (3). Thus, for arbitrary u ∈ Rn with u / ∈ Br (0) there exists a time t1 ∈ T+ such that ϕt1 (u) ∈ Br (0). Example 1.9 Consider the equation of a pendulum ẍ + ε ẋ + sin x = 0 , where ε 0 is a parameter. This equation is equivalent to the planar system ẋ = y , ẏ = −ε y − sin x . (1.26) Since the right-hand side of (1.26) is globally Lipschitz we have global existence and uniqueness of all solutions. Let us denote the dynamical system generated by (1.26) by ({ϕt }t≥0, R2 , | · |). It is well-known that any semi-orbit of this system tends to an equilibrium for t → +∞. The phase portrait is shown in Fig.1.3 It follows that the minimal global attractor AR2,min of (1.26) is the stationary set, i.e. the set of all equilibria C. Consider now a ball Bδ with small radius δ 0 and center at a point u0 = (x0, y0) on the stable manifold of a saddle (Figs.1.3 and 1.4). It is obvious that ϕt (Bδ) converges for t → +∞ to a set consisting of a saddle point and of two heteroclinic orbits coming from this saddle point and going to stable equilibria. It follows that the minimal global B-attractor is the union of the stationary set C and the heteroclinic orbits (Fig.1.5).
- 44. 22 1 Attractors and Lyapunov Functions Fig. 1.3 Minimal global attractor of (1.26) Fig. 1.4 Deformation of a small ball under the flow of (1.26) Fig. 1.5 Minimal global B-attractor of (1.26) In order to apply Proposition 1.10 one has to construct a Lyapunov-type function V satisfying the assumptions (1)–(3). Very often this is a sufficiently difficult problem. In some cases one can avoid this as the next proposition ([32]) shows. Consider the dynamical system ({ϕt }t∈T, Rn , | · |) which is given for continuous time by the ODE ϕ̇ = Aϕ + g(ϕ) , (1.27)
- 45. 1.2 Dissipativity 23 and for discrete time by the map u → Au + g(u) , u ∈ Rn . (1.28) In both cases A is an n × n matrix and g : Rn → Rn is continuous. The matrix A is assumed to be stable, i.e. all eigenvalues of A have negative real part in the continuous-time case, and all eigenvalues have moduli smaller one in the discrete- time case. Proposition 1.11 Suppose that the dynamical system is given by (1.29) resp. (1.30) and g is a bounded map. Then the dynamical system is dissipative. Proof Suppose that |g(u)| ≤ c0 in Rn with some constant c0 0. Any motion ϕ(·) (u) of the dynamical system can be written as ϕt (u) = eAt u + t 0 eA(t−τ) g(ϕτ (u)) dτ , t ≥ 0 , (1.29) in the continuous-time case, and as ϕt (u) = At u + t−1 τ=0 At−τ−1 g(ϕτ (u)) , t = 1, 2, . . . , (1.30) in the discrete-time case. From (1.29), and the stability of A it follows that there exist constants γ 0 and c1 0 such that |ϕt (u)| ≤ c1(e−γ t |u| + c0 ∞ 0 eγ (t−τ) dτ), t ≥ 0 . (1.31) From (1.30) and the stability of A we get with some constants δ ∈ (0, 1) and c2 0 the representation |ϕt (u)| ≤ c2 δt |u| + c0 ∞ τ=0 δ t+τ+1 , t = 1, 2, . . . . (1.32) From (1.31) and (1.32) the assertion follows immediately. Definition 1.3 The equilibrium p of the dynamical system ({ϕt }t∈T, M, ρ) is said to be globally asymptotically stable if p is asymptotically Lyapunov stable and for any q ∈ M we have ϕt (q) → p for t → +∞. The next theorem was proved by E. A. Barbashin and N. N. Krasovskii in [1] for continuous time. For discrete time it was shown in [29].
- 46. 24 1 Attractors and Lyapunov Functions Theorem 1.1 Suppose that p is an equilibrium of the dynamical system ({ϕt }t∈T, Rn , | · |) and there exists a function V : Rn → R (C1 in the continuous-time case and C0 in the discrete-time case) such that the following conditions are satisfied: (1) V (p) = 0 and V (u) 0 for all u ∈ Rn {p} ; (2) V̇ (p) = 0 and V̇ (u) 0 for all u ∈ Rn {p} ; (3) V (u) → +∞ for |u| → +∞ . Then the equilibrium p is globally asymptotically stable. Proof Let for simplicity be p = 0. It follows from the Lyapunov theorem that p = 0 is asymptotically Lyapunov stable. Suppose that ϕ(·) (q) is an arbitrary motion of the dynamical system. Using assumption (3) we choose r so large that q ∈ Br (0) and V (u) V (q) for all |u| ≥ r . (1.33) From assumption (2) we conclude that V (ϕt (q)) ≤ V (q) for all t ∈ T+ . (1.34) Thus, if we take into consideration (1.33) we have |ϕt (q)| r for all t ∈ T+ . Let us put c = lim t→+∞ V (ϕt (q)) and show that c = 0. If we assume that c 0 there exists a number r1 ∈ (0,r) such that |ϕt (q)| ≥ r1 for all t ∈ T+. It follows that r1 ≤ |ϕt (q)| r for t ∈ T+. The proof is complete if we argue as in the Lyapunov theorem. Example 1.10 Let us show that the equilibrium u1 = (0, 0, 0) of the Lorenz system (1.12) is globally asymptotically stable. Take the function V (x, y, z) := 1 2 (x2 + σ y2 + σz2 ) . A direct computation shoes that V̇ (x, y, z) = −σ x2 − (1 + r) xy + y2 + bz2 = −σ 1 − r 2 (x2 + y2 ) + bz2 + 1 + r 2 (x − y)2 ≤ −σ 1 − r 2 (x2 + y2 ) + bz2 . Thus by the continuous-time version of Theorem 1.1 we conclude that u1 = (0, 0, 0) is globally asymptotically stable if 0 r 1.
- 47. 1.3 Existence of a Homoclinic Orbit in the Lorenz System 25 1.3 Existence of a Homoclinic Orbit in the Lorenz System 1.3.1 Introduction InthissectionweconsideragaintheLorenzsystem.Wegiveestimatesfortheshapeof a global B-attractor and prove the existence of homoclinic orbits for certain parameter values. It will be shown that in certain cases these estimates are asymptotically exact. Since all estimates are uniform with respect to the parameters, it becomes possible to prove the existence of a homoclinic orbit using the formulae of asymptotic integration. The Lorenz system ẋ = −σ(x − y), ẏ = rx − y − xz, ż = −bz + xy, (1.35) which is a three-mode approximation of a two-dimensional thermal convection, is now one of the classical models for the transition from global stability to chaotic behaviour and to the generation of attractors with non-integer Hausdorff dimension. Sometimes the phrase “homoclinic explosion” is used to refer to the appearance of various types of chaotic behaviour when parameters are perturbed from the bifurca- tion parameter of a homoclinic orbit. In such a process the role of homoclinic orbits, which appear for bifurcation values of parameters is very important. These orbits and the attractors of the Lorenz system are located in certain domains of the phase space which can be estimated. We shall suppose further that σ, r, b are positive numbers. Let, in addition, r 1 and 2σ b. Note that if one of these restrictions is violated then system (1.35) is convergent, i.e. any its orbit tends to a certain equi- librium when t → +∞ (Example 1.10). Along with system (1.35) we consider the equivalent system ξ̇ = η, η̇ = −μη − ζξ − ϕ(ξ), ζ̇ = −Aζ − Bξη, (1.36) where ϕ(ξ) = −ξ + γ ξ3 , ξ = εx/ √ 2σ , η = ε2 (y − x)/ √ 2 , ζ = ε2 (z − x2 /b), t = t1 √ σ /ε, μ = ε(σ + 1)/ √ σ , A = εb/ √ σ , ε = (r − 1)−1/2 , B = 2 (2σ − b)/b, γ = 2σ/b. 1.3.2 Estimates for the Shape of Global Attractors In this section we shall obtain estimates which are for b ≤ 2 and great r asympto- tically exact for a global B-attractor with respect to the coordinates ξ and η. From these estimates if follows that a global B-attractor of system (1.36) is located in domains which are uniformly bounded with respect to parameter r ∈ (1, +∞). This fact will be used for the demonstration of the existence of homoclinic orbits.
- 48. 26 1 Attractors and Lyapunov Functions It was shown in Sect.1.2 that the surfaces S1 := {(r − z)2 + y2 = l2 + ς} and S2 := {z − x2 /(2σ) = −ς}, where ς 0 and l is given by (1.20), are transversal (“contact-free”) for the solutions of system (1.35). Hence the following inequalities hold on a global attractor of system (1.35): (r − z)2 + y2 ≤ l2 , z ≥ x2 /(2σ) . (1.37) Hence it follows that on a global attractor of system (1.36) − l √ 2 (r − 1) − √ σ ξ √ r − 1 ≤ η ≤ l √ 2 (r − 1) − √ σ ξ √ r − 1 , (1.38) ζ −Bξ2 /2, ∀ ξ = 0 . (1.39) Using estimate (1.37), we introduce the comparison system ([19, 20]) ξ̇ = η, η̇ = −μη + ξ − ξ3 , (1.40) which is equivalent to the first-order equation P d P dξ + μP − ξ + ξ3 = 0. (1.41) Let us consider positive solutions P1(ξ) of (1.41) on the set [0, ξ0) with initial condi- tion P1(ξ0) = 0. They define for system (1.36) in the half-space {ξ ≥ 0} the contact- free surfaces η = P1(ξ), η 0, ξ ∈ [0, ξ0] , {η 0, ξ = ξ0}. (1.42) Negative solutions P2(ξ) of (1.41) on (−ξ0, 0] with the initial condition P2(−ξ0) = 0 define for system (1.36) in the half-space {ξ ≤ 0} the contact-free surfaces η = P2(ξ), η 0, ξ ∈ [−ξ0, 0] , {η 0, ξ = −ξ0}. (1.43) From this and from estimate (1.38) it follows that if the graph of the function η = P1(ξ) intersects the graph of the straight line η = l √ 2 (r − 1) − √ σ √ r − 1 ξ
- 49. 1.3 Existence of a Homoclinic Orbit in the Lorenz System 27 in a certain point ξ1 on the interval (0, ξ0), then the inequalities ξ ξ0, η P1(ξ) for ξ ∈ [ξ1, ξ0] (1.44) hold on a global attractor A of system (1.36). Similarly, if the graph of the function η = P2(ξ) intersects the graph of the straight line η = − l √ 2 (r − 1) − √ σ ξ √ r − 1 in a certain point ξ2 on the interval (−ξ0, 0) then the inequalities ξ −ξ0, η P2(ξ) for [−ξ0, ξ2] (1.45) are true on the global attractor A of system (1.36). Note that the surfaces {ζ = C − Bξ2 /2, C Bξ2 0 /2} are contact-free for system (1.36) in the strip {|ξ| ≤ ξ0}. Hence the estimate ζ ≤ B(ξ2 0 − ξ2 )/2 (1.46) holds on a global attractor of system (1.36). We have thus proved the following result ([24]). Theorem 1.2 Estimates (1.37)–(1.38), (1.44)–(1.46) hold on a global attractor of system (1.35). Let us give now a simple estimate of ξ0. To do this we note that the inequali- ties (1.44) hold if the graph of η = P1(ξ) intersects the graph of the straight line η = l/( √ 2 (r − 1)). From the positiveness of μ in equation (1.41) it follows that P1(ξ)2 (ξ2 − ξ2 0 ) − 1 2 (ξ4 − ξ4 0 ). Therefore, a sufficient condition for the above intersection to take place is that (1 − ξ2 0 ) − 1 2 (1 − ξ4 0 ) = l2 2(r − 1)2 . This inequality implies that ξ0 = 1 + l r − 1 . (1.47) Similar reasoning may also be applied to estimate (1.45). It follows from relation (1.47) that any global attractor of (1.36) lies in a domain which is bounded uniformly with respect to the parameterr ∈ (1, +∞). For a global B-attractor in the case b ≤ 2,
- 50. 28 1 Attractors and Lyapunov Functions the estimates (1.37)–(1.38) are asymptotically the best possible as r → +∞. Indeed, in this case, as r → +∞ the following inequalities hold on the global B-attractor: |η| ≤ 1/ √ 2, |ξ| ≤ √ 2. We recall that a part of a global B-attractor consists of the unstable manifold of the zero equilibrium, which may be represented approximately (for small ε) by the formulae ζ = −Bξ2 /2, η2 = ξ2 − ξ4 /2 . So for large r the global B-attractor has points close to the planes {|ξ| = √ 2}, {|η| = 1/ √ 2}. 1.3.3 The Existence of Homoclinic Orbits Let ξ+ , η+ , ζ+ denote a solution of (1.36) associated with the positive branch of the unstable manifold of the saddle point (0, 0, 0), that goes into the half-plane {ξ 0}, that is, a solution of (1.36) such that lim t→−∞ (ξ+ (t), η+ (t), ζ+ (t)) = (0, 0, 0) and ξ+ (t) 0 for t ∈ (−∞, T ). Here T is a certain number or +∞. It is well-known ([20, 25, 27]) that if the values of σ and b and the value of r are close enough to 1, then T = +∞. Let us consider a smooth path s ∈ [0, 1] → (b(s) , σ(s) , r(s)) in the parameter space {b, σ,r}. The main result of this section is the following theorem ([24]). Theorem 1.3 Suppose that for system (1.36) with parameters b(0), σ(0), r(0) there exist numbers T τ such that the relations ξ+ (T ) = η+ (τ) = 0, ξ+ (t) 0, ∀ t T, (1.48) η+ (t) = 0, ∀ t T, t = τ (1.49) hold. Suppose also that for system (1.36) with parameters b(1), σ(1), r(1) the inequality ξ+ (t) 0, ∀ t ∈ R (1.50) is true. Then there exists a number s0 ∈ [0, 1] such that system (1.36) with parameters b(s0), σ(s0), r(s0) has a solution (ξ+ , η+ , ζ+ ) corresponding to a homoclinic orbit. In order to prove this assertion we shall need the following lemmas.
- 51. 1.3 Existence of a Homoclinic Orbit in the Lorenz System 29 Lemma 1.1 If the conditions η+ (τ) = 0, η+ (t) 0, ∀ t ∈ (−∞, τ), hold for system (1.36), then η̇+ (τ) 0. Proof Suppose the contrary, i.e. η̇+ (τ) = 0. Then we derive from the two last equa- tions of system (1.36) that η̈+ (τ) = Aζ+ (τ) ξ+ (τ) . (1.51) It follows from the relations η+ (t) 0, ξ+ (t) 0, ∀ t ∈ (−∞, τ) and from the last equation of (1.36) that ζ+ (t) 0 , ∀ t ∈ (−∞, τ]. This inequality and (1.51) imply the inequality η̈+ (τ) 0 follows. But this contradicts the assumption η̇+ (τ) = 0 and the conditions of the lemma. This contradiction proves Lemma 1.1. Lemma 1.2 Consider system (1.36). Suppose that the relations (1.48), (1.49) and the inequalities η+ (t) 0, ∀ t ∈ (−∞, τ), η+ (t) ≤ 0, ∀ t ∈ (τ, T ) (1.52) are true. Then inequality (1.49) also holds. Proof Suppose the contrary. Then we conclude that a number ς ∈ (τ, T ), exists such that η+ (ς) = η̇+ (ς) = 0, η̈+ (ς) = Aξ+ (ς)ζ+ (ς) 0, η+ (t) 0, ∀ t ∈ (ς, T ) are valid. Note that the orbit corresponding to the solution (ξ(t), η(t), ζ(t)) = (0, 0, ζ(0) exp(−At)) belongs to the stable manifold of the saddle point (0, 0, 0). Hence, from the conditions (1.48), (1.49) and from the above relations it follows that the positive branch of the unstable manifold corresponding to the solution (ξ+ , η+ , ζ+ ) and the stable manifold intersect. Then the positive branch of the unsta- ble manifold belongs completely to the stable manifold of the saddle, and the relation ξ+ (t) 0, ∀ t ≥ ς is valid. The latter relation contradicts the hypothesis (1.48). This contradiction proves Lemma 1.2. Remark 1.6 It is possible to give the following geometrical interpretation of this proof in the phase space with the coordinates ξ, η, ζ. A piece of the stable manifold of the saddle ξ = η = ζ = 0 is situated “under” the set {ξ 0, η = 0, ζ ≤ 1 − γ ξ2 }. This property does not allow the trajectory with the initial data from the set to reach the plane ξ = 0 if it remains in the quadrant {ξ ≥ 0, η ≤ 0}. Let us consider the polynomial λ3 + aλ2 + bλ + c, (1.53) where a, b, c are positive numbers.
- 52. 30 1 Attractors and Lyapunov Functions Lemma 1.3 Either all zeros of the polynomial (1.53) have negative real parts, or two of them have non-zero imaginary parts. Proof It is well-known ([9]) that all the zeros of (1.53) have negative real parts if and only if ab c. If ab = c the polynomial (1.53) has two pure imaginary zeros. Suppose now that for certain a, b, c with ab c the polynomial (1.53) has only real zeros. Since the coefficients are positive it follows that these zeros are negative. This leads to the inequality ab c which contradicts our assumption. Proof of Theorem 1.3 It is well-known ([12]) that the semi-orbit of system (1.36) {(ξ+ (t), η+ (t), ζ+ (t)) | t ∈ (−∞, t0)} depends continuously on parameter s. Here t0 is an arbitrary fixed number. It follows from this and from Lemma 1.1 that, if conditions (1.48)–(1.49) hold for system (1.36) with parameters b(s1), σ(s1), r(s1) then these conditions also hold for b(s), σ(s), r(s) provided that s ∈ (s1 − δ, s1 + δ). Here δ is a certain sufficiently small number and the numbers τ and T depend on parameter s. It follows from the above reasoning that the relations (1.48)–(1.49) are valid for a certain interval (0, s0). Further we shall assume that (0, s0) is the maximal interval where the relations (1.48)–(1.49) are valid. Let us demonstrate that there exists a certain homoclinic orbit which corresponds to the values b(s0), σ(s0), r(s0). We first note that for these parameters and some value τ η+ (t) 0, ∀ t τ, η+ (t) ≤ 0, ∀ t ≥ τ, ξ+ (t) 0, ∀ t ∈ (−∞, +∞). (1.54) Indeed, if there exist numbers T2 T1 τ, for which ξ+ (t) 0, ∀ t ∈ (−∞, T2), ξ+ (T2) = 0, η+ (T1) 0, η+ (t) 0, ∀ t τ, η+ (τ) = 0, η̇+ (τ) 0 are true then for s sufficiently close to s0 and such that s s0 the inequality η+ (T1) 0 still holds. This contradicts the definition of the number s0. If there exist numbers T1 τ,forwhichη+ (T1) 0, η+ (t) 0,∀ t τ, η+ (τ) = 0, η̇+ (τ) 0, and ξ+ (t) 0, ∀ t ∈ (−∞, +∞), then again for s sufficiently close to s0 and such that s s0 the inequality η+ (T1) 0 remains true and again we have a contradic- tion with the definition of the number s0. If there exist numbers T τ, for which ξ+ (t) 0, ∀ t T , ξ+ (T ) = 0, η+ (t) 0, ∀ t τ, η+ (t) ≤ 0, ∀ t ∈ [τ, T ], then the inequality (1.49) is true according Lemma 1.2. Consequently for s = s0 relations (1.48)–(1.49) are fulfilled and then (0, s0) is not the maximal interval, for which these relations are true. By these contradictions inequalities (1.54) are proven. It follows from (1.54) that for s = s0 only an equilibrium can be the ω-limit set of the orbit of (ξ+ , η+ , ζ+ ). Let us demonstrate that the equilibrium (ξ, η, ζ) = (1/ √ γ , 0, 0) can not be an ω-limit point of this orbit. The linearization in the neighborhood of this equilibrium gives the characteristic polynomial λ3 + (A + μ)λ2 + (Aμ + 2/γ )λ + 2A.
- 53. 1.3 Existence of a Homoclinic Orbit in the Lorenz System 31 Suppose that for s = s0 the ω-limit set of the positive branch of the unstable manifold correspondingtothesolutionξ+ , η+ , ζ+ includesthepoint(ξ, η, ζ) = (1/ √ γ , 0, 0). By Lemma 1.3 and by the fact that the semi-orbits {ξ+ (t), η+ (t), ζ+ (t) | t ∈ (−∞, t0)} depend continuously on the parameter s we obtain the following asser- tion. For the values s which are sufficiently close to s0 the positive branch of the unstable manifold corresponding to (ξ+ , η+ , ζ+ ) either tend to an equilibrium state (ξ, η, ζ) = (1/ √ γ , 0, 0) as t → +∞, or oscillate in some time-interval with chang- ing sign of the coordinate η. Both of these possibilities contradict properties (1.48)– (1.49). Hence, for system (1.36) with parameters b(s0), σ(s0), r(s0) the orbit of (ξ+ , η+ , ζ+ ) tends to the trivial equilibrium state as t → +∞. Remark 1.7 Note that the proof of Theorem 1.3 actually yields a stronger result, which may be formulated as follows. If relations (1.48)–(1.49) hold for s ∈ [0, s0), but not for s = s0, then system (1.36) with parameters b(s0), σ(s0), r(s0) has a homoclinic orbit. Let us apply Theorem 1.3 in various specific cases. Fix the numbers b and σ. It is well-known ([20, 27]) that inequality (1.50) is true for r sufficiently close to 1. We will show that if 3σ − 2b 1 (1.55) and r is sufficiently large, then relations (1.48)–(1.49) will hold. Indeed, consider the system Q dQ dξ = −μQ − Pξ − ϕ(ξ), Q d P dξ = −AP − BQξ, (1.56) which is equivalent to system (1.36) in the sets {ξ ≥ 0, η 0} and {ξ ≥ 0, η 0}, where P and Q are solutions of (1.56) which are functions of ξ. Since Theorem 1.2 implies that the quantities (ξ+ (t), η+ (t), ζ+ (t)) are bounded uniformly with respect to the parameter r, we can carry out an asymptotic integration of the solutions of system (1.56) with a small parameter ε that correspond to the branch of the unstable manifold under consideration. In the first approximation these solutions may be written in the form Q1(ξ)2 = ξ2 − ξ4 2 − 2μ ξ 0 ξ # 1 − ξ2/2 dξ − 2AB ξ 0 ξ 1 − # 1 − ξ2/2 dξ, Q1(ξ) ≥ 0, P1(ξ) = − $ β 2 % ξ2 + AB 1 − # 1 − ξ2/2 ,
- 54. 32 1 Attractors and Lyapunov Functions Q2(ξ)2 = ξ2 − ξ4 2 − 2μ √ 2 ξ ξ 1 − ξ2/2 dξ − 4 3 μ + 2AB √ 2 ξ ξ $ 1 + 1 − ξ2/2 % dξ − 2 3 AB, Q2(ξ) ≤ 0, P2(ξ) = − $ B 2 % ξ2 + AB $ 1 + 1 − ξ2/2 % . It follows from these formulae that if inequality (1.56) holds, then for some T τ relations (1.48)–(1.49) will also hold and at the same time ζ+ (T ) = P2(0) = 2AB, η+ (T ) = Q2(0) = − # 8(AB − μ)/3 = − 8ε(3σ − 2b − 1)/(3 √ σ). Thus, all the conditions of Theorem 1.3 hold for the path s → (b(s), σ(s),r(s)) with b(s) ≡ b, σ(s) = σ,r(0) = r1,r(1) = r2, where r1 is sufficiently large and r2 is sufficiently close to 1. We may therefore formulate the following result. Corollary 1.1 For any positive numbers b and σ satisfying the inequality (1.55) a number r ∈ (1, +∞) exists, such that system (1.36) with these parameters b, σ and r has a solution (ξ+ , η+ , ζ+ ) corresponding to a homoclinic orbit. Remark 1.8 Corollary 1.1 was first obtained in [21, 22] and discussed later in [5, 13, 23]. Now fix σ = 10 and r = 28, and consider the parameter b ∈ (0, +∞). It is well- known ([5]) that for b 3σ − 1 2 condition (1.50) is fulfilled. To analyse system (1.36) for small b, we reduce it to the form ξ̇ = η, η̇ = −μη − uξ + ξ − ξ3 , u̇ = −Au + ε(2σ − b) √ σ ξ2 , (1.57) where u = ζ + Bξ2 /2. Since the semi-orbit {(ξ+ (t), η+ (t), ζ+ (t))|t ∈ (−∞, t0]} depends continuously on the parameter b, it follows that, when b is small, the system (1.57) may be replaced by the system ξ̇ = η, η̇ = − ε(σ + 1) √ σ η − uξ + ξ − ξ3 , u̇ = 2ε √ σ ξ2 . (1.58) Numerical integration of the solution (ξ+ , η+ , ζ+ ) of system (1.58) for σ = 10, r = 28showsthatconditions(1.48)–(1.49)aresatisfied.Hence,theabovearguments, using Theorem 1.3, yield the following.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. S OFFENER BRIEF AN EINEN UNBEKANNTEN EHR geehrter Herr! Ich nehme mir die Freiheit, in aller Öffentlichkeit ein Schreiben an Sie zu richten, weil ich Sie nicht länger darüber im Unklaren lassen möchte, wie unsympathisch Sie mir sind. Mit Erstaunen werden Sie fragen, welche Gründe um alles in der Welt mich, der ich Sie nicht kenne, bewegen, Sie einen mir unsympathischen Menschen zu heißen. So hören Sie denn, daß ich nicht den winzigsten Grund habe, um so mehr, als ich Sie, wie gesagt, nicht kenne. Trotzdem sind Sie mir in tiefster Seele und aus einem, wenn ich mich so ausdrücken darf, allgemeinen Gefühl heraus unausstehlich, und ich versichere laut, daß ich jeden Zug Ihres Wesens, jede Spur Ihres Seins widerlich finde, mögen Sie existieren oder nicht. Ich bin überzeugt, daß Ihre sauber genähten Krawatten mir nicht minder auf die Nerven fallen würden als die Handbewegungen, womit Sie Ihrer jüngsten Tochter, wenn Sie eine hätten, über den Scheitel fahren, wenn sie einen hätte, und daß mich die Geschwulst hinter Ihrem rechten Ohre, gesetzt, Sie hätten eine, ebenso peinlich berühren würde wie die Art, in der Sie über Angelegenheiten der inneren Politik sprechen — wenn Sie darüber sprechen. Warum übrigens in drei Teufels Namen lassen Sie sich jene Geschwulst hinter dem rechten Ohre nicht endlich operieren — für den Fall, Sie haben eine? Sie gelten mir, klipp und klar, in jedweder Hinsicht als vollendeter Typus eines Proleten — herrisch, ordinär, albern, rücksichtslos und seicht, wie Sie hoffentlich sind. Um das Maß voll zu machen, lieben Sie — Sie werden mich darin nicht enttäuschen — das Skatspiel und
- 57. die Lektüre infamer Schmöker, die nicht angeführt sein mögen, und entrüsten sich womöglich als sogenannter Gegner des Fremdwortes, daß ich Wörter wie «Lektüre» und «infam» anwende. Ich gebe zu, daß ich meinem Vorurteil, das am Äußerlichen haftet, allzu willfährig bin und besser daran täte, Ihr Inneres zu prüfen, muß indessen zu meiner Rechtfertigung erklären, daß ich die «Unsympathischkeit» auf den ersten Blick, die sich jederzeit in das Gegenteil verkehren könnte, bei weitem der «Sympathischkeit», um nicht zu sagen «Liebe» auf den ersten Blick den Vorzug gebe, welche kritischen Erschütterungen nur in seltenen Fällen standzuhalten vermag. Mit Freuden bin ich bereit, mich mit Ihnen, den ich gottlob nicht kenne, und von dem ich nicht weiß, ob er überhaupt auf Erden wandelt, an drittem Orte zu treffen, um die wenig erquicklichen Beziehungen, die uns verknüpfen, in erfreulichere oder sogar erfreuliche zu verändern, obwohl ich meine Besorgnis nicht verhehlen möchte, daß Sie gerade bei naher Bekanntschaft und nach Preisgabe Ihres Inwendigen ein gräßliches Subjekt, unter Umständen sogar ein hierorts als «Mistvieh» zu bezeichnendes Individuum abgeben dürften, dem ich besser aus dem Wege trete. Lassen wir es also zu beiderseitigem Vorteile bei der bestehenden Unbekanntschaft verbleiben, und bauen wir auf unser Vorurteil, das sicherlich wohl begründet ist, sei es auch nur gefühlsmäßig. «Unser» Vorurteil schreibe ich, da ich allzu gut weiß, wie wenig Sie Ihrerseits mich leiden mögen — mich, den es gibt. Mit dem Ausdrucke vollkommener Hochachtung bin ich Ihnen, den es nicht gibt, ergeben und schließe mit dem Bemerken, daß die letztgebrauchte Redewendung eine leere Phrase ist und nichts weiter. H. R.
- 58. «W DER OCHSE P e r s o n e n : Hans Kurt Theo AS stehst du da und sinnst?» «Ich sinne nicht. Ich warte auf Theo.» «Wartest du lange?» «Ja, aber er kommt nicht.» «Ich will dir helfen. Du weißt, daß der Ochse kommt, wenn man von ihm spricht?» «Freilich.» «Also laß uns von Theo sprechen.» Hans und Kurt sprechen von Theo, damit der Ochse kommt. Aber er kommt nicht. «Du, unser Sprechen ist für die Katz’. Theo kommt nicht.» «Nein, er kommt nicht.» Theo kommt. Hans und Kurt brechen gleichzeitig in die Worte aus: «Siehst du, er ist d o c h ein Ochse!» «Wer?» fragt Theo. «Du!» lautet die fröhliche Antwort. Theo ist vom Gegenteil überzeugt.
- 59. E VON DEM MANNE, DER AUSZOG, ERDBEEREN ZU SUCHEN UND PFIFFERLINGE MIT HEIMBRACHTE INE sehr schöne Geschichte. Von mir. Und außerdem eine sehr kurze Geschichte. Aber auch kurze Geschichten können schön sein. Ich liebe die kurzen Geschichten, die schön sind. Dies ist eine. Wenigstens meiner Meinung nach. Also: ein Mann ging in den Wald, um Erdbeeren zu suchen. Sogenannte Walderdbeeren. (Weil sie im Walde wachsen!) Aber er fand keine. Aber Pfifferlinge fand er. Einen ganzen Sack voll. Er ging heim mit seinem Sack voller Pfifferlinge oder Pfefferlinge. In Sachsen sagt man «Gehlchen». Die Sachsen müssen immer eine Extrawurst haben. Na, und die schmorte er sich.[1] Und aß sie. Und die schmeckten sehr gut. [1] Die Pilze, meine Verehrten! In Sachsen sagt man «schmeckten sehr s c h ö n ». Die schmeckten also sehr schön. Und da freute sich der Mann schrecklich und vergaß völlig, daß er in den Wald gegangen war, um Erdbeeren zu suchen.
- 60. * * * Das ist die ganze Geschichte. Ist sie nicht schön?
- 61. U DIE WAHRHEIT M es ganz aufrichtig und ehrlich zu sagen, so halte ich — menschlich — jeden beliebigen Kaufmann für tausendmal wertvoller als irgendeinen Künstler. Man wird mir diesen Satz nicht glauben — um so weniger, als ich heftig beteuere, ihn durchweg ernst zu meinen. Aber: ich halte zehn gute Kaufleute, Gott straf mich, für tausendmal wichtiger — menschlich — als einen halben Gymnasiallehrer. Auch diesen Satz wird mir niemand glauben. Nun denn, ganz aufrichtig und ehrlich: ich halte weder Kaufmann noch Lehrer für wichtig, geschweige denn für wertvoll. Den Künstler erst recht nicht. Dies ist voller Ernst und mein letztes Wort in dieser Sache. Punktum.
- 62. A KEIN SCHÖNRER TOD IST AUF DER WELT . . . LS es 418 (418!) Tage lang, 418 Tage lang hintereinander, 418 Tage lang ununterbrochen hintereinander geregnet hatte, 418 Tage lang geregnet hatte, waren alle Wesen des Lebens überdrüssig. Und der hochbetagte Bibliothekar Stibulke sprach zu seiner Frau: «Rosa, weißt du was, wir ersäufen uns!» Das war aber gar nicht mehr nötig; denn — siehe — in demselben Augenblicke wurde das Ehepaar von den eindringenden Fluten hinweggespült.
- 63. S SERENISSIMUS JAGT SCHMETTERLINGE ERENISSIMUS jagt Schmetterlinge. Für seine Sammlung. — Hat eine Schmetterlings-Sammlung. — Lauter Schmetterlinge. Und Käfer. — Und Briefmarken. — Alles durcheinander. — Auch Strumpfbänder. Weibliche. — Souvenirs. — Namentlich Strumpfbänder. — Nebenbei auch einige Schmetterlinge. — Zwei oder drei. — Oder einen? — Ja, e i n e n . Einen einzigen. Tja. Aber einen ganz seltenen! — Ein Mistpfauenauge. Oder so ähnlich. Ganz drolliges Viech. — Sieht aus wie en Käfer. — Tja. — Ist auch en Käfer. Heißt genau genommen Mistpfauenkäfer. — Oder so ähnlich. — Oder Mistkäfer. — Ja: Mistkäfer. — Geschmacklos. — Warum nich Guanokäfer? Oder Kloakenkäfer? — Tja. — Ein entzückender Kloakenkäfer. — Schillert in allen Farben. — Täuschend imitiert. — Sieht aus wie echt. Wie wenn er lebte. — Tja. — War ooch teuer genug! Zierte Lisas Strumpfbänder, die Katze. — Z w e i waren es sogar. Eigentlich. Ursprünglich. — Na, der e i n e ist gerettet. — Apartes Andenken. An die verflossene Lisa. — Saßen auf dem Strumpfband, die beiden Käfer. Oder vielmehr: auf d e n Strumpfbändern. Auf jedem einer. — Lisa mußte zweie haben. — Dolles Weib. T, t, t, t. — Viel Geld gekostet. — Tja. — Na, egal. — War die Sache wert. — Süßer Käfer. — Hat Karriere gemacht. — Nach unten. — Bis in den Rinnstein. — Ooch en Kloakenkäfer geworden. Oder Mistkäfer. — Hähä, blendender Witz. — Jaja, feines Köppchen! — Tja. — Na, wolln ma sehn, was sich tun läßt. Serenissimus stelzt über ein Stoppelfeld. Das Schmetterlingsnetz in der Hand. Er will seine Sammlung bereichern. Schmetterlinge jagen ist sein neuster Sport.
- 64. Serenissimus ist passionierter Schmetterlingsjäger. Absolut einwandfrei edles Weidwerk. Totschick! — Heissa, hussa! Serenissimus stelzt über das Stoppelfeld. Mit sagenhaft elastischen Schritten. Einem Schmetterling ist er auf den Fersen. Einem Sauerkohlweißling. Der schillert so angenehm rötlich. Vielleicht gar en Rotkohlweißling? Oder en Sauerkohlrötling? Vertrackt schwierige Kiste, Schmetterlinge jagen. Die Tiere flattern in der Luft herum. Sind gar nich en bißchen zutraulich. Na, wern den Kerl schon kriegen! — Serenissimus stelzt über die Stoppeln. Dem Weißling hinterher. Da geschieht etwas durchaus Unerwartetes. Eine Dampfwalze kommt in rasendem Tempo auf Serenissimus zugeschossen. Wie ein Pfeil. Serenissimus, der bei e i n e m H a a r e den Weißling im Netz hatte, springt — juchopps — mit einem Fluch beiseite. Himmelherrgottspappedeckel, Klabund und Wolkenbruch!! — — — Die Dampfwalze prescht wie besessen an dem verdatterten Ferschten vorüber . . . . Da bemerkt Serenissimus dort, wo die Dampfwalze ihren Weg genommen hat, einen rotgelben Tupfen: den zu Brei gequetschten Sauerkohlrotweißling. Er hebt ihn auf und steckt ihn ins Netz. Das Netz schultert er und geht heim. Serenissime. S o f i n g S e r e n i s s i m u s s e i n e n e r s t e n S c h m e t t e r l i n g . * * * Daraus geht hervor: Um einem Serenissimo dienstbar zu sein, scheuen die himmlischen Gewalten weder Kosten noch Mühe.
- 65. L DAS ZIMMER INKS eine Wand. Rechts eine Wand. Vorn eine Wand. Hinten eine Wand. Oben die Decke. Unten die Diele. — In der linken Wand eine Tür, in der rechten Wand zwei Fenster, in der vorderen Wand nichts, in der hinteren Wand nichts. — An allen vier Wänden Tapete. — In der Mitte der Diele ein Tisch, darauf eine Vase. Um den Tisch drei Stühle. An der rechten Wand zwischen den Fenstern ein Büchergestell. An der linken Wand über der Tür ein Haussegen. An der vorderen Wand ein Ofen, ein Waschtisch, ein Bett, ein Spiegel. An der hinteren Wand ein Sofa, ein Schreibtisch mit Lehnsessel, ein Schrank; über dem Sofa ein großes Bild. An der Decke eine Lampe. Dies ist ein Zimmer. — Was ist ein Zimmer? — Ein Selbstmordmotiv. Öde, kahl, ekel. — — — Laß an den Fenstern Gardinen anbringen, und in der Dämmerstunde stell auf den Tisch die duftenden Reseden: — das Zimmer ist traut und wohnlich. Und liegt ein sündhaft schönes Weib im Bett, der Teufel hole dich, wenn du das Zimmer nicht mit Lust beziehst.
- 66. D HAND UND AUGE (Ein Reise-Erlebnis) P e r s o n e n : Die anmutige Dame Der stattliche Herr O r t : Eisenbahn-Abteil 2. Klasse ER Herr: «Darf ich das Fenster öffnen?» Die Dame: «Ja.» Der Herr: «Stört es Sie, wenn ich eine Zigarette rauche?» Die Dame: «Nein.» Der Herr: «Darf ich fragen, wohin Ihre Reise geht?» Die Dame: «Ja. Nach Danzig.» Der Herr: «Wie sich das trifft! Ausgerechnet nach Danzig fahre auch i c h !» Der Herr: «Ist es Ihnen unangenehm, mit mir im selben Abteil fahren zu müssen?» Die Dame: «Nein.» Der Herr: «Fahren Sie gern Eisenbahn?» Die Dame: «Nein.» — — Ein Gespräch kommt nicht zustande. Es ist frostern im Abteil. Die Dame ist zugeknöpft. Der Herr versucht es mit einem Gewaltmittel: «Schauen Sie», spricht er, «ich hab’ ein Glasauge!» und nimmt sein linkes Auge heraus. Die Dame taut auf: «Ach!? — Ist das echt?» «Jawohl — es ist ein echtes nachgemachtes Auge.»
- 67. «Gott, wie goldig!» «Nicht wahr?» «Und o h n e das Auge sehen Sie gar nichts?» «Nein, nicht das mindeste.» «Und m i t dem Auge?» «Sehe ich auch nichts!» «Ja, ist denn das Auge nicht durchsichtig?» «Doch — aber womit sollte ich hindurchsehen?» «Haben Sie das Auge verloren?» «Ja — ein Fräulein hat es mir mit der Hutnadel ausgestochen.» «Wie gemein!» «Ich habe mich gebührend gerächt.» «Inwiefern?» «Ich habe das Fräulein geheiratet.» Die Dame rückt ab und knöpft sich wiederum zu. Der Herr hat seinen Reiz zur guten Hälfte verloren. Er ist verheiratet! Der Herr steckt sein Auge ein. Die Dame — nach langer Pause —: «Sie tragen ja gar keinen Trauring?» «Nein, warum? Ich bin ja nicht verheiratet.» «Sie sagten doch . . .» «Ein Scherz.» «Aber das falsche Auge ist doch wenigstens e c h t , wie?» «Völlig echt, meine Gnädige.» «Darf ich es mal sehen?» «Mit Vergnügen.» Der Herr reicht der Dame das echte falsche Auge. Die Dame nimmt es in die linke Hand. Sie faßt das Auge scharf ins Auge und spricht: «Es ist täuschend imitiert. Besser als diese meine linke Hand.» «Was ist mit der Hand?» «Sie ist künstlich. Aus Marmor.» «Seltsam. Ein falsches Auge in falscher Hand!» «Ich finde das weniger seltsam, als wenn ein echtes Auge in einer echten Hand läge.» «So? Wäre das seltsamer?»
- 68. «Es wäre nicht nur seltsamer, es wäre u n m ö g l i c h .» «Es ist nicht unmöglich. — Mein Auge ist kein Glasauge. — Das Auge ist mein wirkliches, echtes Auge.» Die Dame läßt erschreckt das Auge fallen. Das Auge blickt die Dame wehmutig an. Die Dame greift gerührt mit ihrer Linken nach dem Auge — — — die Hand füllt sich mit Leben, Blut durchrinnt sie, Puls klopft auf. Das Auge zwinkert bedeutsam. Der Herr sieht die marmornen Finger der Dame sich regen; «Ihre Hand, Gnädige, scheint lebend zu sein!» Die Dame krümmt die Finger — und ist selbst betroffen über die Verwandlung. Sie streicht mit der Rechten über das Auge in ihrer Linken, und das Auge schläft ein. Der Herr nimmt es und steckt es in seine Höhle zurück. Die Dame kann nicht anders, sie drückt einen Kuß auf das Auge. Der Herr küßt der Dame die linke Hand. Das Auge öffnet sich und blickt dankbar. Die Linke der Dame streichelt die Wange des Herrn. «D a n z i g —!»
- 69. A TROPFEN AUS HEITERM HIMMEL UF der Wiese steht ein Greis und will eine Kneippkur machen. Er ist barfuß und barhaupt. Über ihm hängt ein wunderschöner, blauer, wolkenloser Himmel. Der Greis hält Ausschau nach einer Kuh, die fern am Waldrande Bedürfnis über Bedürfnis verrichtet. Da tropft dem Greis etwas aufs Haupt. Ein dicker Tropfen. Der Greis greift mit der Hand auf seinen Schädel und wischt den Tropfen ab. Dann lugt er auf zum Himmel. Der Himmel glänzt in seidiger Bläue. «Wie?» denkt der Greis, «ein Tropfen aus heiterm Himmel?» Und er begibt sich von dem Flecke, auf dem er gestanden, weg und pflanzt sich anderswo auf. Daselbst hält er wiederum Ausschau nach jener bedürfnisstrotzenden Kuh. Er steht nicht lange — der Greis —, so kleckt ihm ein zweiter Tropfen aufs Haupt. Aufschauend zum Himmel, wundert er sich ins Fäustchen und wischt sodann den nassen Tropfen sich vom Schädel. Der Himmel lacht. Mit Recht. «Wenn das so weitergeht,» denkt unser Greis bei sich, «das kann ja gut werden!» Und er bleibt stehen, wo er steht. Er will herauskriegen, wo die Tropfen herkommen; auch will er wissen, ob ihrer noch mehr herunterklecken. Abermals wendet er sein Augenmerk nach jener fladenden Kuh und vergißt über sie das Tropfen.
- 70. Es währt nur kurze Zeit, so tropft dem Greis ein dritter Tropfen auf den Kopf. Der Greis runzelt die Stirn und betrachtet den Himmel. Der thront unschuldig und engelisch-rein über der Szenerie. Der Greis legt sich ins grüne Gras und läßt den Himmel nicht aus dem Auge. Es kleckt kein Tropfen mehr vom Himmel. «Aha,» denkt sich der Greis, «dies geschieht, weil ich Obacht gebe». Und er paßt auf. Er wendet keinen Blick vom Himmel. — — — — — Auf der Wiese liegt ein Greis. Er hat eine Kneippkur machen wollen, aber er muß aufpassen, ob es tropft. Er ist überzeugt, daß in dem Augenblicke, wo er den Himmel außer acht läßt, ein Tropfen ihm aufs Haupt kleckt. Der Greis schläft darüber ein. Er träumt, daß ihm ein Tropfen auf den Kopf kleckt. Er stellt sich anderswohin, und ein zweiter Tropfen kleckt. Er bleibt stehen, und ein dritter Tropfen kleckt. Da legt er sich ins grüne Gras und spannt auf den Himmel. — Dies träumt der Greis. Die Kuh möhkt plötzlich dicht bei ihm. Davon erwacht der Greis, erhebt sich ächzend und begibt sich an die Kneippkur. Ihm ist, als seien drei Tropfen auf seinen Kopf gekleckt. Dies ist jedoch völlig unmöglich. Denn der Himmel ist blau, heiter und wolkenlos. Hat der Greis geträumt?
- 71. D DAS ALTER P e r s o n e n : Der gutgelaunte Vorgesetzte Der wie auf den Kopf gefallene Bewerber ER Vorgesetzte läßt den Bewerber eintreten und ersucht ihn, Platz zu greifen. Es entspinnt sich eine Unterredung, die auf einem gewissen halbtoten Punkt stehen bleibt: Der Vorgesetzte möchte Einzelheiten aus dem Privatleben des Bewerbers wissen. Er fragt zuvörderst nach dem Alter. «Wie alt sind Sie denn?» «Ich werde 32.» «Wie alt Sie sind?» «Ich werde 32.» «Ich will nicht wissen, wie alt Sie w e r d e n ; ich will wissen, wie alt Sie s i n d .» Der Bewerber schweigt kopfscheu. «Na wie alt s i n d Sie denn?» «Ich bin 31 gewesen.» «Guter Mann, hm, wenn Sie 31 g e w e s e n sind, so sind Sie zur Zeit 32. Soeben behaupten Sie jedoch, Sie w ü r d e n erst 32.» «Ja, das stimmt.» «Nee, das stimmt nicht. Wenn Sie 32 w e r d e n , können Sie nicht 32 s e i n .» «Nein, so nicht, — ich bin nicht 32. Ich w e r d e 32.» «Schön. Demnach dürften Sie 31 sein.» «Ja natürlich. Ich bin 31!» «Also Sie sind 31. — Wann ist Ihr Geburtstag?» «Am 5. April.» «Das wäre heute in 6 Wochen?» «Zu dienen.» «Wie alt werden Sie heute in 6 Wochen?»
- 72. Der Bewerber, zaghaft und scheu: «32 . .» «Richtig.» «Ihr wievielter Geburtstag ist das?» «Mein 32. selbstredend.» «Durchaus nicht! — Ihr 33.!» «Das verstehe ich nicht.» «Nein? — Merken Sie auf: Als Sie zur Welt kamen, begingen Sie Ihren ersten Geburtstag. An jenem ersten Geburtstage waren Sie null Jahre alt. — Als Sie Ihren zweiten Geburtstag feierten, vollendeten Sie das erste Jahr, d. h. Sie wurden am z w e i t e n Geburtstag e i n Jahr alt. — Sehen Sie das ein?» Der Bewerber, gänzlich verwirrt: «Oh ja!» «Nun also. — Sie s i n d 30 g e w e s e n , s i n d 31, w e r d e n 32 und feiern in Kürze den 33. Geburtstag.» Der Bewerber bricht ohnmächtig zusammen. Die Unterredung ist beendet.
- 73. D ALLE WEGE FÜHREN NACH ROM IESES Sprichwort ist eine hundsgemeine Lüge. Der Privatdozent Kladderosinenzagel mußte es am eigenen Leibe erfahren. Er, den wir um der Kürze willen K. nennen wollen, machte sich an einem Ferientage auf die denn doch nicht mehr so eigentlich ganz naturfarbig genannt werden dürfenden Socken, um gen Rom zu fahrten. Er, K., fußte auf dem Sprichwort: Alle Wege führen nach Rom. K. wanderte, mit reichlichem Mundvorrate und einer leeren Thermosflasche ausgestattet, einen vollen Nachmittag lang. Reiseziel: Rom. Es führen aber mitnichten alle Wege nach Rom. D e r W e g , den K. einzuschlagen für ratsam befunden hatte, hörte plötzlich auf, ein Weg zu sein und verwandelte sich in eine Wiese, auf welcher notgedrungen sieben Kühe — die Verkörperung der fetten Jahre — sich an ihrem Anblicke und dem saftigen Grün weideten. Und K. stand hinter einer Tafel, die von vorn zu besichtigen er nicht umhinkonnte. Die Tafel bezog sich auf den Weg, welchen K. zurückgelegt hatte, und trug die Aufschrift: «Verbotener Weg». In einem Lande, wo die Polizei so auf dem Damme ist wie in Deutschland, führt zwar mancher Weg nach Rom, aber er ist verboten. K. mußte umkehren und sich des Planes, auf natürlichem Wege nach Rom zu gelangen, entschlagen.
- 74. «HÖHENLUFT» Ein Roman aus den Tiroler Bergen von Paul Grabein ist im Okt. 1916 als Ullstein-Buch — 1 M.! — erschienen. Ich habe das Buch gelesen — unter Aufgebot größter Energie. Ein paar Worte darüber und dazu. Die Personen des Buches sind: Karl Gerboth, Maler, Hilde, seine Tochter, Franz Hilgers, Maler, Günther Marr, Leutnant. Handlung: Franz hat seinen Jugendfreund Günther eingeladen. Günther leistet der Einladung — Erholungsurlaub — Folge. Auf Seite 19 trifft er, nach dem Dörfchen, in dem Franz wohnt, wandernd, eine Dame. Dies ist Hilde Gerboth. Sofort weiß man «alles», und es kommt auch tatsächlich «alles» so. Franz ist der einzige Schüler Karl Gerboths und Bräutigam eben jener Hilde, freilich, ohne daß diese darum weiß. Der alte Gerboth hat sich von der Welt zurückgezogen und schafft in aller Stille. Hilde wird von ihm behütet und betreut, daß es eine Art hat. Sie ist die Tochter einer Dame, die — als Gattin Gerboths — Temperament und etliches darüber hinaus besaß. Aus Angst, Hilde könne ihrer Mutter nachschlagen, läßt sie der alte Gerboth nicht von sich. Sie ist absolut naiv und ahnungslos. Sie weiß nicht Musik, Tramway, Kino, Theater, Börse, Bordell, Liebe, Geld, Börse (absichtlich 2 Mal) — kurz: was Leben ist. Das weiß sie nicht. Sie ist 20 Jahre alt. Und Franz ist ein Schwächling, ein thraniger, limonadiger Hampelmann. Er muß kurz nach Günthers Ankunft
- 75. verreisen. Infolgedessen Solo-Szene zwischen Günther und Hilde. Aussprache — er schildert ihr die Welt und das Leben. Sie — die Freiheit lockt — verliebt sich in ihn. Sie will hinaus — in die sogenannte Welt. Sagt’s ihrem Vater. Der refüsiert. Hilde knickt zusammen. Günther trifft sie — tatsächlich durch Zufall! (Ich glaub’s! Wer noch?) — ein zweites Mal. Er redet ihr energisch zu. Franz kehrt zurück (aber das ging fix!) und erfährt durch Günther selbst, daß er, G., Hilde liebt und überhaupt: daß was los war. Franz zum alten Gerock oder Gehrock oder Gerboth: Höre mal, so und so — — und Gerboth spricht gründlich mit seinem Töchting. Klamauk. Sie will Franz nicht. Sie will Günther. Und in die Welt hinaus. Bon. Am Tag drauf hält Günther um ihre Hand an beim alten Klopstock. Der sagt Nein. Da sagt Günther: Dann heirat ich Ihre Hilde gegen Ihren Willen. Bumms. Aber der Alte — philosophisch! — gestattet eine letzte Aussprache zwischen Hilde und Günther, worin sie ihm erklärt, er dürfe hoffen, wenn er vor sie hinträte. Am nächsten Tag reist Günther nicht ab, oh nein. Er kann nicht: eine richtige Lawine hat sich bemüht, herniederzugehen, und das ist ihr auch gelungen. Aber die gute Hilde, die irgendeinen Schafhirten hat retten wollen vom Hungertode, gerät mitsamst ihrem Freßkörbchen und dem Bernhardiner (aha!) in sie (die Lawinije) hinein. Na, und Günther rettet sie selbstredend. Na, und dann kriegen sie sich. Na, und das ist ja die Hauptsache. Das Buch schließt (auf Seite 253!) mit den Worten Günthers: «Wagen wir es denn zusammen, Hilde!» Und nun sind sie glücklich, und uns entpullert eine Träne. Ich setze das Romänchen fort: Am 12. Sept. 1916 fällt Günther in der Sommeschlacht (das Buch spielt nämlich direktemang im Weltkrieg). Daraufhin begeht seine Frau einen ganz totsicheren Selbstmord. Daraufhin kriegt ihr Vater einen geharnischten Schlaganfall. Sela.
- 76. M EHE ANN und Frau faulenzen auf dem Diwan. Der Mann ist am Einschlafen. Die Frau wird von Halbträumen umfangen. Eine Fliege summt. Die Glocken einer fernen Kirche baumeln. — — — Der Mann ächzt, räkelt sich, fragt: «Sind das Glocken?» Die Frau horcht. «Das sind doch keine Glocken. — Das ist eine Fliege.» «Unsinn. Das ist doch keine Fliege. — Das sind Glocken.» «Das ist eine Fliege.» «Das sind Glocken.» Beide horchen. Der Mann: «Selbstredend sind das Glocken. — Warum wird denn geläutet?» Die Frau: «Ich werde doch Glocken von einer Fliege unterscheiden können! Ich höre keine Glocken. Das ist eine Fliege.» «Das sind Glocken.» «Wenn ich dir sage, das ist eine Fliege.» «Herrgott, das sind Glocken. Das ist doch keine Fliege!» «Das i s t eine Fliege!» «Das sind G l o c k e n !» «Na, da bleib’ bei deinem Glauben.» «So etwas Dummes! Ich bin doch nicht verrückt. Natürlich sind das Glocken. — Ganz deutlich.» «Eine Fliege ist es.» «Wo ich genau die einzelnen Glocken heraushöre.» «Was d u alles fertig bringst. — Ich höre bloß eine Fliege. — Warum sollten denn jetzt die Glocken läuten?!» «Ja, das möchte ich eben gerne wissen.» «Du kannst dich drauf verlassen, das ist eine Fliege.»
- 77. Beide horchen. Die Glocken haben aufgehört, zu summen. Auch die Fliege läutet nicht mehr. Der Mann denkt: Ekelhaft. So macht sie’s immer. Bei jeder Gelegenheit. Da ist einfach nichts zu wollen. Zum Auswachsen. — Eine Fliege! Lachhaft. — Aber da kann sie niemand davon abbringen. Sie bleibt bei ihrer Fliege. Es ist eine Fliege. Und wenn die Glocken hier in der Stube vor ihrer Nase läuteten, — — es ist eben eine Fliege. Albern. Wenn sie sich etwas einbildet, bleibt sie dabei. — Selbstredend waren es Glocken. — — — Mir einstreiten zu wollen, daß es eine Fliege war . . . . Er schläft. Die Frau denkt: Wenn es nicht zufällig mein Mann wäre, ich konnte ihn ohrfeigen. Das Schaf. Immer recht haben. Immer recht haben. Muß er. — Ich höre deutlich die Fliege summen. Nein, es sind eben Glocken. — — Ich kann sagen, was ich will: er bleibt bei seinen Glocken. — Jetzt, um die Zeit Glocken! — — — So ein Schaf! — — — Aber das ist jeden Tag so. — — — Das Kamel . . . . Sie schläft. Sie träumt von einer Fliege, die hoch auf dem Kirchturme geläutet wird. Der Mann träumt von Glocken, die ihm über das Gesicht krabbeln. Ganz leise fängt die Fliege wieder an, zu summen. Es klingt wie fernes Glockenläuten.
- 78. I ICH BIN, ICH WAR CH bin eine Blume. Ich blühe auf der Heide. Ich bin eine Blume und blühe auf der Heide. Da kommt eine Kuh und frißt mich ab. Nun bin ich eine Blume gewesen. Nun bin ich keine Blume mehr. Wie bin ich traurig! — — — — — Ich bin eine Kuh und grase. Niemand merkt mir an, daß ich traurig bin. Grasen ist fade, Kuhsein ist fade; als Blume hatte ich es besser. Aber muß man als Kuh nicht stoisch sein und tragen, was man aufgebürdet kriegt? Geduldig sein und grasen und sich fassen, möh. — Es ist schließlich gar nicht so traurig, Kuh zu sein. Die Sonne scheint, die Wiese duftet, der Himmel bläut — und da soll ich traurig sein? Ich bin lustig. Aber es ist nicht die Blumenlustigkeit, die mich durchglüht, es ist die Lustigkeit der Kühe. Ich mache mutwillige Sprünge und möhe und muhe. Die Welt ist schön, muh. Muh, schön ist die Welt. Und ich bin doch traurig! (Ich war eine Blume!!) — — — Da kommen zwei vermummte Kerle. Die fackeln nicht lange: Einer packt mich hinterrücks und ringelt mir den Schwanz zusammen, das tut weh. Der andere schlingt mir eine Kette ums Gehörn und knufft
- 79. mich. Sein Spießgeselle peitscht auf mich ein. Ich weiß nicht, was gehauen und gestochen ist. (Einst war ich eine Blume.) Man führt mich hinweg von meiner Wiese. Ade, du Wiese, ade! — — — In der Abendstunde erreichen wir ein Gehöft. Einst war ich eine Blume, ich denke dran. Blume bin ich nimmer; bin eine armselige, wehrlose Kuh, muh. (Hilft mir der Stoizismus etwas?) Rasch tritt der Tod die Kühe an: Eine Ledermaske mit einem bösen Stirnbolzen wird mir aufgestülpt — — — ein Schlag, und ich stürze hin. Da hilft kein Muhen. Mit einem Rohrstock pfählt man mir das arme Hirn. Das macht mich traurig. Oder lustig? Ich weiß nicht, ich glaube, ich bin tot. Kuh bin ich gewesen. Blume bin ich gewesen. Ich entsinne mich wirr . . . es ist mir, ja . . . vor langer, langer Zeit — war ich ein Falter. Aber ich weiß es nicht. Daß ich Blume war, weiß ich mit Sicherheit. Ich lege meinen Huf dafür ins Feuer. Es ist vorbei. Bin weder Kuh noch Blume mehr. — — — — — Bin Wurst. Salamiwurst. Ich koste das Pfund 1.80 M.[2] Ich bin erstklassige Ware, elektrisch hergestellt. Den Stoizismus habe ich behalten. Dennoch stimmt es trübe, Wurst sein zu müssen, wenn man Blume hat sein dürfen. Ich bin mir Wurst. Ich nehme es hin. Muh. (Eigentlich dürfte ich als Wurst nimmer muhen. Ich nehme das Muh als anachronistisch zurück.) Ich habe keine Freude mehr auf der Welt. Ich bin eine kalte Wurst. Nichts tangiert mich. Wenn ich mein Leben überdenke, so muß ich frank gestehen: Wurst sein, das ist das Schlimmste nicht. Mensch sein ist weitaus schlimmer!
- 80. Doch Kuh sein, das ist schöner als Wurst sein. Das Allerallerschönste freilich war: Blume sein, Blume gewesen sein, Blume sein gedurft zu haben. Mir war’s verstattet. [2] Wer’s glaubt. Ich war Blume, ich war Blume! O Blumen, ihr seid glücklicher als Kuh und Wurst! O Blumen, nichts auf Erden ist glücklicher denn ihr. O Blumen — — * * * Vom wurstigen Standpunkt gesehen, ist es vielleicht das Vorteilhafteste, Kuh zu sein. Die Kuh ist besser dran als die Blume. Denn während eine Kuh sehr wohl Blumen fressen kann, kann eine Blume nichts fressen. Und eine Wurst kann auch nichts fressen: nicht Kuh, nicht Blume. Kuh gewesen sein gedurft zu haben ist also — mit Vorbehalt — noch erhebender als Blume gewesen sein gedurft zu haben. Ich wünsch’ euch eine gute Nacht und mir, wieder Kuh werden zu dürfen.
- 81. E MÄRCHEN S war einmal ein Frosch, der konnte sich gewaltig giften, wenn seine Frau zu ihm quakte: «I, sei doch kein Frosch!» Infolgedessen quakte die Fröschin den Satz bei jeder Gelegenheit. Der Frosch getraute sich überhaupt nichts mehr zu äußern. Sagte er etwas, so mußte er als Antwort hören: «I, sei doch kein Frosch!» Da raffte er sich auf und nahm seine Ehefrau ernstlich ins Gebet, sie solle es fürderhin gefälligst unterlassen, den albernen Satz zu quaken. «I, sei doch kein Frosch!» stereotypte die Fröschin. Es war mit ihr nichts anzufangen. Sie war in der Ehe verblödet. Da verfiel der Frosch, der keiner sein sollte, auf einen Ausweg: Er kam seiner Frau mit der Redensart zuvor und apostrophierte sie, wo immer er ihrer ansichtig wurde, mit dem Satze: «I, sei doch keine Fröschin!» Er antwortete mit nichts anderem als mit diesem Satze. Er sagte nichts als diesen Satz. Er verkehrte mit seiner Frau nur noch auf Grund und unter Zuhilfenahme dieses Satzes. Die Fröschin zeigte sich der Situation nicht gewachsen und ersäufte sich. Der Frosch war kein Frosch und holte sich eine andere heim. M o r a l : Ihr Frauen, reizet eure Männer nicht zum Äußersten und lasset sie gewähren, selbst wenn sie Frösche sind.
- 82. D AUF DER OALM, DOA GIBT’S EINEM ON DIT ZUFOLGE KOA SÜAND! IE weitverbreitete Meinung, auf der Alm gäbe es ka Sünd, hat ihren Ursprung in dem sprichwortgewordenen Liedertext: «Auf der Alm, da gibt’s ka Sünd». Selbstverständlich gibt es auf der Alm a Sünd. Das wäre ja n o c h schöner, wenn es auf der Alm ka Sünd geben täte! Von ka Sünd kann gar keine Rede nicht sein. A Sünd gibt’s überall — namentlich auf der Alm. Ich möchte sogar so weit gehen, zu behaupten: Wenn es überhaupt a Sünd gibt, so vor allem auf der Alm. . . . . . . . . . . Plötzlich erschallt draußen unter meinem Fenster das Gerassel und Gebimmel der Feuerwehr. Ich armer, schwacher Mensch unterbreche mein Schreiben und stehe eilends auf, um nachzusehen, wo es brennt. . . . . . . . . . . Es war weiter nichts. Ein Pferd ist gestützt. Ich kann also in meinem Schreiben fortfahren. Aber ich habe, offen gestanden, nicht mehr die rechte Lust dazu und stecke es auf. Ein ander Mal. Der Zensor würde die Geschichte ohnehin gestrichen haben; denn es geht toll zu auf der Alm. I c h h a b e B e w e i s e .
- 83. P PETERLE Ein Märchen ETERLE war ein gutes Kind und machte dennoch seinen Eltern großen Kummer. Wie ist das möglich? Es lag an Peterle. Peterle hätte nicht soviel träumen sollen, bei Nacht nicht und bei hellerlichtem Tag nicht. Peterle träumte, wo sie ging und stand; wo sie lag und saß. Sie träumte immerfort. Nichts war mit ihr anzufangen, kein vernünftiges Wort mit ihr zu reden. Sie spielte nicht die Spiele ihresgleichen; sie spielte nicht mit anderen und nicht für sich allein — sie puppelte nicht einmal! Nein, von Puppen mochte sie gar nichts wissen. Und was das Tollste ist: Peterle wollte durchaus ein Junge sein, obwohl sie doch ein Fräulein war. Sie behauptete, sie sei ein Junge namens Peterle, und damit holla! Sie und ein Mädchen — haha! «Ich bin ein Junge» verkündete sie jedem, der es wissen wollte, und beharrte eigensinnig auf diesem ihrem Vorurteil. Peterle hatte ihre lustigen Seiten. Nicht nur die, daß sie ein Junge sein wollte, sondern vor allem ihre Person, ihre «Erscheinung», ihr «Äußeres». Peterle war winzig klein, aber dafür dick wie ein Moppel. Sie hatte eine kurze, umgestülpte Nase, zwei wasserblaue Guckaugen und einen verschmitzten Mund. Aber das Putzigste an ihr war die Frisur: sie trug die spärlichen, bindfadendünnen Zöpfchen in zwei Schnecken prätentiös über die Ohren geringelt! Und die Zöpfe waren strohgelb. Und doch war sie den Eltern ein Persönchen — Gegenstand kann man wohl nicht sagen — argen Kummers.
- 84. Während andere Eltern prahlten und Stolzes voll die Taten, Antworten und sonstigen Äußerungen ihrer «aufgeweckten» Kinder zum besten gaben, empfanden Peterles Eltern schmerzliche Beschämung, wenn sie von ihrem Mädelchen nichts aussagen konnten als: «Sie träumt.» Peterle tat nämlich nichts als Träumen. Stundenlang saß sie hinterm Ofen oder auf dem Boden und träumte für sich hin. Wovon sie träumte, das erfuhr kein Mensch; denn sie teilte sich nicht mit, sondern behielt alles fein im Herzen. Aber sie war nun schon fünf Jahre alt und sollte über ein dreiviertel Jahr bereits zur Schule. Noch hatte sie große Ferien. Waren die erst einmal verstrichen, diese sechsjährigen großen Ferien, dann stand es bös. Ach, es würden trübe Zeiten kommen für Peterle; denn war sie erst schulpflichtig, mußte die Träumerei ein Ende nehmen. Die Eltern wußten sich keinen Rat und hätten ihr Kind am liebsten der Schule ferngehalten. Da erschien eines Tages — und zwar an jenem, der jenem, an welchem sie ihr fünftes Lebensjahr vollendete, vorausging — dem Peterle eine Fee. Keine großartige, sondern eine ganz gewöhnliche Fee, wie sie täglich dutzendweise den braven Kindern erscheinen. Diese Fee stellte dem Peterle einen Wunsch frei. Sie dürfe sich zu ihrem morgigen Geburtstage etwas wünschen — gleichviel was —, der Wunsch werde in Erfüllung gehen. Peterle schwankte keinen Augenblick, obwohl sich tausend Wünsche auf ihre niedliche Zunge drängen wollten. Sie wünschte sich das Schönste, das sie sich je hatte ersinnen können: Schnee. — Sie wünschte sich Schnee. — Sie wünschte, daß zu ihrem Geburtstage Schnee fiele. Die Fee runzelte die Stirn, aber da sie sich keine Blöße geben wollte, sprach sie: «Es wird geschehen; was du wünschest. An deinem Wiegenfeste soll es schneen.» Und verschwand, nicht ohne einen merklich holden Duft zu hinterlassen. Klein-Peterle hüpfte nicht und tanzte nicht vor Freuden, sondern träumte weiter in sich hinein — wenn auch in einer mäßig
- 85. aufgeregten Erwartung und Neugier. Sie träumte dem Geburtstage entgegen. Die Fee setzte schleunigst alle Hebel in Bewegung; denn es war kein Kleines, des Peterles Wunsch zu erfüllen und Schnee fallen zu lassen. Es sei eine kurze Unterbrechung verstattet: w a n n beginnt ein Geburtstag? Zweifellos in der Sekunde, womit der Geburtstag selbst anhebt, mithin nach Ablauf der zwölften Stunde des Vortages. Es hätte demzufolge unmittelbar auf den zwölften, mitternächtigen Glockenschlag desselben Tages, an dem die Fee bei Peterle vorsprach, zu schneen einsetzen müssen. Indes sind Feen und Kinder nicht so spitzfindig wie die Herren Juristen, die gewißlich zunächst untersucht haben würden, ob die Äußerung des Wunsches jenes Kindes namens Peterle (unvorbestraft, besondere Merkmale: prätentiöse Schnecken) die Bedingung in sich geschlossen habe, daß es den g e s c h l a g e n e n G e b u r t s t a g oder nur ü b e r h a u p t am Geburtstage schneen solle usw., — und daher zerbrach sich die Fee ihren anmutig geformten Kopf nicht über Dinge, die das Kopfzerbrechen nicht verlohnen, sintemal ihr aus der eigenen Jugend wohl bewußt war, daß für jegliches Kind der Geburtstag dann anfängt, wenn es erwacht und sich der Tatsache, daß heut’ Geburtstag ist, bewußt wird. Peterle erwachte erst gegen neun Uhr. Ihr erster Blick fiel durch das Fenster auf die Straße hinaus. Peterle jubilierte: Schnee! Es schneete wirklich! Und zwar in glitzrigen, silbrigen Flöckchen, in zierlichen. Peterle freute sich unbändig. Nicht, weil es schneete; auch nicht, weil die Fee den Wunsch erfüllt hatte, sondern, weil sie — Peterle — den Schnee (indirekt) s e l b s t «gemacht» hatte. Es war i h r Schnee, der da draußen fiel. Sie ließ zu ihrem Geburtstage Schnee fallen. Schnee — zu ihrem Geburtstage! Ihr meint, das sei nichts Besonderes?
- 86. Oho, da muß ich sehr bitten: das ist etwas ganz besonders Besonderes! Peterle ist nämlich am elften Juni zur Welt gekommen. Nun stellt Euch vor: an einem elften Juni schneete es! War das nicht Grund genug für Peterle, sich des Schnees zu freuen und den ganzen Geburtstag am Fenster zu kauern und in den Schnee zu gucken? Ich denke doch. Peterle saß denn auch am elften Juni unerschütterlich am Fenster und war glücklich über den vielen, vielen Schnee, der da vom Himmel heruntergeschüttet wurde. — — Es ist nichts mehr von Peterle zu erzählen. Sie hat ihren Schnee gehabt und weiter geträumt, bis sie zur Schule mußte. Und der Rohrstock des Lehrers erwies sich — bezüglich der Träumereien — als ein besserer Pädagog als die verhätschelnde Liebe der Eltern. Es wäre vielleicht dem oder jenem Leser angenehm gewesen, wenn sich herausgestellt hätte, daß Klein-Peterle Fieber gehabt hätte und an ihrem Geburtstage (nach Erledigung der «Schnee-Vision») ein Englein geworden sei. Sozusagen: der «tragische» Tod eines Kindes. Oh nein! Peterle hat kein Fieber gehabt — und der Schnee war wirklicher, e c h t e r Schnee. Meine Eltern wohnten damals in derselben Straße wie Peterles Eltern, und ich bin Zeuge — ich erinnere mich noch deutlich —, daß es im Jahre 18 . ., am elften Juni den lieben, langen Tag über ununterbrochen geschneet hat. Allerdings nur in u n s e r e r Straße und sonst nirgends. Das war damals ein allgemeines Verwundern und Kopfschütteln in Klotzsche — in Klotzsche hat sich der Schneefall begeben! —, und meine Eltern und wir alle haben nichts damit anzufangen gewußt, bis mir vierzehn Jahre später Peterle selbst von ihrem Geburtstagswunsche und der Fee berichtet hat. Peterle ist nämlich meine Frau geworden. Aber eine Fee ist ihr nicht wieder erschienen. Ich glaube, daran bin i c h schuld.
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