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Library of Congress Cataloging-in-Publication Data
Names: Awrejcewicz, J. (Jan)
Title: Deterministic chaos in one dimensional continuous systems /
Jan Awrejcewicz [and three others].
Description: New Jersey : World Scientific, 2015. | Series: World Scientific
series on nonlinear science. Series A ; vol. 90 | Includes bibliographical
references and index.
Identifiers: LCCN 2015035631| ISBN 9789814719698 (hardcover : alk. paper) |
ISBN 9814719692 (hardcover : alk. paper)
Subjects: LCSH: Engineering mathematics. | Buildings--Vibration. | Strength
of materials--Mathematics. | Building materials.
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Preface vii
process includes the transition from governing PDEs to a set of finite
(relatively large) number of ODEs, or finally the reduction is aimed
at decreasing of an order of the governing input PDEs. Therefore, in
the case of beams and rods two coordinates can be removed while
bending requires the introduction of numerous hypotheses. The men-
tioned approaches have been initiated by Bernoulli, Euler, Rayleigh,
Timoshenko, Sheremetev, Pelekh, Ambartsumian, Grigoliuk and oth-
ers. In the case of shell-type structures, two main classes of problems
can be studied: (i) 2D spatial problems, where the generalized beam
hypotheses can be applied; (ii) axially symmetric shells. It is observed
that the mentioned modeling yields the counterpart 1D problems
but on the other hand, approximates the shell-type constructions. In
what follows, we show how the problem regarding thickness of the
shell-type constructions can be overcome.
Recall that in classical theories of Euler and Bernoulli developed
in the XVIII Century [Euler (1757)] the following main assumptions
were introduced: (1) transversal cross sections of a beam that are flat
and perpendicular to the beam axis before deformation remain plain
and perpendicular to the beam axis during the deformation process
in time; (2) Normal stresses on the fibers located in parallel to the
rod axis are small and can be neglected, i.e. the longitudinal cross
sections induce resistance against deflection independently, without
any interaction between them; (3) inertial effects of rotation of rod
elements are omitted within the deflection process.
In Rayleigh’s theory proposed in 1873 [Strutt (1899)], inertial
effects introduced by the beam element rotation while a rod under-
goes simulteneous bending have been taken into account, which
implies a reconstruction of the form of the beam kinetic energy.
In addition, in the theory proposed by Timoshenko in 1921 [Tim-
oshenko (1921a)] it is assumed that the transversal cross sections
remain flat but not perpendicular to the beam axis. The latter
hypothesis implies an occurrence of additional terms in the formula
governing the rod potential energy.
In the published works of B.L. Pelekh and M.P. Sheremetev in
1964 [Pelekh (1973, 1978); Pelekh and Teters (1968); Sheremetev
and Pelekh (1964)] the following generalizations of the Timoshenko
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viii Deterministic Chaos in One-Dimensional Continuous Systems
theory have been proposed: (i) a transversal cross section being per-
pendicular to the beam axis before deformation is not only perpen-
dicular to the beam axis after the rod deformation process but also
curved. This hypothesis implies the occurrence of a principally novel
set of PDEs governing rod dynamics. Around 17 years after publish-
ing the results by Pelekh and Sheremetev, similar conclusions have
been published by Levinson and Reddy [Levinson (1981); Reddy,
(1984a,1984b)].
The mathematical model introduced by Pelekh–Sheremetiev has
been applied and developed in the works of V.A. Krysko and his
co-workers [Krysko (1976)] (we cite only the monograph while the
remaining papers have been published in Russian).
This brief description given in the above explains why beam-type
structures and axially symmetric shells are interpreted as 1D spatial
structures in this book.
The book is organized in the following manner — it consists
of nine chapters. Each chapter begins with a short introductory
overview of the chapter contents. Chapter 1 presents a short litera-
ture critical overview emphasizing on the recent results devoted to
analysis and control of nonlinear dynamics of beams, plates, panels
and shells, and their interplay with nonlinear problems of stability,
buckling, bifurcation and chaos. The first few chapters concentrate
on introduction to fractal dynamics (Chapter 2), definitions of chaos
and scenarios of transition from regular to chaotic dynamics as well as
an introduction of the classical Fourier analysis (FFT) versus wavelet
transform approaches (Chapter 3). Simple chaotic systems are briefly
displayed and commented in Chapter 4, and the illustrated examples
of strange chaotic attractors are then revisited in the remaining chap-
ters which are purely devoted to the study of structural members.
Modeling of dissipative systems and factors met in real-world pro-
cesses for both discrete (lumped mechanical systems) and continu-
ous structural members are briefly introduced in Chapter 5. Chaotic
dynamics of Euler–Bernoulli beams including geometric and phys-
ical nonlinearities, taking into account thermal effects and elastic-
plastic deformations, has been modeled, illustrated and discussed
in Chapter 6. Chapter 7 presents a study of the Timoshenko and
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x Deterministic Chaos in One-Dimensional Continuous Systems
(iv) All the presented results devoted to structural members include
numerous novel nonlinear phenomena and are based either
on previously/recently published or unpublished authors’
results.
(v) Applied mathematicians and physicists should be attracted
by the fascinating and rich dynamics exhibited by simple
structural members and by the solution properties of the gov-
erning 1D nonlinear PDEs. On the other hand, the engineering-
oriented researchers and graduate students will sometimes find
unexpected nonlinear phenomena, which are a waiting experi-
mental validation.
(vi) Our book is based only on numerical computations, but a
reader may find other analytical or semi-analytical approaches
based on asymptotic theories in the following supplemen-
tary books [Andrianov et al. (2004, 2014); Awrejcewicz et al.
(1998)];
(vii) The book endeavours to utilize and extend our earlier results
presented in the monographs [Awrejcewicz and Krysko (2003,
2008); Awrejcewicz et al. (2007, 2004)].
(viii) The book covers a wide variety of the studied PDEs, the way of
their validated reduction to ODEs, classical and non-classical
methods of analysis, influence of various boundary conditions
and control parameters, as well as the illustrative presentation
of the obtained results in the form of 2D and 3D figures and
vibration type charts and scales.
(ix) The book contains originally discovered, illustrated and dis-
cussed novel and/or modified classical scenarios of transition
from regular to chaotic dynamics exhibited by 1D structural
members. It shows a way to control chaotic and bifurcational
dynamics as well as gives directions to study other dynamical
systems modelled by chains of nonlinear oscillators.
(x) The book presents numerous challenges for civil/mechanical
engineers showing that more sophisticated tools are required
to understand and predict real structural responses.
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 3
Bifurcational and Chaotic Dynamics of Simple Structural Members 3
original Euler–Bernoulli theory holds for infinitesimal strains and
small rotations and can be extended to fit problems of moderately
large rotations with inclusion of the von Kármán strains.
There exists a vast literature devoted to the Euler–Bernoulli beam
dynamics and its state-of-the-art that will be omitted here. In what
follows, our review of the literature is mainly aimed at the bifurca-
tional and chaotic dynamics of the Euler–Bernoulli beams.
Abhyankar et al. [Abhyankar et al. (1993)] have applied an explicit
finite difference scheme to study partial differential equations gov-
erning the Euler–Bernoulli beam dynamics, aimed at analysis of the
beam chaotic vibrations. The space–time spectral method has been
employed to solve a simply supported modified Euler–Bernoulli non-
linear beam exhibiting lateral forced vibrations [Bar–Yoseph et al.
(1996)]. The authors used a generalized Galerkin method for the
temporal discretization as well as a discontinuous mixed Galerkin
method for the temporal discretization. The obtained solutions have
been compared visually with the reference solution of the Duffing
equation. The classical Euler–Bernoulli theory has been employed
to study microtubules, i.e. proteins organized in a network [Civalek
and Demir (2011)]. Eringen’s non-local elasticity theory has been
used to include the size effect, but only static analysis has been pre-
sented. Nonlinear responses of a clamped–clamped buckled Euler–
Bernoulli beam governed by a PDE with a cubic nonlinearity has
been studied by Barari et al. [Barari et al. (2011)]. However, the
problem has been strongly truncated and reduced to that of the
Duffing equation. The nonlinear transverse vibrations of a simply
supported Euler–Bernoulli beam under both principal parametric
and internal resonances have been analyzed in reference [Sahoo
et al. (2013)]. Periodic, quasiperiodic and chaotic dynamics matched
with resonances have been carried out using the method of multiple
scales.
Recently, attention has been paid to microbeams widely applied
in micro-electro-mechanical systems (MEMS). Avsec [Avsec (2011)]
has studied vibrations of microbeams and nanotubes, whereas Batra
et al. [Batra et al. (2008)] analyzed vibrations of narrow microbeams
initially predeformed by an electric field. Pull-in instability of
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 4
4 Deterministic Chaos in One-Dimensional Continuous Systems
structural members including microbeams has been analyzed in
references [Kacem et al. (2012); Krylov (2007); Moghimi and Ahma-
dian (2009, 2010); Vyasarayani et al. (2011)]. Chaotic electrostatic
microbeam oscillator has been studied by Towfighian et al. [Tow-
fighian et al. (2010)], and bifurcation diagrams in the plane voltage
amplitude-frequency have been constructed. A micro Euler–Bernoulli
beam, under electro-statically actuated voltage has been analyzed in
reference [Sedighi and Shirazi (2013)] taking into account the von
Kármán nonlinearity, and the pull-in instability behavior has been
investigated. However, only the first mode approximation has been
taken into account while applying the Bubnov–Galerkin procedure.
The Euler–Bernoulli simply supported beam taking into account the
von Kármán geometric nonlinearity under an extremal excitation
has been studied in reference [Dai and Sun (2014)]. They carried
out a comparison of a chaotic and a multi-dimensional model and
concluded that in order to control chaotic beam dynamics the con-
tribution of high order vibrations should be taken into account.
In spite of Euler–Bernoulli beam theory, there exists the widely
applied Timoshenko beam theory [Rosinger and Ritchie (1977); Tim-
oshenko (1921a, 1922, 1932)] developed by S. Timoshenko (1878–
1972). The Timoshenko model is particularly suitable for analysis
of a short beam, composite beams or beams under high-frequency
excitation, since it takes into account rotational inertia effects and
shear deformation. The Timoshenko beam model can be viewed as
more general with respect to the Euler–Bernoulli model because if
the shear modulus of the beam material tends to infinity (beam is
rigid in shear) and if rotational inertia phenomena can be neglected,
the Timoshenko model takes the form of the Euler–Bernoulli model.
Since the Euler–Bernoulli beam does not include a shear deforma-
tion yielding rotation, it is stiffer compared to the Timoshenko beam.
However, if a real beam has large length and small thickness, then a
difference between two models is small.
It should be noted that the effect of rotary inertia was proposed by
Rayleigh in 1894, and in literature there exists the Rayleigh beam the-
ory. Recently, the Euler–Bernoulli and Timoshenko models have been
reconsidered with respect to the two-dimensional elasticity model
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 5
Bifurcational and Chaotic Dynamics of Simple Structural Members 5
[Labuschagne et al. (2009)]. Using the example of a cantilever beam,
the authors conclude that the Timoshenko model is close to that con-
structed with the help of the 2D theory for models of practical use.
The Euler–Bernoulli beam theory corresponds to the first approxi-
mation, whereas the Timoshenko theory stands for the refined beam
theory.
Li [Li (2008)], using examples of functionally graded beams with
the rotary inertia and shear deformation included, showed that the
Euler–Bernoulli and Rayleigh beam theories can be derived from
the Timoshenko beam theory. Nonlinear flexural waves and chaotic
vibrations of a Timoshenko beam have been analyzed by Zhang and
Liu [Zhang and Liu (2010)]. The finite-deflection and the axial iner-
tia are included into the derived PDEs governing flexural waves in
the beam. Then the problem has been strongly reduced into a non-
linear ordinary equation. The latter non-autonomous equation has
been analyzed following the classical Melnikov method to define
the threshold condition of the occurrence of a transverse hetero-
clinic point, and to predict the deterministic chaos. He’s homo-
topy perturbation method has been applied to study nonlinear
free vibrations of clamped-clamped and clamped-free Timoshenko
microbeams by Moeenfard et al. [Moeenfard et al. (2011)]. Again,
the original problem of infinite dimension has been strongly trun-
cated to consideration of only one nonlinear ordinary differential
equation.
1.2 Plates
The study of vibrations of plates has a long history in mechanics
and applied mathematics and numerous books and papers have been
published dedicated to this issue. In this brief state-of-the-art review,
we aim only at recent results matching nonlinear vibrations of plates
with bifurcation and chaotic phenomena.
Both local and global bifurcations of parametrically excited
nearly square plates have been studied by Yang and Sethna
[Yang and Sethna (1991)], putting emphasis on occurrence of the
Smale horseshoe manifold. This approach based mainly on the
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6 Deterministic Chaos in One-Dimensional Continuous Systems
averaging procedure has been extended to analyze vibrations of
nearly square plates to the anti-symmetric case [Yang and Sethna
(1992)]. Shilnikov-type homoclinic orbits, bifurcations and chaotic
vibrations of thin plates subjected to parametric excitation have been
analyzed in reference [Feng and Sethna (1993)] using the so-called
global perturbation method.
The multiple scales method has been applied to study modal
interaction of nonlinear clamped plates and harmonic excitation by
Hadian and Nayfeh [Hadian and Nayfeh (1990)]. A double mode
approach has been employed by Shu et al. [Shu et al. (1999)], and
chaotic vibrations have been detected using the Melnikov method.
The global perturbation method has been applied in references
[Samoylenko and Lee (2007); Yeo and Lee (2006)] to study global
vibrations of an imperfect circular plate for the case of 1:1 inter-
nal resonance, and the chaotic orbits. A similar analytical technique
has been used by Yu and Chen [Yu and Chen (2010)] to study global
bifurcations of a simply supported rectangular metallic plate under a
transverse harmonic excitation. Both Melnikov’s approach and aver-
aging procedure have been employed by Zhang and Li [Zhang and Li
(2010)] to detect resonant chaotic vibrations of a simply supported
rectangular plate excited externally and periodically. The extended
Melnikov-type analytical approach has been applied by Zhang et al.
[Zhang et al. (2008, 2010)] to study the global bifurcations and multi-
pulse chaotic vibrations of a buckled thin plate as well as a laminated
composite piezoelectric rectangular plate. The so-called energy-phase
method has been utilized by Yao and Zhang [Yao and Zhang (2007)]
to analyze the Shilnikov-type multi-pulse heteroclinic orbits and
chaotic vibrations of a parametrically and externally excited rectan-
gular thin plate. The similar-like approach has been used to examine
the Shilnikov-type multi-pulse homoclinic orbits exhibited by a circu-
lar plate excited harmonically [Yu and Chen (2010b)]. More recently,
Yao and Zhang [Yao and Zhang (2014)] have investigated multi-pulse
global heteroclinic bifurcations and chaotic dynamics of a simply sup-
ported rectangular thin plate in the resonant case using the extended
Melnikov method.
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Bifurcational and Chaotic Dynamics of Simple Structural Members 7
The so far reported references deal with a strong truncation of the
studied PDEs governing dynamics of plates, and hence many non-
linear phenomena have been omitted during those studies. There are
also papers mainly based on numerical simulations, which are closer
to the investigation topic of this book. The fractal dimensions, the
maximum Lyapunov exponents and bifurcation diagrams have been
employed to study bifurcational and chaotic vibrations of a simply
supported thermoelastic plate with variable thickness in reference
[Yeh et al. (2003)]. The global bifurcation and chaotic dynamics of
a rectangular thin plate have been studied by Zhang [Zhang (2001)].
Touzé et al. [Touzé et al. (2011)] have studied the transition from
periodic to chaotic vibrations in free-edge, perfect and imperfect cir-
cular plates based on the von Kármán PDEs for thin plates includ-
ing geometric nonlinearity. The obtained numerical results have been
confirmed with experimental investigations. High dimensional chaos
versus the framework of wave turbulence have been illustrated and
discussed.
Recently, a challenging interest coming from an industry has
been devoted to new functionally graded materials (FGMs), being
inhomogeneous composites made of a mixture of metals and ceram-
ics. FGM plates have been widely used in various branches of the
industry since their material properties allow for a smooth and con-
tinuous change from one surface to another, and hence the prob-
lem of thermal stress concentrations can be withdrawn. The non-
linear transient thermoelastic analysis of FGM plates under pres-
sure loading and temperature fields has been carried out by Parveen
and Reddy [Parveen and Reddy (1998)]. Yang and Shen [Yang and
Shen (2002)] studied free and forced vibrations of FGM plates in
a thermal environment, and the material parameters have been
temperature-dependent. Dynamics of the pre-stressed graded layer
and two surface-mounted piezoelectric actuator layer have been
examined by Yang et al. [Yang et al. (2003)]. Nonlinear dynamics
of FGM plates embedded in thermal environment including heat
conduction and temperature-variable material properties have been
studied by Huang and Shen [Huang and Shen (2004)]. The effect
of transverse shear strains and rotary inertia have been taken into
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 8
8 Deterministic Chaos in One-Dimensional Continuous Systems
account while analyzing nonlinear vibrations of FGM plates with
initial stress by Chen [Chen (2005)]. The nonlinear dynamics of a
simply supported FGM rectangular plate under both transversal and
in-plane excitation in a thermal environment has been investigated
by Hao et al. [Hao et al. (2008)]. The problem has been reduced
to the study of a two-degree-of-freedom system with the quadratic
and cubic nonlinear terms. The cases of internal and principal para-
metric resonances have been analyzed using asymptotic perturbation
methods. Periodic, quasi-periodic and chaotic vibrations have been
studied. A 3D-exact solutions for free and forced vibration of sim-
ply supported FGM rectangular plates has been derived by Vel and
Batra [Vel and Batra (2004)]. Nonlinear vibrations of FGM plates
with randomness of the material properties and plates with geomet-
ric imperfections have been considered by Kitipornchai et al. [Kiti-
pornchai et al. (2004, 2006)].
Yang and Huang [Yang and Huang (2007)] dealt with a semi-
analytical investigation of geometrical imperfections on the nonlinear
vibrations of simply supported FGM plates. Nonlinear vibration,
bifurcation and chaos of viscoelastic cracked plates have been investi-
gated by Hu and Fu [Hu and Fu (2007)]. The von Kármán plate the-
ory and the linear isotropic constitutive theory have been employed
to a nonlinear integral-partial differential equation for a rectangular
plate with an all-over part-through crack. In particular, the effects of
the depth and the position of the crack and the viscoelastic material
parameters’ impact on the bifurcational and chaotic dynamics with
movable simply-supported boundary conditions have been analyzed.
Onozato et al. [Onozato et al. (2009)] carried out laboratory
experiments on chaotic vibrations of a rectangular plate under
in-plane elastic constraint at clamped edges. Chaotic responses
have been examined using the Fourier spectra, the Poincaré projec-
tions, the maximum Lyapunov exponents and the Karhunen–Loéve
method. Zhang et al. [Zhang et al. (2010)] investigated nonlinear
vibrations and chaos of a FGM rectangular plate in thermal environ-
ment and under parametric and external excitations. The govern-
ing PDEs are derived based on the Reddy third-order shear defor-
mation plate theory and the Hamilton’s principle. Application of
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Bifurcational and Chaotic Dynamics of Simple Structural Members 9
the Galerkin procedure yielded a three-degree-of-freedom nonlinear
systems, which has been further studied via multiple scales and clas-
sical numerical approaches.
Bifurcation and chaotic phenomena exhibited by an axially mov-
ing plate under external and parametric excitations have been exam-
ined by Liu et al. [Liu et al. (2012)]. The derived coupled PDEs of
transverse deflection and stress have been reduced to a set of ODEs,
which has been studied numerically. A relevance between the onset of
chaos with the corresponding linear instability range has been illus-
trated and discussed. The global bifurcations and multiple chaotic
vibrations of a simply supported laminated composite piezoelectric
rectangular thin plate subjected to parametric and transverse excita-
tions have been investigated by Zhang and Zhang [Zhang and Zhang
(2011)]. Both von Kármán plate theory and the first-order piston
theory are used to derive PDEs governing nonlinear dynamics of
cantilever plate in supersonic flow in reference [Xie et al. (2014)].
The Rayleigh–Ritz procedure has been employed to get ODEs being
then solved numerically. Chaotic, prechaotic and postchaotic regimes
have been identified, and the routes to chaos have been studied.
1.3 Panels
In this section, we proceed to a brief review of the published papers
contributing to bifurcational and chaotic vibrations of panels.
Maestrello et al. (1992) studied the dynamic response of an air-
craft panel forced at resonance and off-resonance by plane acous-
tic waves. Period doubling bifurcations and chaotic panel vibrations
have been detected, when the sound pressure level of the excitation
increased. Good agreement between the experimental and numerical
results has been obtained.
Yamaguchi and Nagai [Yamaguchi and Nagai (1997)] have pre-
sented numerical results devoted to chaotic vibrations of a shal-
low cylindrical shell-panel subjected to harmonic lateral excitation.
Based on the Donnell–Mushtari–Vlasov theory, the problem has
been reduced to that of a multiple-degree-of freedom system by the
Galerkin procedure. Chaotic vibrations combined with a dynamic
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10 Deterministic Chaos in One-Dimensional Continuous Systems
snap-through have been monitored with the help of Lyapunov expo-
nents and Poincaré maps. The effect of the in-plane elastic constraint
on the chaos of the shell-panel has been illustrated and discussed.
Chaotic vibrations of a panel forced by turbulent boundary layer
and sound have been studied by Maestrello [Maestrello (1999)]. The
panel response combined with period-doubling bifurcations makes a
transition to chaos when forced by the boundary layer, which has
been associated with quasi-periodic dynamics as the wave loses the
spatial homogeneity. Nonlinear bifurcations and chaotic dynamics
of fluttering panel in post-critical domain have been analyzed by
Bolotin et al. [Bolotin et al., (1998a,1998b)]. Chaotic vibrations of
a panel forced by buffeting aerodynamic loads have been demon-
strated and investigated by Epureanu et al. [Epureanu et al. (2004)].
The finite difference method has been employed to study coherent
structures of the panel nonlinear dynamics, which have been identi-
fied by a proper orthogonal decomposition. Nagai et al. [Nagai et al.
(2004)] have presented a shallow cylindrical shell-panel with a con-
centrated mass under periodic excitation. Furthermore, Nagai et al.
[Nagai et al. (2007)] carried out the detailed experimental and analyt-
ical analyses on chaotic vibrations of a shallow cylindrical shell-panel
under gravity and periodic excitation. The detected chaotic dynam-
ics has been monitored and quantified by Fourier spectra, Poincaré
maps, maximum Lyapunov exponents and Lyapunov dimension. It
has been shown that the dominant chaotic dynamics is yielded by
the responses of the sub-harmonic resonance of 1/2 order and of the
ultra-sub-harmonic resonance of 2/3 order. The similar-like approach
has been extended to study a modal interaction in chaotic vibra-
tions of a shallow double-curved shell-panel by Maruyama et al.
[Maruyama et al. (2008)]. The shell-panel with square boundary and
initial geometric imperfection has been simply supported, whereas
the in-plane displacement at the boundary has been constrained
elastically.
Similar to the case of the plates, the FGM panel has been recently
extensively studied. It has been shown that owing to the aero-thermo-
elastic interactions at high Mach numbers, skin panels may exhibit
divergence and flutter instability behavior. The combined action of
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 11
Bifurcational and Chaotic Dynamics of Simple Structural Members 11
thermal and aerodynamic loads on both static and dynamic stabil-
ities of FGM panels have been analyzed by Sohn and Kim [Sohn
and Kim (2008)]. Ibrahim et al. [Ibrahim et al. (2008)] investigated
the nonlinear flutter and thermal buckling of a FGM panel using
FEM (Finite Element Method). The aero-thermo-elastic post-critical
and vibration characteristics of temperature-dependent FGM panels
in a supersonic flow have been studied by Hosseini and Fazelzadeh
[Hosseini and Fazelzadeh (2010)]. The von Kármán theory has been
employed, whereas the material properties have been considered as
temperature-dependent and graded in the thickness direction. PDEs
have been converted to ODEs and then classical numerical tools have
been implemented to study chaotic vibrations.
Hosseini et al. [Hosseini et al. (2011)] investigated chaotic
and bifurcation dynamics of FGM curved panels subjected to
aero-thermal loads. The panel has been infinitely long and sim-
ply supported, and the material properties have been taken as
temperature-dependent and varying through the thickness direction.
The governing PDE has been truncated to a set of nonlinear ODEs
solved numerically. Regular and chaotic vibrations regimes have been
detected and monitored via Poincaré maps, time histories, frequency
spectra and Lyapunov exponents. The effect of the geometrically
imperfect curved skin panel parameters on the flutter behavior has
been investigated by Abbas et al. [Abbas et al. (2011)]. The thermal
degradation and Kelvin’s model of structural damping have been
included into the analysis.
Aginsky and Gottlieb [Aginsky and Gottlieb (2012)] studied a
nonlinear bifurcation structure of panels under periodic acoustic
fluid-structure interaction. In particular, an intricate bifurcation
structure near the fifth-mode panel resonance including coexisting
symmetric and asymmetric periodic solutions has been detected. In
addition, the emergence of a non-stationary spatio-temporal chaotic
solution has been found.
Li et al. [Li et al. (2012)] addressed the problem of the aero-elastic
stability and bifurcation structure of subsonic nonlinear thin pan-
els subjected to external excitation. The von Kármán large deflec-
tion model and Kelvin’s structural damping have been utilized while
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 12
12 Deterministic Chaos in One-Dimensional Continuous Systems
deriving the governing equation of the simply supported 2D-panel.
The problem has been reduced to that of finite number of nonlinear
ODEs, which then have been solved numerically using the fourth
order Runge–Kutta method. It has been shown that: (i) the panel
lost its stability by divergence; (ii) periodic and chaotic zones appear
alternately; (iii) a route to chaos is realized via period-doubling bifur-
cation. The similar-like approach has been extended to study bifurca-
tion structure and scaling properties of a subsonic periodically driven
panel with geometric nonlinearity by Li et al. [Li et al. (2015)]. In
particular, the scaling properties of the bifurcation structure are dis-
cussed in terms of discrete mapping and based on linear approxima-
tion. However, only one-mode reduction has been employed during
the studies.
1.4 Shells
In the case of shells chaotic and bifurcational dynamics, the num-
ber of reports is rather limited. Period-doubling bifurcations of an
infinitely long cylindrical shell under the condition of internal res-
onance has been studied by Nayfeh and Raouf [Nayfeh and Raouf
(1987)]. Chaotic vibrations of non-shallow arches loaded at their
crown by a vertical harmonic force have been investigated by Thom-
sen [Thomsen (1992)]. The quasi-periodic break-up, intermittency
and long transient behavior have been detected as routes to chaotic
dynamics. Chaotic energy pumping through auto-parametric reso-
nance in cylindrical shell has been investigated by Popov et al. [Popov
et al. (2001)]. Transient and steady-state instabilities matched with
chaotic vibrations exhibited by pressure-loaded shallow spherical
shells have been analyzed in reference [Soliman and Goncalves
(2003)]. Amabili [Amabili (2005)] studied chaotic vibrations of dou-
bly curved shallow shells. Nagai and Yamaguchi [Nagai and Yam-
aguchi (1995)] analyzed chaotic vibrations of a shallow cylindrical
shell with rectangular boundary under cyclic excitation.
Sheng and Wang [Sheng and Wang (2011)] have investigated a
nonlinear response of FGM cylindrical shells under both mechan-
ical and thermal loads within the von Kármán nonlinear theory.
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![February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 13
Bifurcational and Chaotic Dynamics of Simple Structural Members 13
The influence of temperature change, fraction exponent of FGM
and geometry parameters has been addressed. Krasnopolskaya et al.
[Krasnopolskaya et al. (2013)] introduced two new mathemati-
cal models of cross-wave generation in fluid free surface between
two cylindrical shells. Two eigenmodes approximations have been
employed, and periodic, quasi-periodic and chaotic regimes have been
detected and illustrated. Alijani and Amabili [Alijani and Amabili
(2012)] introduced sub-harmonic, quasi-periodic and chaotic dynam-
ics of FGM doubly curved shells under concentrated harmonic load.
Gear’s backward differentiation formula has been applied to get bifur-
cation diagrams, Poincaré maps and time histories, as well as the
Lyapunov spectrum has been computed.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 14
Chapter 2
Introduction to Fractal Dynamics
In this chapter, in order to get a deeper understanding of nonlinear
dynamical phenomena covered by this book, the fundamental con-
cepts of one-dimensional (1D) maps and fractal sets are briefly
reviewed and illustrated. First notions of Cantor’s set, Cantor’s dust
and Koch’s snowflake are presented. Then 1D maps are considered
putting emphasis on their regular and chaotic dynamics, the cobweb
diagrams and the period doubling bifurcation routes to chaos includ-
ing estimation of the Feigenbaum constant are also briefly revisited.
The Sharkovsky theorem is addressed with presentation of its advan-
tages and limitations. Next, the Julia, Fabout and Mandebrot sets
are shortly described and illustrated.
2.1 Cantor Set and Cantor Dust
The term Cantor dust was introduced by German mathematician
Georg Cantor in 1883 [Cantor (1883)], though it was discovered ear-
lier by H. J. Smith [Smith (1874)].
This set has been considered as a set of zero Lebesque measure.
Fractal properties of the Cantor set have important meaning since
many of the known functions are similar families of this set. The
Cantor ternary set is constructed by repeatedly removing the open
middle part of the unit interval [0, 1], assuming that it is divided into
three parts.
We begin (the first step) by deleting the open middle third
(1/3, 2/3) leaving two line segments [0, 1/3] ∪ [2/3, 1]. Proceeding in
14
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 15
Introduction to Fractal Dynamics 15
Fig. 2.1 Construction of the Cantor set.
a similar way with all the remaining parts, we obtain (see Fig. 2.1)
C0 = [0, 1] ,
C1 =
0, 1
/3
∪
2
/3, 1
,
C2 =
0, 1
/9
∪
2
/9, 1
/3
∪
2
/3, 7
/9
∪
8
/9, 1
,
Cn =
Cn−1
3
∪
2
3
+
cn−1
3
.
(2.1)
The obtained remaining segments/points can be presented via the
following general formula
C =
∞
m=1
3m−1
k=0
0,
3k + 1
3m
∪
3k + 2
3m
, 1
, (2.2)
which has been proved by Soltanifar [Soltanifar (2006)].
Limiting set C, which contains all of the non-deleted sets Cn,
n = 0, 1, 2, . . . , is called the classical Cantor dust. In what follows,
we discuss a few important properties of the Cantor set.
1. Cantor set is a self-similar fractal of dimension
d = log(2)
/log(3) ≈ 0.6309, (2.3)
where Nrd = 1 is satisfied for N = 2 and r = 1/3; N is the num-
ber of equal parts; 1/2 — decrease of 1/2 times; d — fractal (non-
integer) dimension or dimension similarity d = log N
log 1/2
. Logarithm
can be taken with any basis, for instance, using e ≈ 2, 7183 . . . .
The Hausdorf dimension of the Cantor set is ln 2/ ln 3 = log3 2,
whereas its Lebesque measure is zero.
2. Cantor set does not include intervals of integer length.
3. Sum of removed intervals while constructing the set C is equal
to 1.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 16
16 Deterministic Chaos in One-Dimensional Continuous Systems
This statement can be proved in the following way. Length of the
first interval is 1/3. In order to get C2, we remove two intervals
of length 1/32 . In the next step, we remove 22 intervals, each of
length 1/33 , and so on. Therefore, the sum of removed intervals is
S =
1
3
+
2
32
+
22
33
+ . . . +
2n−1
3n
+ . . . , (2.4)
or equivalently
S =
1
/3 · 1 + 2
/3 +
2
/3
2
+
2
/3
3
+ . . . . (2.5)
Let us recall the geometric series
1
1 − x
= 1 + x + x2
+ x3
+ . . . |x| 1, (2.6)
and hence we get
S =
1
/3 ·
1
1 − 2/3
= 1. (2.7)
4. Comparison of the Cantor set with the interval [0, 1] shows that
powers of these two sets are equal, i.e. power of the Cantor set
C is equal to the power of continuum [0, 1].
5. Classical Cantor dust presents the example of a compact fully
discontinuous set.
The Cantor set cannot have any interval of non-zero length. Since
the subsequent subsets of the Cantor set construction do not remove
end points, the Cantor set is not empty and contains an uncountably
infinite number of points. Another observation is that all algebraic
irrational numbers are normal, whereas the Cantor set members are
either rational or transversal (they are not normal).
The Cantor set can serve as the archetype of a fractal. It exhibits
the left and right self-similarity transformations fl(x) = x/3 and
fr(x) = (2 + x)/3, which acting on the Cantor set yield fl(C) ∼
=
fr(C) ∼
= C. It means that the Cantor set is invariant with respect
to the introduced homeomorphism. The set {fl, fr} together with
the function composition creates the dyadic monoid. On the other
hand, the Cantor dust is a multi-dimensional Cantor set, i.e. it is
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 17
Introduction to Fractal Dynamics 17
constructed as a finite Cartesian product of the Cantor set with itself
by yielding a Cantor space also with zero measure.
The method of subdividing a shape into copies of itself simultane-
ously removing a few of such copies has been extended to 2D objects.
In spite of the Cantor dust, one may construct the Sierpiński triangle
discovered by W. Sierpiński in 1916 [Sierpiński (1916)] (an equilat-
eral triangle is subdivided into four equilateral triangles removing
the middle triangle) and the Menger sponge reported by K. Menger
in 1928 [Menger (1928)] (a cube is taken; each face is divided into
nine squares; the smaller cube in the middle of each face is removed
and also the smaller cube in the center of the large cube is removed
to get a void cube; the last two steps are repeated for the remaining
smaller cubes and the described iterations are repeated to infinity).
It is remarkable that each face of the Menger sponge is a Sierpiński
carpet, and its Lebesque measure is zero (uncountable set).
However, the Lebesque covering dimension of the Menger sponge
is one. It can be shown that every curve is homeomorphic to a subset
of the Menger sponge including trees and graphs, vertices and closed
loops, etc. giving the right to understand the Menger sponge as a
universal curve.
On the other hand, the Sierpiński carpet can be viewed as a
universal curve for all planar curves, i.e. projection of the Menger
universal curves onto a plane yields the Sierpiński universal curves.
The Hansdorff dimension of the Sierpiński carpet is log 8/ log 3 ∼
=
1.893, whereas the Hansdorff dimension of the Menger sponge is
log 20/ log 3 ∼
= 2.727.
2.2 Koch Snowflake
In 1904, the Swedish mathematician Helge von Koch presented a
way to construct the so-called Koch snowflake, known as the Koch
island and Koch star, as a mathematical curve being a prototype
for a fractal [Peitgen et al. (1926)]. Boundary of Koch’s snowflake is
described by a curve consisting of three similar fractals of dimension
d ≈ 1.2618. Each third part of the snowflake is constructed in an
iterative way, beginning from one side of an equilateral triangle.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 21
Introduction to Fractal Dynamics 21
Fig. 2.4 Cobweb diagram of the map f (x) = x2
for the following initial condi-
tions: (a) x0 = 1.1, (b) x0 = −0.9.
Another case of iterations is a limit-cycle. It represents a periodic
dynamics, since the orbit tends to a series of points that repeats itself
(a number of iterations needed to repeat itself is called a period of
the orbit). In our case x0 = 0 and x0 = 1 are fixed points of the map
y = x2. If x0 1, then the orbit tends to + ∞, if 0 x0 1 or
−1 x0 0, then the orbit tends to a fixed point 0. If x0 = 1, then
the orbit is [−1, 1 1 1 . . .], i.e. it is periodic, and finally if x0 −1,
then xn → ∞ and the orbit diverges. In the given case, the fixed point
0 is attractive, whereas the fixed point 1 is repelling. The functions
f (x) = x2 and f (x) = x2 −1 are particular cases of the map f (x) =
x2 + c, and they are widely used in the theory of dynamical systems
[Sharkovsky et al. (1997)]. Although f (x) = x2+c is only a quadratic
function, it is widely applied.
We consider a real case, i.e. when x and c are real numbers. We
solve equation x2 +c = x, and we get ξ = 1+
√
1−4c
2 ; η = 1−
√
1−4c
2 . We
deal with a fixed point for 1 − 4c ≥ 0 (c ≤ 1
4 ), if −ξ η ξ, and
besides, f (−ξ) = ξ. Orbits for x0 ξ and x0 −ξ tend to + ∞.
For −3/4 c 1/4, the fixed point η is attractive, i.e. |f (x)| 1
and all orbits tend to η. For c −3/4 |f (x)| 1, i.e. η becomes
a repeller. When c transits over the value −3/4, the system exhibits
a period doubling bifurcation. A second period doubling bifurcation
takes place for c = −5/4. If c −5/4, then an attractive periodic
orbit of period 4 appears. Decreasing c, attractive orbits of length 8,
16 and 32 are observed, i.e. after period doubling bifurcations chaos
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 22
22 Deterministic Chaos in One-Dimensional Continuous Systems
occurs. For c = −2, there exist periodic orbits of f (x) with periods
2, 3, 4, . . . .
Feigenbaum studied intervals between period doubling bifurca-
tion of the square function f (x) = cx (1 − x), also known as the
logistic map [Feigenbaum (1990); Thunberg (2001)]. The obtained
diagram of orbits is similar to that of the function f (x) = x2 + c.
The fundamental meaning of the Feigenbaum analysis relies on the
detection of the mechanism’s universality of a route to chaos via
period doubling bifurcation, which is exhibited not only by the logis-
tic map f (x) = cx (1 − x), but also by the maps f (x) = x2 + c,
f (x) = c sin (πx) , f (x) = cx2 sin (πx) defined on appropriate inter-
vals. The aforementioned class includes functions f (x), defined on
the interval [0, 1] and achieving a maximum at point xm ∈ (0, 1)
for f(xm) = 0, whereas f (x) on [0, xm] and [xm, 1], as well
as its Schwartz derivative Sf (x) = f(x)
f(x) − 3
2
f(x)
f(x)
2
is negative
∀ x ∈ [0, 1].
We denote by c0, c1, c2, . . . , bifurcation points on the orbits’
diagram (Fig. 2.5), i.e. those points, where iterations f (x) = x2 + c
change the attracting orbit of period 2n into attracting orbit 2n+1.
One may check that c∞ = lim
n→∞
= −1.401155 . . . , i.e. we deal with
Fig. 2.5 Diagram exhibiting period doubling bifurcation of the map f (x) =
x2
+ c.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 23
Introduction to Fractal Dynamics 23
the Feigenbaum type process. For c = −1.7548777 . . . period-3 orbit
exists, which corresponds to the more lighted part of the diagram.
Further, we have d = lim
n→∞
cn−cn−1
cn+1−cn
= 4.669162 . . . , which defines the
Feigenbaum constant.
It should be noted that it is, in general, difficult to find analyti-
cally bifurcation points cn for any given function, for instance x2 +c,
and hence to define the Feigenbaum constant. However, there exists
another approach. For each pair of bifurcation points cn and cn+1
there exists a point c∗
n, which corresponds to an orbit with period
2n. For this value of c, the critical point x0 of the function fc satisfies
the equation f
(n)
c = x0. Then, the Feigenbaum constant d can be
found through the following equation
d = lim
n→∞
c∗
n − c∗
n−1
c∗
n+1 − c∗
n
, (2.15)
where c∗
n is found numerically with the help of the Newton method.
The stretching and folding exhibited by the logistic map produces
sequences of iterates, which finally yield an exponential divergence
validated by the Lyapunov exponent computation. The logistic map
correlation dimension is 0.5, and its Hausdorff dimension is 0.538.
The logistic map as well as other chaotic maps have found applica-
tions in developing image encryption schemes [Pareek et al. (2006);
Wong et al. (2008)].
2.4 Sharkovsky’s Theorem
Diagram of orbits shown in Fig. 2.5 presents attracting periodic
orbits for fc(x). For 1/4 c −3/4, there is an attracting orbit
of period 1. For −3/4 c −9/4, we deal with the attracting
orbit of period 2, which bifurcates into attracting period-4 orbit, for
c = −5/4.
On some intervals, the diagram exhibits windows. For instance,
for c ≈ −1, 75 we have a white zone with period-3 orbits. A nat-
ural question appears: Do other periodic orbits exist? Those orbits
should be repellers, since in the diagram we observe only attracting
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 24
24 Deterministic Chaos in One-Dimensional Continuous Systems
orbits. It is shown that the occurrence of a period-3 orbit implies the
occurrence of orbits with periods n = 1, 2, 3, . . . .
In 1964, A. N. Sharkovsky published a general theorem, regarding
a map onto itself valid for the real functions, which had an important
impact on the development of the theory of nonlinear dynamical
systems [Sharkovsky (1964)].
A brief description of Sharkovsky’s theorem follows. Let I be finite
or infinite interval in R. We assume the map f : I → I is continuous.
We define the number x as a periodic point of period m if fm(x) = x
and fk(x) = x for all 0 k m. Let us construct the following
ordering of the positive integers
3, 5, 7, 9, . . .
2 · 3, 2 · 5, 2 · 7, 2 · 9, . . .
22
· 3 , 22
· 5 , 22
· 7 , 22
· 9 , . . .
23
· 3 , 23
· 5 , 23
· 7 , 23
· 9 , . . .
. . . . . . . . .
. . . , 2n
, . . . , 23
, 22
, 21
, 1.
(2.16)
We start with the construction of the odd numbers in the increas-
ing order, followed by 2 times the odds, 22 times the odds, 23 times
the odds, and so on, whereas at the end we take powers of two in
the decreasing order. Then, if f has a periodic point of least period
m and m precedes in the ordering (2.16), f also has a periodic point
of least period n. Assuming now that f has only a finite number of
periodic points, then those points have periods numbers of which are
powers of 2. If there exists a periodic point of period-3, then there
exist also periodic points of all other periods. It should be emphasized
that the Sharkovsky theorem does not concern with the stability of
periodic orbits but rather their existence. In other words, if one takes
the logistic map, for a wide range of the control parameter only one
period-3 orbit is detected, which is an attractor. The remaining orbits
of all periods are not visible, since they are unstable and cannot be
detected through the cobweb construction described so far.
The Sharkovsky Theorem is valid only for a real function defined
on a real interval. If, for instance, function f is given by a rotation
of each point lying on a circle about the angle 2/n, then the orbits
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 25
Introduction to Fractal Dynamics 25
of each point have the same period n. In this case, there is lack of
any additional periods, and Sharkovsky’s theorem is not applicable.
On the other hand, if there exists an orbit of the odd period larger
than one, then a number of different periods tends to infinity.
2.5 Julia Set
We denote by C a set of complex numbers a + bi = z, where a
and b are respectively real and imaginary parts of z, i.e. a = Re (z),
b = Im(z), |z| =
√
a2 + b2. Series of complex numbers {zn}∞
n=1 →
lim
n→∞
zn = ∞, which means that ∀ M 0, ∃ N 0, ∀ n N :
|zn| M, i.e. all points zn lie inside a circle of radius M for suffi-
ciently large n and it is necessary that absolute magnitudes increase
to infinity.
Let f(z) = anzn + an−1zn−1 + . . . + a1z + a0, an = 0, be a poly-
nomial of order n ≥ 2, and its coefficients an, an−1, , . . . , a1, a0
are complex numbers (in particular case, they are real). The Julia
set of a function f (known as J(f) set) is the limit of a set of
points z, approaching infinity while iterating f(z). This set honors
the French mathematician Gaston Julia [Julia (1918)] (1893–1978),
who simultaneously with Pierre Fabout [Fatou (1917)] (1878–1929)
in 1917–1919 wrote the fundamental papers devoted to the iteration
of the functions of a complex variable. The factor set of the func-
tion exhibits a property that all nearby values behave in a similar
manner under the action of repeated iterations of functions. On the
other hand, the Julia set exhibits values in which perturbations yield
drastic changes in the sequence of iterated function values.
A simple example of Julia’s set is: f(z) = z2. In what follows, we
consider the following Julia set fc (z) = z2 + c, where c is constant
in C. In Figs. 2.6 and 2.7, a few Julia sets are presented.
In order to proceed with a more rigorous statement, we consider
a complex rational function f(z) = p(z)/g(z) from the plane onto
itself, where p(z), g(z) are complex polynominals. There exists a
finite number of open sets F1, F2, . . . that are invariant owing to the
action of f(z) having two properties: (i) a set of Fi is dense in the
plane; (ii) f(z) behaves regularly on each of the sets Fi. One may
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 26
26 Deterministic Chaos in One-Dimensional Continuous Systems
Fig. 2.6 The Julia set for f (z) = z2
− 1, where z is a complex number (adapted
from [Douady (1986)]).
Fig. 2.7 The Julia set for f (z) = z2
− 0.20 + 0.75i (adapted from [Sierpiński
(1916)]).
distinguish between attracting cycles and neutral cycles. The sets Fi
are Fatou domains of f(z), and each of the domains has at least one
critical point of f(z). The complement of F(f) (Fatou set) is the
Julia set J(f). The latter one is invariant by f(z), and the iteration
is repelling in a neighborhood of z, which yields chaotic deterministic
iterations of f(z) on the Julia set. The Julia set may include a finite
number of regular points, i.e. those whose sequence of operation is
finite. There exist various equivalent descriptions of the Julia set.
It can be also proved that the Julia set and the Fatou set of f are
completely invariant under iterations of the holomorphic function f,
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 27
Introduction to Fractal Dynamics 27
i.e.
f−1
(J(f)) = f(J(f)) = J(f), f−1
(F(f)) = f(F(f)) = F(f).
(2.17)
More detailed description of Fatou and Julia sets can be found in
references [Barnsley (1988); Douady (1986); Lauwerier (1991); Pait-
gen and Richter (1986)]. There are also a few proposals for the direct
application of Julia/Fatou set theories to pure and applied sciences.
The Newton’s-secants method for finding numerical solutions to the
nonlinear equation F(z) = z2 − C = 0 has been proposed and then
extended to solve Newton’s-secants and Tchebishev’s-secants imag-
inary problems based on the Julia set theory by Tomova [Tomova
(2001)]. Another application of Julia sets to switched dynamical pro-
cesses has been presented by Lakhatakia [Lakhtakia (1991)]. The
Julia set theory has been employed to transient chaos detection in
process control systems [Russell and Alpigini (1996)]. The quaternion
Julia set has been applied to generate real-time-based symmetric
keys for cryptography in the reference [Rubesh Anand et al. (2009)].
2.6 Mandelbrot’s Set
Mandelbrot extended investigations of French mathematicians Fatou
and Julia by studying the parameter space of quadratic polynomials
in 1980 [Mandelbrot (1980)]. The Mandelbrot set is governed by the
quadratic recurrence equation
zn+1 = z2
n + z0, (2.18)
where the orbit zn does not tend to infinity (remains bounded).
Namely, a complex number z0 = c belongs to the Mandelbrot set
M, when the absolute value of zn remains bounded. For example,
for z0 = 1 we get the sequence {0, −1, 0, −1, 0, . . . } being bounded,
whereas for z0 = 1 we obtain the sequence {0, 1, 2, 5, 26, . . . }, which
is unbounded. The Mandelbrot set boundary also exhibits a smaller
version of the main shape (see Fig. 2.8), and hence the fractal prop-
erty of self-similarity is applied to the entire set. A zone of period-3
Mandelbrot’s set is shown in Fig. 2.9.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 28
28 Deterministic Chaos in One-Dimensional Continuous Systems
Fig. 2.8 Mandelbrot’s set fc(z) = z2
+c (a) and the window (b) of Mandelbrot’s
set M around point c = −1.75+0i (points belonging to set M are colored black);
adapted from [Mandelbrot (1980)].
More precisely, the Mandelbrot set can be defined in the following
way. Set M for fc(z) = z2+c is defined as {c} ∈ C, for which the orbit
of point 0 is bounded, i.e.: (i) M = {c ∈ C : {f
(n)
c (0)}∞
n=0 bounded}
or equivalently (ii) M = {c ∈ C : f
(n)
c (0) → ∞ for n → ∞}.
Equivalence of definition 1 and definition 2 is yielded by the
observation that lim
z→∞
z2+c
z = ∞, i.e. ∃ R 0 we have |z| R →
|fc(c)| 2|z|. If ∀n0 ∃|f
(n0)
c (0)| R, ∀n n0 : |f
(n0)
c (0)| 2n−n0 R,
i.e. f
(n)
c (0) → ∞. Point zero is the only one point where the deriva-
tive is equal to zero. Problems regarding boundaries of the orbits
should be justified using the following theorem.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 29
Introduction to Fractal Dynamics 29
Fig. 2.9 Area of period-3 of Mandelbrot’s set (it should be considered simulta-
neously with Figs. 2.8 and 2.9); adapted from [Mandelbrot (1980)].
If |c| 2 and |z| ≥ c, then on orbit z → ∞. In particular, this
implies that points c /
∈ M. Point c = −2 is the only one point of the
circle |c| = 2, which belongs to Mandelbrot’s set.
In Fig. 2.10, certain parts of the Mandelbrot’s set corresponding
to the existence of the attractive periodic orbits of different orbits are
presented. The presented diagram of the orbits shows what happens
on the real axis of the Mandelbrot set. Each bifurcation corresponds
to a new frame, which intersects the abscissa axis, and the period
corresponds to a number of branches of the orbital diagram. A value
of c, for which periodic attracting points of period-2 exist in the
Julia set lie inside the circle |c + 1| = 1/4. It has been impossible so
far to find an analytical dependence between a period and a frame
for periods larger than two.
The Hausdorff dimension of the Mandelbrot set boundary is
equal to two [Shishikura (1998)]. There is a strong correspondence
between the geometry of the Mandelbrot set at a given point and
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 30
30 Deterministic Chaos in One-Dimensional Continuous Systems
Fig. 2.10 Periods of frames (Mandelbrot’s set); adapted from [Mandelbrot
(1980)].
the associated Julia set. Properties and various peculiarities of the
Mandelbrot set are described and illustrated in numerous publica-
tions including [Lei (1990, 2000)]. The real-world applications of
fractal and Mandelbrot sets can be found in physics, mechanics, eco-
nomics, biomechanics and financial mathematics.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 32
32 Deterministic Chaos in One-Dimensional Continuous Systems
Therefore, Ovidius understood chaos as the infinite space existing
before the appearance of all other matters. Romes understood chaos
as the initial disordered matter, where a Creator introduced order
and harmony, i.e. the same description for chaos holds for Romes
and Greeks. Nothing has been changed up till now: G. Schuster,
for instance, describes chaos as a state of disorder and irregularity
[Schuster (1998)].
Nowadays, more attention is paid to the investigation of the struc-
tural stochasticity (chaos is born from order), which can be exhib-
ited by the deterministic dynamics of structural members in their
pre-critical states. Stochasticity is implied by a complex intrinsic
system dynamics and is not given as a result of the noise or fluc-
tuation input. It can be treated, in some cases, as the occurrence
of a turbulent behavior. It is well known that there exists relatively
large amount of the scenarios of transitions into turbulent behavior
being (in majority of cases) governed by the Navier–Stokes equa-
tions, describing dynamics of the uncompressed fluid, which have
the following form
∂u
∂ t
+ (u · ∇) u − ν∇2
u = −
1
ρ
∇p + f, (3.1)
div u = 0, (3.2)
u = 0 on D, (3.3)
where u = u(xi, t), i = 1, 2, 3, p is pressure, D is boundary of the
space containing a fluid, ρ is fluid density, f is external force, ν is
kinematic viscosity. Energy dissipation is governed by the term ν∇2u.
Equation (3.1) presents a 3D partial differential equation regarding
u (velocities) with respect to a fixed system of coordinates (Euler’s
approach), whereas Eq. (3.2) is responsible for the condition of the
uncompressed fluid behavior, and Eq. (3.3) governs the boundary
conditions.
Note that there is lack of proof of the turbulent solution exis-
tence to Eqs. (3.1)–(3.3), when time goes to infinity. However, there
is a proof of the turbulent solutions existence for 2D equations.
On the other hand, physical properties described by the Navier–
Stokes equations are relatively well investigated and understood.
The first work in this field was published by Reynolds [Reynolds
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 33
Introduction to Chaos and Wavelets 33
(1883)]. He introduced the following non-dimensional parameters:
Reynold’s number R = UL
ν , u/U — non-dimensional velocities,
xi/L — non-dimensional coordinates, t = L/U — non-dimensional
time, p = p/ρU2 — non-dimensional pressure, and he considered the
following partial differential equation
∂u
∂t
+ (u · ∇) u −
1
R
∇2
u = −∇p. (3.4)
Reynolds showed that an increase in R may change the fluid
motion qualitatively from a regular (laminar) to disordered chaotic
(turbulent) flow.
In hydrodynamics, the word “turbulence” is used as a descrip-
tion of the spatio-temporal chaos. It implies that chaos in a fluid is
exhibited in all scales in space and time. However, the mathemati-
cal description of this state belongs to one of the most tedious and
complicated problems to be solved. Up to now, it is not clear how a
stochastic attractor of the turbulent flow should be constructed. On
the other hand, more important results have been recently obtained
for the investigation of simple dynamical systems yielded, for exam-
ple, by truncation of the original continuous system, and being gov-
erned by ordinary differential equations as well as maps (difference
equations). However, truncated systems include only timing chaos.
It allows, in the first approximation, to analyze a turbulence birth,
i.e. the case when the velocity field began to fluctuate in time in a
disordered manner.
One of the aims of the book is to illustrate and discuss various
scenarios of transition of the space structural members as beams into
spatio-temporal chaos for different mathematical models taken into
account. In what follows, we address a few examples of transitions
from regular to chaotic dynamics exhibited by simple dynamical sys-
tems, in order to introduce the reader briefly to the topics covered
by the book contents.
3.1.2 On chaos definitions
In the beginning, we introduce a few definitions of chaos exhibited
by dynamical systems.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 34
34 Deterministic Chaos in One-Dimensional Continuous Systems
(i) Devaney’s definition [Devaney (1989); Eckmann and Ruelle
(1985); Stewart (1989)].
The Devaney three features of chaos are formulated for a continuous
map f : X → X, where X is a metric space. First (1) Devaney’s
condition states that for all non-empty open subsets U and V of X
there exists k (natural number) such that fk(U) ∩ V is non-empty,
which defines f as transitive. Second (2) Devaney’s condition exhibits
the so-called “element of regularity”, where there are periodic points
of f, forming a dense subset of X. Third (3) Devaney’s condition
states, that f satisfies the property of sensitive dependence on initial
conditions, which means that if there is a positive real number δ such
that for every x in X and every neighborhood N of X there exists
a point y in N and n (nonnegative integer) such that the iterates
fn(x), fn(y) are more than distance δ apart.
Hence, the fundamental characteristics of chaos require three con-
ditions: essential dependence on the initial conditions; mixing caused
by transitivity; regularity condition implied by density of periodic
points. There is a theorem saying that if f : X → X is transitive
and has dense periodic points then f has sensitive dependence on ini-
tial conditions, which has been proved in the reference [Banks et al.
(1992)].
Let us present Devaney’s definition in a more rigorous way: let
there be a metric space (x, d).
Now we explain the meaning of the already introduced three con-
ditions and the introduced metric space (X, d).
1. Let x ∈ X and U be an open set containing x. Map f depends
essentially on the initial conditions if
(∀δ 0), (∃n 0), (y ∈ U): d(f(n)
(x), f(n)
(y)) δ. (3.5)
2. Let f is transitive, then ∀(u, v) of open sets
(∃n 0) f(n)
(u)Πv = 0. (3.6)
3. Density of the periodic points means that at any arbitrary neigh-
bourhood of any point in X there exists at least one periodic
point.
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 35
Introduction to Chaos and Wavelets 35
(ii) Baker’s map ([Driebe (1999); Fox (1997); Hasegawa and Saphir
(1992)]).
A map from the unit square onto itself is called the Baker map,
which is a chaotic map. The name comes from the bakers’ operation.
Namely, kneading is a process applied by bakers in the making of the
dough. During kneading, the dough is folded in half and compressed.
The horseshoe map is topologically conjugate to the baker’s map. It
preserves the 2D Lebesgue measure, it is strong topological mixing
and it can be understood as a two-sided operator of the symbolic
dynamics of a 1D lattice.
(iii) One may apply the following simple properties of chaos, i.e.
there either exists the essential sensitivity on the initial conditions
or one of the Lyapunov exponents measuring divergence of the neigh-
borhood trajectories is positive.
3.1.3 Landau–Hopf (LH) scenario
The first scenario was proposed by Landau [Landau (1944)] in 1944,
and then by Hopf [Hopf (1948)] in 1948. When Reynold’s number R
regarding parameters characterizing flow activity achieves the critical
value R, the previously stationary flow loses its stability. For R → ∞,
the velocity of occurrence of new frequencies increases (Pn0; 1), and
the solution can be presented in the following form:
u(x, t) =
∞
n=1
Am(x)i m (ω t+δ)
, (3.7)
where ω = {ω1, ω2, . . . , ωn}; n → ∞; R → ∞. Although the
frequency ratio is irrational, the spectrum becomes a broad band
and similar to chaotic one, i.e. an infinite quasi-periodic process of
“turbulence” is observed, which has not been verified by experiments
(the Couette flow and the Rayleigh–Bernard convection).
The so far described LH scenario (Fig. 3.1) is associated with
Hopf theory of bifurcations, which we will briefly present now. Let a
motion be described by ordinary differential equations:
dx
dt
= Fp(x), x = x1, . . . , xk, (3.8)
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![February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 36
36 Deterministic Chaos in One-Dimensional Continuous Systems
Fig. 3.1 The LH scenario.
Fig. 3.2 (a) Spiral type trajectories approaching a stable critical point; (b) Spiral
type trajectories approaching a stable limit cycle.
where p is a system parameter (for example, p can be an amplitude
of excitation). Critical points of equation (3.8) are the points x = xc,
where
dxc
dt
= 0 , i.e. Fp(xc
) = 0. (3.9)
Stability of the points defined by (3.9) is estimated via the asso-
ciated linearized equations and their characteristic values λ = λ(p).
Assuming that λ lies in the left half-plane, i.e. they have negative
real parts, the investigated critical point is stable [Fig. 3.2(a)].
The Hopf bifurcation takes place under the condition that a com-
plex conjugated pair of the eigenvalues moves from the left-hand side
of the complex plane to its right-hand side. For the critical values of
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![L O N D O N:
Printed for J. Wilford, at the Three Flower-de-luces, behind
the Chapter-house, St. Pauls.
[Price One Shilling.]
1733
Reduced Leaf in original, 8.5 × 12.62 inches.
It was about this date, I suppose, that I read Bishop Butler's
Analogy; the study of which has been to so many, as it was to
me, an era in their religious opinions. Its inculcation of a
visible church, the oracle of truth and a pattern of sanctity, of
the duties of external religion, and of the historical character
of Revelation, are characteristics of this great work which
strike the reader at once; for myself, if I may attempt to
determine what I most gained from it, it lay in two points
which I shall have an opportunity of dwelling on in the
sequel: they are the underlying principles of a great portion
of my teaching.
Newman](https://image.slidesharecdn.com/5623646-250521193017-fe217c26/85/Deterministic-Chaos-In-Onedimensional-Continuous-Systems-Jan-Awrejcewicz-61-320.jpg)
![And sold by M. Cooper in Pater-noster-Row. 1751.
[Price Six-pence.]
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Deterministic Chaos In Onedimensional Continuous Systems Jan Awrejcewicz
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- 6. WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES* Volume 75: Discrete Systems with Memory R. Alonso-Sanz Volume 76: A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science (Volume IV) L. O. Chua Volume 77: Mathematical Mechanics: From Particle to Muscle E. D. Cooper Volume 78: Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations A. M. Samoilenko & O. Stanzhytskyi Volume 79: Robust Chaos and Its Applications Z. Elhadj & J. C. Sprott Volume 80: A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science (Volume V) L. O. Chua Volume 81: Chaos in Nature C. Letellier Volume 82: Development of Memristor Based Circuits H. H.-C. Iu & A. L. Fitch Volume 83: Advances in Wave Turbulence V. Shrira & S. Nazarenko Volume 84: Topology and Dynamics of Chaos: In Celebration of Robert Gilmore’s 70th Birthday C. Letellier & R. Gilmore Volume 85: A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science: (Volume VI) L. O. Chua Volume 86: Elements of Mathematical Theory of Evolutionary Equations in Banach Spaces A. M. Samoilenko & Y. V. Teplinsky Volume 87: Integral Dynamical Models: Singularities, Signals and Control D. Sidorov Volume 88: Wave Momentum and Quasi-Particles in Physical Acoustics G. A. Maugin & M. Rousseau Volume 89: Modeling Love Dynamics S. Rinaldi, F. D. Rossa, F. Dercole, A. Gragnani & P. Landi Volume 90: Deterministic Chaos in One-Dimensional Continuous Systems J. Awrejcewicz, V. A. Krysko, I. V. Papkova & A. V. Krysko * To view the complete list of the published volumes in the series, please visit: http://www.worldscientific.com/series/wssnsa Lakshmi - Deterministic Chaos.indd 1 2/11/2015 2:42:27 PM Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 7. NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO World Scientific NONLINEAR SCIENCE WORLD SCIENTIFIC SERIES ON Series Editor: Leon O. Chua Series A Vol. 90 DeterministicChaosin One-DimensionalContinuousSystems Jan Awrejcewicz Lodz University of Technology, Poland Vadim A Krysko Irina V Papkova Saratov State Technical University, Russia Anton V Krysko Saratov State Technical University, Russia Cybernetic Institute, National Research Tomsk Polytechnic University, Russia International Institute for Applied Systems Analysis 9775hc_9789814719698_tp.indd 2 22/2/16 2:38 PM Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 8. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Awrejcewicz, J. (Jan) Title: Deterministic chaos in one dimensional continuous systems / Jan Awrejcewicz [and three others]. Description: New Jersey : World Scientific, 2015. | Series: World Scientific series on nonlinear science. Series A ; vol. 90 | Includes bibliographical references and index. Identifiers: LCCN 2015035631| ISBN 9789814719698 (hardcover : alk. paper) | ISBN 9814719692 (hardcover : alk. paper) Subjects: LCSH: Engineering mathematics. | Buildings--Vibration. | Strength of materials--Mathematics. | Building materials. Classification: LCC TA342 .D48 2015 | DDC 624.1/7011857--dc23 LC record available at http://lccn.loc.gov/2015035631 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore Lakshmi - Deterministic Chaos.indd 2 2/11/2015 2:42:28 PM Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 9. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page v Preface There is no hope, at least in the coming future, to solve in full the prob- lem devoted to vibrations of 3D thermo-elastic structural elements, which are widely applied in mechanical and civil engineering (bridges, silos, oil platforms), civil and military industries (aircrafts, space launchers, missiles, tanks, satellites), biomechanics and biomedical engineering (stents, surgical devices), maritime and offshore engineer- ing (ships, boats, sea platforms, tubes and pipelines), cars and motor- cycle factories, as well as MEMS engineering. In general, the proposed book deals with nonlinear vibrations of structural members (beams, plates, panels, shells), where the dynam- ical problems can be reduced to that of one spatial variable and time, and therefore they are further named as 1D systems. Our aim is not to synthesize and overview the general literature devoted to this problem as it is very easy (using Google, Wikipedia, and so on) nowa- days to find a lot of material and descriptions including the history as well as state-of-the art information about the mechanics/dynamics of structural members. The roots of structural mechanics come from Cauchy (1828), Poisson (1829), Kirchhoff (1850) and von Kármán (1910). The first- order shear deformation utilized by the Reissner–Mindlin theory was extended by Reddy (1990), who developed the higher order shear deformation theory including cubic terms. On the other hand, the Donnell–Mushtari–Vlasov nonlinear shallow shell theory has been proposed, which is validated for very thin shells by neglecting the in-plane inertia, transverse shear deformation and rotary inertia. Nonlinear theories for moderate and large deformations of thin elastic shells have been presented by Mushtari and Galimov (1957), Vlasov v Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 10. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page vi vi Deterministic Chaos in One-Dimensional Continuous Systems (1944) and Vorovich (1999). The tensor notation has been applied by Sanders (1968) and Koiter (1966) yielding the Sanders–Koiter equations governing the nonlinear shells vibrations. Another track in the development shell theory was added by the Flügge–Lur’e–Byrne’s modification, which was further generalized by Novozhilov (1953). Additional important contributions to shell theories have been given by Naghdi and Nordgren (1963), Librescu (1987), and Libai and Simmonds (1988). As it has already been mentioned, investigation of stability, vibra- tions and buckling of mechanical structures should be considered as a 3D time-dependent nonlinear problem taking into account various nonlinear factors such as geometrical nonlinearity, physical nonlinearity, elastic-plastic properties, cyclic loadings, damping and material rheological properties, relaxation and hysteresis phenom- ena, fatigue resistance among others. Note that the mentioned com- plex problem cannot be solved fully in spite of the recent remarkable development of numerical approaches. The main barrier stopping the investigation is associated with a so-called curse of dimension, since three independent spatial variables and time are involved in the stud- ied dynamical process. On the other hand, engineering and physical processes require reliable results obtained in engineering-accepted simulation time intervals, which requires many novel proposals of modeling of particular problems of dynamics devoted to simple struc- tural members like beams, strings, plates, rods, panels, shells, as well as their dynamical interactions. Nowadays, engineering constructions can be viewed as a collec- tion of interacting sets consisting of the mentioned simple structures usually modelled either as 1D or 2D time-dependent spatial prob- lems. The mathematical theory of elasticity takes into account var- ious aspects of static and dynamic problems of deflection of beams and shells in a 3D formulation. The same problem, from an engineer- ing approach, can be approximated via reduction of sets of 3D PDEs to that of 1D and 2D problems. The carried out reduction is based on physical motivations and imaginations being analogous to a formal mathematical approach aimed at reduction of problems with infi- nite dimension to finite one, or equivalently, the employed reduction Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 11. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page vii Preface vii process includes the transition from governing PDEs to a set of finite (relatively large) number of ODEs, or finally the reduction is aimed at decreasing of an order of the governing input PDEs. Therefore, in the case of beams and rods two coordinates can be removed while bending requires the introduction of numerous hypotheses. The men- tioned approaches have been initiated by Bernoulli, Euler, Rayleigh, Timoshenko, Sheremetev, Pelekh, Ambartsumian, Grigoliuk and oth- ers. In the case of shell-type structures, two main classes of problems can be studied: (i) 2D spatial problems, where the generalized beam hypotheses can be applied; (ii) axially symmetric shells. It is observed that the mentioned modeling yields the counterpart 1D problems but on the other hand, approximates the shell-type constructions. In what follows, we show how the problem regarding thickness of the shell-type constructions can be overcome. Recall that in classical theories of Euler and Bernoulli developed in the XVIII Century [Euler (1757)] the following main assumptions were introduced: (1) transversal cross sections of a beam that are flat and perpendicular to the beam axis before deformation remain plain and perpendicular to the beam axis during the deformation process in time; (2) Normal stresses on the fibers located in parallel to the rod axis are small and can be neglected, i.e. the longitudinal cross sections induce resistance against deflection independently, without any interaction between them; (3) inertial effects of rotation of rod elements are omitted within the deflection process. In Rayleigh’s theory proposed in 1873 [Strutt (1899)], inertial effects introduced by the beam element rotation while a rod under- goes simulteneous bending have been taken into account, which implies a reconstruction of the form of the beam kinetic energy. In addition, in the theory proposed by Timoshenko in 1921 [Tim- oshenko (1921a)] it is assumed that the transversal cross sections remain flat but not perpendicular to the beam axis. The latter hypothesis implies an occurrence of additional terms in the formula governing the rod potential energy. In the published works of B.L. Pelekh and M.P. Sheremetev in 1964 [Pelekh (1973, 1978); Pelekh and Teters (1968); Sheremetev and Pelekh (1964)] the following generalizations of the Timoshenko Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 12. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page viii viii Deterministic Chaos in One-Dimensional Continuous Systems theory have been proposed: (i) a transversal cross section being per- pendicular to the beam axis before deformation is not only perpen- dicular to the beam axis after the rod deformation process but also curved. This hypothesis implies the occurrence of a principally novel set of PDEs governing rod dynamics. Around 17 years after publish- ing the results by Pelekh and Sheremetev, similar conclusions have been published by Levinson and Reddy [Levinson (1981); Reddy, (1984a,1984b)]. The mathematical model introduced by Pelekh–Sheremetiev has been applied and developed in the works of V.A. Krysko and his co-workers [Krysko (1976)] (we cite only the monograph while the remaining papers have been published in Russian). This brief description given in the above explains why beam-type structures and axially symmetric shells are interpreted as 1D spatial structures in this book. The book is organized in the following manner — it consists of nine chapters. Each chapter begins with a short introductory overview of the chapter contents. Chapter 1 presents a short litera- ture critical overview emphasizing on the recent results devoted to analysis and control of nonlinear dynamics of beams, plates, panels and shells, and their interplay with nonlinear problems of stability, buckling, bifurcation and chaos. The first few chapters concentrate on introduction to fractal dynamics (Chapter 2), definitions of chaos and scenarios of transition from regular to chaotic dynamics as well as an introduction of the classical Fourier analysis (FFT) versus wavelet transform approaches (Chapter 3). Simple chaotic systems are briefly displayed and commented in Chapter 4, and the illustrated examples of strange chaotic attractors are then revisited in the remaining chap- ters which are purely devoted to the study of structural members. Modeling of dissipative systems and factors met in real-world pro- cesses for both discrete (lumped mechanical systems) and continu- ous structural members are briefly introduced in Chapter 5. Chaotic dynamics of Euler–Bernoulli beams including geometric and phys- ical nonlinearities, taking into account thermal effects and elastic- plastic deformations, has been modeled, illustrated and discussed in Chapter 6. Chapter 7 presents a study of the Timoshenko and Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 13. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page ix Preface ix the Pelekh–Sheremetev beams along with numerous examples of rich nonlinear dynamics with application of wavelet transforms and com- putations of Lyapunov exponents. Bifurcational dynamics as well as chaos, hyperchaos, hyper–hyper chaos and deep chaos exhibited by the rectangular plate-strips and the cylindrical panels are inves- tigated in Chapter 8. The interplay of the obtained results with the Sharkovsky series and his theorem is illustrated among others, and the realibility of the obtained results is addressed. Chapter 9 presents numerous novel bifurcation and chaotic phenomena exhib- ited by spherical and conical shells with constant and variable thick- ness emphasizing on spatio-temporal dynamics and control of chaotic vibrations with the help of the wavelet-based analysis. The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of univer- sities and companies dealing with nonlinear dynamical systems, as well as theoretically inclined mechanical and civil engineers. The book has the following unique and original features which distinguishes it from other books existing in the market: (i) The state-of-the-art nonlinear dynamics of structural mem- bers is briefly addressed allowing a reader to follow the main book track, which spans across various research topics such as vibrations of beams, plates, panels and shells with nonlinearity, bifurcation and chaos. (ii) Novel methods versus classical approaches are presented to study nonlinear phenomena exhibited by continuous systems with the help of Lyapunov exponents (exponents up to four are estimated), the wavelet-based analysis as well as neural network approaches to achieve realiable numerical results in a faster way. (iii) Our approach, contrary to majority of analyses (see the lit- erature overview given in Chapter 1), relies on truncation of nonlinear PDEs governing the dynamics of structural mem- bers in a way that gives reliable and validated results. In other words, the approximated set contains a relatively large number of nonlinear ODEs modelling real objects with infinite degrees of freedom. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 14. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page x x Deterministic Chaos in One-Dimensional Continuous Systems (iv) All the presented results devoted to structural members include numerous novel nonlinear phenomena and are based either on previously/recently published or unpublished authors’ results. (v) Applied mathematicians and physicists should be attracted by the fascinating and rich dynamics exhibited by simple structural members and by the solution properties of the gov- erning 1D nonlinear PDEs. On the other hand, the engineering- oriented researchers and graduate students will sometimes find unexpected nonlinear phenomena, which are a waiting experi- mental validation. (vi) Our book is based only on numerical computations, but a reader may find other analytical or semi-analytical approaches based on asymptotic theories in the following supplemen- tary books [Andrianov et al. (2004, 2014); Awrejcewicz et al. (1998)]; (vii) The book endeavours to utilize and extend our earlier results presented in the monographs [Awrejcewicz and Krysko (2003, 2008); Awrejcewicz et al. (2007, 2004)]. (viii) The book covers a wide variety of the studied PDEs, the way of their validated reduction to ODEs, classical and non-classical methods of analysis, influence of various boundary conditions and control parameters, as well as the illustrative presentation of the obtained results in the form of 2D and 3D figures and vibration type charts and scales. (ix) The book contains originally discovered, illustrated and dis- cussed novel and/or modified classical scenarios of transition from regular to chaotic dynamics exhibited by 1D structural members. It shows a way to control chaotic and bifurcational dynamics as well as gives directions to study other dynamical systems modelled by chains of nonlinear oscillators. (x) The book presents numerous challenges for civil/mechanical engineers showing that more sophisticated tools are required to understand and predict real structural responses. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 15. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page xi Preface xi The authors wish to express their thanks to M. Kaźmierczak, R. Kepiński and O. Szymanowska for their help in the book preparation. Finally, J. Awrejcewicz acknowledges the financial support of National Science Centre of Poland under the grant MAESTRO 2, No. 2012/04/A/ST8/00738, during 2013–2016. Lodz and Saratov, 2015 Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 16. May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PST˙ws This page intentionally left blank This page intentionally left blank Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 17. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page xiii Contents Preface v 1. Bifurcational and Chaotic Dynamics of Simple Structural Members 1 1.1 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Panels . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Shells . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. Introduction to Fractal Dynamics 14 2.1 Cantor Set and Cantor Dust . . . . . . . . . . . . . . 14 2.2 Koch Snowflake . . . . . . . . . . . . . . . . . . . . . 17 2.3 1D Maps . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Sharkovsky’s Theorem . . . . . . . . . . . . . . . . . 23 2.5 Julia Set . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Mandelbrot’s Set . . . . . . . . . . . . . . . . . . . . 27 3. Introduction to Chaos and Wavelets 31 3.1 Routes to Chaos . . . . . . . . . . . . . . . . . . . . 31 3.2 Quantifying Chaotic Dynamics . . . . . . . . . . . . 46 4. Simple Chaotic Models 86 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Autonomous Systems . . . . . . . . . . . . . . . . . 87 4.3 Non-Autonomous Systems . . . . . . . . . . . . . . . 102 xiii Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 18. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-fm page xiv xiv Deterministic Chaos in One-Dimensional Continuous Systems 5. Discrete and Continuous Dissipative Systems 105 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 Linear Friction . . . . . . . . . . . . . . . . . . . . . 106 5.3 Nonlinear Friction . . . . . . . . . . . . . . . . . . . 109 5.4 Hysteretic Friction . . . . . . . . . . . . . . . . . . . 116 5.5 Impact Damping . . . . . . . . . . . . . . . . . . . . 117 5.6 Damping in Continuous 1D Systems . . . . . . . . . 118 6. Euler–Bernoulli Beams 129 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 130 6.2 Planar Beams . . . . . . . . . . . . . . . . . . . . . . 140 6.3 Linear Planar Beams and Stationary Temperature Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.4 Curvilinear Planar Beams and Stationary Temperature Fields . . . . . . . . . . . . . . . . . . . 207 6.5 Flexible Curvilinear Beam in Stationary Temperature and Electrical Fields . . . . . . . . . . 233 6.6 Beams with Elasto-Plastic Deformations . . . . . . . 238 6.7 Multi-Layer Beams . . . . . . . . . . . . . . . . . . . 269 7. Timoshenko and Sheremetev–Pelekh Beams 307 7.1 The Timoshenko Beams . . . . . . . . . . . . . . . . 307 7.2 The Sheremetev–Pelekh Beams . . . . . . . . . . . . 324 7.3 Concluding Remarks . . . . . . . . . . . . . . . . . . 339 8. Panels 340 8.1 Infinite Length Panels . . . . . . . . . . . . . . . . . 341 8.2 Cylindrical Panels of Infinite Length . . . . . . . . . 438 9. Plates and Shells 459 9.1 Plates with Initial Imperfections . . . . . . . . . . . 460 9.2 Flexible Axially-Symmetric Shells . . . . . . . . . . . 491 Bibliography 530 Index 551 Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by 37.9.46.49 on 11/23/16. For personal use only.
- 19. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 1 Chapter 1 Bifurcational and Chaotic Dynamics of Simple Structural Members: Literature Review There are numerous papers and books devoted to dynamics of struc- tural members, i.e. beams, plates, panels and shells. The aim of this chapter is to overview the literature and state-of-the art research devoted to dynamics of the aforementioned structural members and their interplay with bifurcation and chaotic phenomena. This is a novel challenging research track, and hence not many papers and books are published in this topic. In general, from a mathematical point of view, the book deals with nonlinear PDEs and the devel- oped methods for analysis of their solutions with respect to stability, bifurcations, buckling, as well as regular and chaotic dynamics of the modeled continuous objects. On the other hand, the problem is always reduced (though by different ways) to a study of a set of large amount of nonlinear ODEs. In the case of a few first-order nonlinear ODEs, they may govern dynamics of simple nonlinear autonomous and non-autonomous oscillators (two or three first-order ODEs) or dynamics of coupled oscillators. This is why in the first four chapters a background about bifurcational and chaotic dynamics of difference and differential equations has been given to introduce the reader with basic knowledge devoted to dynamics of lumped mechanical systems and beyond. This effort allows for a smooth transition from nonlinear dynamics exhibited by simple dynamical systems to that of simple continuous systems, which can be also understood as 1D or 2D chains of nonlinear oscillators. 1 Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 20. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 2 2 Deterministic Chaos in One-Dimensional Continuous Systems 1.1 Beams Though Galileo Galilei and Leonardo da Vinci are among the first scientists who considered modeling of a beam, their theories have not been completed due to incorrect assumptions and hypotheses. Jacob Bernoulli (1654–1705) observed that the elastic beam curvature at any of its point is proportional to the bending moment at that point. This idea has been extended by Daniel Bernoulli (1700–1782), who derived the partial differential equation governing beam dynamics. This theoretical background has been used and extended by Leon- hard Euler (1707–1783), who rigorously investigated elastic beams subjected to various loads. From an engineering point of view, a beam is a mechanical (civil engineering) structure with one of its dimensions significantly larger than the remaining two dimensions. The so far introduced rough definition of a beam can be directly applied to beam-like struc- tures such as shafts, manipulator arms, airplane wings, long-span bridges, flexible satellites, fuselages, etc. Nowadays, there exist sev- eral beam theories originating from solid mechanics. However, the concepts introduced by Bernoulli and Euler are simple and accept- able in engineering, and this theory is commonly known as Euler– Bernoulli beam theory, the classical beam theory, Euler beam theory, Bernoulli beam theory and Bernoulli–Euler beam theory. The Euler–Bernoulli beam theory relies on the main assumption that the cross-section of the beam is infinitely rigid in its own plane (in-plane displacement field is composed of two rigid body trans- lations and one rigid body rotation). The second assumption says that the cross-section of a beam remains plane after deformation. The third assumption is that the cross-section remains normal to the deformed axis of the beam. The given assumptions are known as the Euler–Bernoulli assumptions for beams or as kinematic assumptions for Euler–Bernoulli beams. The Euler–Bernoulli beam can be extended to include an analysis of curved beams, buckling beam phenomena, composite beams, geo- metrically nonlinear beams, 3D transverse loading beams, as well as beams under viscoelastic/plastic deformations. In spite of that the Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 21. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 3 Bifurcational and Chaotic Dynamics of Simple Structural Members 3 original Euler–Bernoulli theory holds for infinitesimal strains and small rotations and can be extended to fit problems of moderately large rotations with inclusion of the von Kármán strains. There exists a vast literature devoted to the Euler–Bernoulli beam dynamics and its state-of-the-art that will be omitted here. In what follows, our review of the literature is mainly aimed at the bifurca- tional and chaotic dynamics of the Euler–Bernoulli beams. Abhyankar et al. [Abhyankar et al. (1993)] have applied an explicit finite difference scheme to study partial differential equations gov- erning the Euler–Bernoulli beam dynamics, aimed at analysis of the beam chaotic vibrations. The space–time spectral method has been employed to solve a simply supported modified Euler–Bernoulli non- linear beam exhibiting lateral forced vibrations [Bar–Yoseph et al. (1996)]. The authors used a generalized Galerkin method for the temporal discretization as well as a discontinuous mixed Galerkin method for the temporal discretization. The obtained solutions have been compared visually with the reference solution of the Duffing equation. The classical Euler–Bernoulli theory has been employed to study microtubules, i.e. proteins organized in a network [Civalek and Demir (2011)]. Eringen’s non-local elasticity theory has been used to include the size effect, but only static analysis has been pre- sented. Nonlinear responses of a clamped–clamped buckled Euler– Bernoulli beam governed by a PDE with a cubic nonlinearity has been studied by Barari et al. [Barari et al. (2011)]. However, the problem has been strongly truncated and reduced to that of the Duffing equation. The nonlinear transverse vibrations of a simply supported Euler–Bernoulli beam under both principal parametric and internal resonances have been analyzed in reference [Sahoo et al. (2013)]. Periodic, quasiperiodic and chaotic dynamics matched with resonances have been carried out using the method of multiple scales. Recently, attention has been paid to microbeams widely applied in micro-electro-mechanical systems (MEMS). Avsec [Avsec (2011)] has studied vibrations of microbeams and nanotubes, whereas Batra et al. [Batra et al. (2008)] analyzed vibrations of narrow microbeams initially predeformed by an electric field. Pull-in instability of Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 22. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 4 4 Deterministic Chaos in One-Dimensional Continuous Systems structural members including microbeams has been analyzed in references [Kacem et al. (2012); Krylov (2007); Moghimi and Ahma- dian (2009, 2010); Vyasarayani et al. (2011)]. Chaotic electrostatic microbeam oscillator has been studied by Towfighian et al. [Tow- fighian et al. (2010)], and bifurcation diagrams in the plane voltage amplitude-frequency have been constructed. A micro Euler–Bernoulli beam, under electro-statically actuated voltage has been analyzed in reference [Sedighi and Shirazi (2013)] taking into account the von Kármán nonlinearity, and the pull-in instability behavior has been investigated. However, only the first mode approximation has been taken into account while applying the Bubnov–Galerkin procedure. The Euler–Bernoulli simply supported beam taking into account the von Kármán geometric nonlinearity under an extremal excitation has been studied in reference [Dai and Sun (2014)]. They carried out a comparison of a chaotic and a multi-dimensional model and concluded that in order to control chaotic beam dynamics the con- tribution of high order vibrations should be taken into account. In spite of Euler–Bernoulli beam theory, there exists the widely applied Timoshenko beam theory [Rosinger and Ritchie (1977); Tim- oshenko (1921a, 1922, 1932)] developed by S. Timoshenko (1878– 1972). The Timoshenko model is particularly suitable for analysis of a short beam, composite beams or beams under high-frequency excitation, since it takes into account rotational inertia effects and shear deformation. The Timoshenko beam model can be viewed as more general with respect to the Euler–Bernoulli model because if the shear modulus of the beam material tends to infinity (beam is rigid in shear) and if rotational inertia phenomena can be neglected, the Timoshenko model takes the form of the Euler–Bernoulli model. Since the Euler–Bernoulli beam does not include a shear deforma- tion yielding rotation, it is stiffer compared to the Timoshenko beam. However, if a real beam has large length and small thickness, then a difference between two models is small. It should be noted that the effect of rotary inertia was proposed by Rayleigh in 1894, and in literature there exists the Rayleigh beam the- ory. Recently, the Euler–Bernoulli and Timoshenko models have been reconsidered with respect to the two-dimensional elasticity model Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 23. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 5 Bifurcational and Chaotic Dynamics of Simple Structural Members 5 [Labuschagne et al. (2009)]. Using the example of a cantilever beam, the authors conclude that the Timoshenko model is close to that con- structed with the help of the 2D theory for models of practical use. The Euler–Bernoulli beam theory corresponds to the first approxi- mation, whereas the Timoshenko theory stands for the refined beam theory. Li [Li (2008)], using examples of functionally graded beams with the rotary inertia and shear deformation included, showed that the Euler–Bernoulli and Rayleigh beam theories can be derived from the Timoshenko beam theory. Nonlinear flexural waves and chaotic vibrations of a Timoshenko beam have been analyzed by Zhang and Liu [Zhang and Liu (2010)]. The finite-deflection and the axial iner- tia are included into the derived PDEs governing flexural waves in the beam. Then the problem has been strongly reduced into a non- linear ordinary equation. The latter non-autonomous equation has been analyzed following the classical Melnikov method to define the threshold condition of the occurrence of a transverse hetero- clinic point, and to predict the deterministic chaos. He’s homo- topy perturbation method has been applied to study nonlinear free vibrations of clamped-clamped and clamped-free Timoshenko microbeams by Moeenfard et al. [Moeenfard et al. (2011)]. Again, the original problem of infinite dimension has been strongly trun- cated to consideration of only one nonlinear ordinary differential equation. 1.2 Plates The study of vibrations of plates has a long history in mechanics and applied mathematics and numerous books and papers have been published dedicated to this issue. In this brief state-of-the-art review, we aim only at recent results matching nonlinear vibrations of plates with bifurcation and chaotic phenomena. Both local and global bifurcations of parametrically excited nearly square plates have been studied by Yang and Sethna [Yang and Sethna (1991)], putting emphasis on occurrence of the Smale horseshoe manifold. This approach based mainly on the Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 24. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 6 6 Deterministic Chaos in One-Dimensional Continuous Systems averaging procedure has been extended to analyze vibrations of nearly square plates to the anti-symmetric case [Yang and Sethna (1992)]. Shilnikov-type homoclinic orbits, bifurcations and chaotic vibrations of thin plates subjected to parametric excitation have been analyzed in reference [Feng and Sethna (1993)] using the so-called global perturbation method. The multiple scales method has been applied to study modal interaction of nonlinear clamped plates and harmonic excitation by Hadian and Nayfeh [Hadian and Nayfeh (1990)]. A double mode approach has been employed by Shu et al. [Shu et al. (1999)], and chaotic vibrations have been detected using the Melnikov method. The global perturbation method has been applied in references [Samoylenko and Lee (2007); Yeo and Lee (2006)] to study global vibrations of an imperfect circular plate for the case of 1:1 inter- nal resonance, and the chaotic orbits. A similar analytical technique has been used by Yu and Chen [Yu and Chen (2010)] to study global bifurcations of a simply supported rectangular metallic plate under a transverse harmonic excitation. Both Melnikov’s approach and aver- aging procedure have been employed by Zhang and Li [Zhang and Li (2010)] to detect resonant chaotic vibrations of a simply supported rectangular plate excited externally and periodically. The extended Melnikov-type analytical approach has been applied by Zhang et al. [Zhang et al. (2008, 2010)] to study the global bifurcations and multi- pulse chaotic vibrations of a buckled thin plate as well as a laminated composite piezoelectric rectangular plate. The so-called energy-phase method has been utilized by Yao and Zhang [Yao and Zhang (2007)] to analyze the Shilnikov-type multi-pulse heteroclinic orbits and chaotic vibrations of a parametrically and externally excited rectan- gular thin plate. The similar-like approach has been used to examine the Shilnikov-type multi-pulse homoclinic orbits exhibited by a circu- lar plate excited harmonically [Yu and Chen (2010b)]. More recently, Yao and Zhang [Yao and Zhang (2014)] have investigated multi-pulse global heteroclinic bifurcations and chaotic dynamics of a simply sup- ported rectangular thin plate in the resonant case using the extended Melnikov method. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 25. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 7 Bifurcational and Chaotic Dynamics of Simple Structural Members 7 The so far reported references deal with a strong truncation of the studied PDEs governing dynamics of plates, and hence many non- linear phenomena have been omitted during those studies. There are also papers mainly based on numerical simulations, which are closer to the investigation topic of this book. The fractal dimensions, the maximum Lyapunov exponents and bifurcation diagrams have been employed to study bifurcational and chaotic vibrations of a simply supported thermoelastic plate with variable thickness in reference [Yeh et al. (2003)]. The global bifurcation and chaotic dynamics of a rectangular thin plate have been studied by Zhang [Zhang (2001)]. Touzé et al. [Touzé et al. (2011)] have studied the transition from periodic to chaotic vibrations in free-edge, perfect and imperfect cir- cular plates based on the von Kármán PDEs for thin plates includ- ing geometric nonlinearity. The obtained numerical results have been confirmed with experimental investigations. High dimensional chaos versus the framework of wave turbulence have been illustrated and discussed. Recently, a challenging interest coming from an industry has been devoted to new functionally graded materials (FGMs), being inhomogeneous composites made of a mixture of metals and ceram- ics. FGM plates have been widely used in various branches of the industry since their material properties allow for a smooth and con- tinuous change from one surface to another, and hence the prob- lem of thermal stress concentrations can be withdrawn. The non- linear transient thermoelastic analysis of FGM plates under pres- sure loading and temperature fields has been carried out by Parveen and Reddy [Parveen and Reddy (1998)]. Yang and Shen [Yang and Shen (2002)] studied free and forced vibrations of FGM plates in a thermal environment, and the material parameters have been temperature-dependent. Dynamics of the pre-stressed graded layer and two surface-mounted piezoelectric actuator layer have been examined by Yang et al. [Yang et al. (2003)]. Nonlinear dynamics of FGM plates embedded in thermal environment including heat conduction and temperature-variable material properties have been studied by Huang and Shen [Huang and Shen (2004)]. The effect of transverse shear strains and rotary inertia have been taken into Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 26. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 8 8 Deterministic Chaos in One-Dimensional Continuous Systems account while analyzing nonlinear vibrations of FGM plates with initial stress by Chen [Chen (2005)]. The nonlinear dynamics of a simply supported FGM rectangular plate under both transversal and in-plane excitation in a thermal environment has been investigated by Hao et al. [Hao et al. (2008)]. The problem has been reduced to the study of a two-degree-of-freedom system with the quadratic and cubic nonlinear terms. The cases of internal and principal para- metric resonances have been analyzed using asymptotic perturbation methods. Periodic, quasi-periodic and chaotic vibrations have been studied. A 3D-exact solutions for free and forced vibration of sim- ply supported FGM rectangular plates has been derived by Vel and Batra [Vel and Batra (2004)]. Nonlinear vibrations of FGM plates with randomness of the material properties and plates with geomet- ric imperfections have been considered by Kitipornchai et al. [Kiti- pornchai et al. (2004, 2006)]. Yang and Huang [Yang and Huang (2007)] dealt with a semi- analytical investigation of geometrical imperfections on the nonlinear vibrations of simply supported FGM plates. Nonlinear vibration, bifurcation and chaos of viscoelastic cracked plates have been investi- gated by Hu and Fu [Hu and Fu (2007)]. The von Kármán plate the- ory and the linear isotropic constitutive theory have been employed to a nonlinear integral-partial differential equation for a rectangular plate with an all-over part-through crack. In particular, the effects of the depth and the position of the crack and the viscoelastic material parameters’ impact on the bifurcational and chaotic dynamics with movable simply-supported boundary conditions have been analyzed. Onozato et al. [Onozato et al. (2009)] carried out laboratory experiments on chaotic vibrations of a rectangular plate under in-plane elastic constraint at clamped edges. Chaotic responses have been examined using the Fourier spectra, the Poincaré projec- tions, the maximum Lyapunov exponents and the Karhunen–Loéve method. Zhang et al. [Zhang et al. (2010)] investigated nonlinear vibrations and chaos of a FGM rectangular plate in thermal environ- ment and under parametric and external excitations. The govern- ing PDEs are derived based on the Reddy third-order shear defor- mation plate theory and the Hamilton’s principle. Application of Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 27. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 9 Bifurcational and Chaotic Dynamics of Simple Structural Members 9 the Galerkin procedure yielded a three-degree-of-freedom nonlinear systems, which has been further studied via multiple scales and clas- sical numerical approaches. Bifurcation and chaotic phenomena exhibited by an axially mov- ing plate under external and parametric excitations have been exam- ined by Liu et al. [Liu et al. (2012)]. The derived coupled PDEs of transverse deflection and stress have been reduced to a set of ODEs, which has been studied numerically. A relevance between the onset of chaos with the corresponding linear instability range has been illus- trated and discussed. The global bifurcations and multiple chaotic vibrations of a simply supported laminated composite piezoelectric rectangular thin plate subjected to parametric and transverse excita- tions have been investigated by Zhang and Zhang [Zhang and Zhang (2011)]. Both von Kármán plate theory and the first-order piston theory are used to derive PDEs governing nonlinear dynamics of cantilever plate in supersonic flow in reference [Xie et al. (2014)]. The Rayleigh–Ritz procedure has been employed to get ODEs being then solved numerically. Chaotic, prechaotic and postchaotic regimes have been identified, and the routes to chaos have been studied. 1.3 Panels In this section, we proceed to a brief review of the published papers contributing to bifurcational and chaotic vibrations of panels. Maestrello et al. (1992) studied the dynamic response of an air- craft panel forced at resonance and off-resonance by plane acous- tic waves. Period doubling bifurcations and chaotic panel vibrations have been detected, when the sound pressure level of the excitation increased. Good agreement between the experimental and numerical results has been obtained. Yamaguchi and Nagai [Yamaguchi and Nagai (1997)] have pre- sented numerical results devoted to chaotic vibrations of a shal- low cylindrical shell-panel subjected to harmonic lateral excitation. Based on the Donnell–Mushtari–Vlasov theory, the problem has been reduced to that of a multiple-degree-of freedom system by the Galerkin procedure. Chaotic vibrations combined with a dynamic Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 28. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 10 10 Deterministic Chaos in One-Dimensional Continuous Systems snap-through have been monitored with the help of Lyapunov expo- nents and Poincaré maps. The effect of the in-plane elastic constraint on the chaos of the shell-panel has been illustrated and discussed. Chaotic vibrations of a panel forced by turbulent boundary layer and sound have been studied by Maestrello [Maestrello (1999)]. The panel response combined with period-doubling bifurcations makes a transition to chaos when forced by the boundary layer, which has been associated with quasi-periodic dynamics as the wave loses the spatial homogeneity. Nonlinear bifurcations and chaotic dynamics of fluttering panel in post-critical domain have been analyzed by Bolotin et al. [Bolotin et al., (1998a,1998b)]. Chaotic vibrations of a panel forced by buffeting aerodynamic loads have been demon- strated and investigated by Epureanu et al. [Epureanu et al. (2004)]. The finite difference method has been employed to study coherent structures of the panel nonlinear dynamics, which have been identi- fied by a proper orthogonal decomposition. Nagai et al. [Nagai et al. (2004)] have presented a shallow cylindrical shell-panel with a con- centrated mass under periodic excitation. Furthermore, Nagai et al. [Nagai et al. (2007)] carried out the detailed experimental and analyt- ical analyses on chaotic vibrations of a shallow cylindrical shell-panel under gravity and periodic excitation. The detected chaotic dynam- ics has been monitored and quantified by Fourier spectra, Poincaré maps, maximum Lyapunov exponents and Lyapunov dimension. It has been shown that the dominant chaotic dynamics is yielded by the responses of the sub-harmonic resonance of 1/2 order and of the ultra-sub-harmonic resonance of 2/3 order. The similar-like approach has been extended to study a modal interaction in chaotic vibra- tions of a shallow double-curved shell-panel by Maruyama et al. [Maruyama et al. (2008)]. The shell-panel with square boundary and initial geometric imperfection has been simply supported, whereas the in-plane displacement at the boundary has been constrained elastically. Similar to the case of the plates, the FGM panel has been recently extensively studied. It has been shown that owing to the aero-thermo- elastic interactions at high Mach numbers, skin panels may exhibit divergence and flutter instability behavior. The combined action of Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 29. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 11 Bifurcational and Chaotic Dynamics of Simple Structural Members 11 thermal and aerodynamic loads on both static and dynamic stabil- ities of FGM panels have been analyzed by Sohn and Kim [Sohn and Kim (2008)]. Ibrahim et al. [Ibrahim et al. (2008)] investigated the nonlinear flutter and thermal buckling of a FGM panel using FEM (Finite Element Method). The aero-thermo-elastic post-critical and vibration characteristics of temperature-dependent FGM panels in a supersonic flow have been studied by Hosseini and Fazelzadeh [Hosseini and Fazelzadeh (2010)]. The von Kármán theory has been employed, whereas the material properties have been considered as temperature-dependent and graded in the thickness direction. PDEs have been converted to ODEs and then classical numerical tools have been implemented to study chaotic vibrations. Hosseini et al. [Hosseini et al. (2011)] investigated chaotic and bifurcation dynamics of FGM curved panels subjected to aero-thermal loads. The panel has been infinitely long and sim- ply supported, and the material properties have been taken as temperature-dependent and varying through the thickness direction. The governing PDE has been truncated to a set of nonlinear ODEs solved numerically. Regular and chaotic vibrations regimes have been detected and monitored via Poincaré maps, time histories, frequency spectra and Lyapunov exponents. The effect of the geometrically imperfect curved skin panel parameters on the flutter behavior has been investigated by Abbas et al. [Abbas et al. (2011)]. The thermal degradation and Kelvin’s model of structural damping have been included into the analysis. Aginsky and Gottlieb [Aginsky and Gottlieb (2012)] studied a nonlinear bifurcation structure of panels under periodic acoustic fluid-structure interaction. In particular, an intricate bifurcation structure near the fifth-mode panel resonance including coexisting symmetric and asymmetric periodic solutions has been detected. In addition, the emergence of a non-stationary spatio-temporal chaotic solution has been found. Li et al. [Li et al. (2012)] addressed the problem of the aero-elastic stability and bifurcation structure of subsonic nonlinear thin pan- els subjected to external excitation. The von Kármán large deflec- tion model and Kelvin’s structural damping have been utilized while Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 30. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 12 12 Deterministic Chaos in One-Dimensional Continuous Systems deriving the governing equation of the simply supported 2D-panel. The problem has been reduced to that of finite number of nonlinear ODEs, which then have been solved numerically using the fourth order Runge–Kutta method. It has been shown that: (i) the panel lost its stability by divergence; (ii) periodic and chaotic zones appear alternately; (iii) a route to chaos is realized via period-doubling bifur- cation. The similar-like approach has been extended to study bifurca- tion structure and scaling properties of a subsonic periodically driven panel with geometric nonlinearity by Li et al. [Li et al. (2015)]. In particular, the scaling properties of the bifurcation structure are dis- cussed in terms of discrete mapping and based on linear approxima- tion. However, only one-mode reduction has been employed during the studies. 1.4 Shells In the case of shells chaotic and bifurcational dynamics, the num- ber of reports is rather limited. Period-doubling bifurcations of an infinitely long cylindrical shell under the condition of internal res- onance has been studied by Nayfeh and Raouf [Nayfeh and Raouf (1987)]. Chaotic vibrations of non-shallow arches loaded at their crown by a vertical harmonic force have been investigated by Thom- sen [Thomsen (1992)]. The quasi-periodic break-up, intermittency and long transient behavior have been detected as routes to chaotic dynamics. Chaotic energy pumping through auto-parametric reso- nance in cylindrical shell has been investigated by Popov et al. [Popov et al. (2001)]. Transient and steady-state instabilities matched with chaotic vibrations exhibited by pressure-loaded shallow spherical shells have been analyzed in reference [Soliman and Goncalves (2003)]. Amabili [Amabili (2005)] studied chaotic vibrations of dou- bly curved shallow shells. Nagai and Yamaguchi [Nagai and Yam- aguchi (1995)] analyzed chaotic vibrations of a shallow cylindrical shell with rectangular boundary under cyclic excitation. Sheng and Wang [Sheng and Wang (2011)] have investigated a nonlinear response of FGM cylindrical shells under both mechan- ical and thermal loads within the von Kármán nonlinear theory. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 31. February 17, 2016 13:50 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch01 page 13 Bifurcational and Chaotic Dynamics of Simple Structural Members 13 The influence of temperature change, fraction exponent of FGM and geometry parameters has been addressed. Krasnopolskaya et al. [Krasnopolskaya et al. (2013)] introduced two new mathemati- cal models of cross-wave generation in fluid free surface between two cylindrical shells. Two eigenmodes approximations have been employed, and periodic, quasi-periodic and chaotic regimes have been detected and illustrated. Alijani and Amabili [Alijani and Amabili (2012)] introduced sub-harmonic, quasi-periodic and chaotic dynam- ics of FGM doubly curved shells under concentrated harmonic load. Gear’s backward differentiation formula has been applied to get bifur- cation diagrams, Poincaré maps and time histories, as well as the Lyapunov spectrum has been computed. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 06/25/16. For personal use only.
- 32. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 14 Chapter 2 Introduction to Fractal Dynamics In this chapter, in order to get a deeper understanding of nonlinear dynamical phenomena covered by this book, the fundamental con- cepts of one-dimensional (1D) maps and fractal sets are briefly reviewed and illustrated. First notions of Cantor’s set, Cantor’s dust and Koch’s snowflake are presented. Then 1D maps are considered putting emphasis on their regular and chaotic dynamics, the cobweb diagrams and the period doubling bifurcation routes to chaos includ- ing estimation of the Feigenbaum constant are also briefly revisited. The Sharkovsky theorem is addressed with presentation of its advan- tages and limitations. Next, the Julia, Fabout and Mandebrot sets are shortly described and illustrated. 2.1 Cantor Set and Cantor Dust The term Cantor dust was introduced by German mathematician Georg Cantor in 1883 [Cantor (1883)], though it was discovered ear- lier by H. J. Smith [Smith (1874)]. This set has been considered as a set of zero Lebesque measure. Fractal properties of the Cantor set have important meaning since many of the known functions are similar families of this set. The Cantor ternary set is constructed by repeatedly removing the open middle part of the unit interval [0, 1], assuming that it is divided into three parts. We begin (the first step) by deleting the open middle third (1/3, 2/3) leaving two line segments [0, 1/3] ∪ [2/3, 1]. Proceeding in 14 Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 33. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 15 Introduction to Fractal Dynamics 15 Fig. 2.1 Construction of the Cantor set. a similar way with all the remaining parts, we obtain (see Fig. 2.1) C0 = [0, 1] , C1 = 0, 1 /3 ∪ 2 /3, 1 , C2 = 0, 1 /9 ∪ 2 /9, 1 /3 ∪ 2 /3, 7 /9 ∪ 8 /9, 1 , Cn = Cn−1 3 ∪ 2 3 + cn−1 3 . (2.1) The obtained remaining segments/points can be presented via the following general formula C = ∞ m=1 3m−1 k=0 0, 3k + 1 3m ∪ 3k + 2 3m , 1 , (2.2) which has been proved by Soltanifar [Soltanifar (2006)]. Limiting set C, which contains all of the non-deleted sets Cn, n = 0, 1, 2, . . . , is called the classical Cantor dust. In what follows, we discuss a few important properties of the Cantor set. 1. Cantor set is a self-similar fractal of dimension d = log(2) /log(3) ≈ 0.6309, (2.3) where Nrd = 1 is satisfied for N = 2 and r = 1/3; N is the num- ber of equal parts; 1/2 — decrease of 1/2 times; d — fractal (non- integer) dimension or dimension similarity d = log N log 1/2 . Logarithm can be taken with any basis, for instance, using e ≈ 2, 7183 . . . . The Hausdorf dimension of the Cantor set is ln 2/ ln 3 = log3 2, whereas its Lebesque measure is zero. 2. Cantor set does not include intervals of integer length. 3. Sum of removed intervals while constructing the set C is equal to 1. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 34. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 16 16 Deterministic Chaos in One-Dimensional Continuous Systems This statement can be proved in the following way. Length of the first interval is 1/3. In order to get C2, we remove two intervals of length 1/32 . In the next step, we remove 22 intervals, each of length 1/33 , and so on. Therefore, the sum of removed intervals is S = 1 3 + 2 32 + 22 33 + . . . + 2n−1 3n + . . . , (2.4) or equivalently S = 1 /3 · 1 + 2 /3 + 2 /3 2 + 2 /3 3 + . . . . (2.5) Let us recall the geometric series 1 1 − x = 1 + x + x2 + x3 + . . . |x| 1, (2.6) and hence we get S = 1 /3 · 1 1 − 2/3 = 1. (2.7) 4. Comparison of the Cantor set with the interval [0, 1] shows that powers of these two sets are equal, i.e. power of the Cantor set C is equal to the power of continuum [0, 1]. 5. Classical Cantor dust presents the example of a compact fully discontinuous set. The Cantor set cannot have any interval of non-zero length. Since the subsequent subsets of the Cantor set construction do not remove end points, the Cantor set is not empty and contains an uncountably infinite number of points. Another observation is that all algebraic irrational numbers are normal, whereas the Cantor set members are either rational or transversal (they are not normal). The Cantor set can serve as the archetype of a fractal. It exhibits the left and right self-similarity transformations fl(x) = x/3 and fr(x) = (2 + x)/3, which acting on the Cantor set yield fl(C) ∼ = fr(C) ∼ = C. It means that the Cantor set is invariant with respect to the introduced homeomorphism. The set {fl, fr} together with the function composition creates the dyadic monoid. On the other hand, the Cantor dust is a multi-dimensional Cantor set, i.e. it is Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 35. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 17 Introduction to Fractal Dynamics 17 constructed as a finite Cartesian product of the Cantor set with itself by yielding a Cantor space also with zero measure. The method of subdividing a shape into copies of itself simultane- ously removing a few of such copies has been extended to 2D objects. In spite of the Cantor dust, one may construct the Sierpiński triangle discovered by W. Sierpiński in 1916 [Sierpiński (1916)] (an equilat- eral triangle is subdivided into four equilateral triangles removing the middle triangle) and the Menger sponge reported by K. Menger in 1928 [Menger (1928)] (a cube is taken; each face is divided into nine squares; the smaller cube in the middle of each face is removed and also the smaller cube in the center of the large cube is removed to get a void cube; the last two steps are repeated for the remaining smaller cubes and the described iterations are repeated to infinity). It is remarkable that each face of the Menger sponge is a Sierpiński carpet, and its Lebesque measure is zero (uncountable set). However, the Lebesque covering dimension of the Menger sponge is one. It can be shown that every curve is homeomorphic to a subset of the Menger sponge including trees and graphs, vertices and closed loops, etc. giving the right to understand the Menger sponge as a universal curve. On the other hand, the Sierpiński carpet can be viewed as a universal curve for all planar curves, i.e. projection of the Menger universal curves onto a plane yields the Sierpiński universal curves. The Hansdorff dimension of the Sierpiński carpet is log 8/ log 3 ∼ = 1.893, whereas the Hansdorff dimension of the Menger sponge is log 20/ log 3 ∼ = 2.727. 2.2 Koch Snowflake In 1904, the Swedish mathematician Helge von Koch presented a way to construct the so-called Koch snowflake, known as the Koch island and Koch star, as a mathematical curve being a prototype for a fractal [Peitgen et al. (1926)]. Boundary of Koch’s snowflake is described by a curve consisting of three similar fractals of dimension d ≈ 1.2618. Each third part of the snowflake is constructed in an iterative way, beginning from one side of an equilateral triangle. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 36. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 18 18 Deterministic Chaos in One-Dimensional Continuous Systems Fig. 2.2 Subsequent step of construction of Koch’s curve. Let K0 be the initial part. We remove the third middle part and we add two new parts of the same length, as shown in Fig. 2.2. The obtained Koch’s curve has an infinite length, since each iteration adds one-third of the previous length. In addition, it is continuous everywhere but differentiable nowhere. The obtained set is called K1. We repeat the described procedure a few times changing in each step the middle third part by new parts. Denoting by Kn the figure obtained from the nth step for each of the triangle side, we get a Koch snowflake (three Koch’s curves contribute to the snowflake), see Fig. 2.3. The fractal dimension of the Koch curve is d = log 4 /log 3 ≈ 1.2618. (2.8) Intuitively, the series {Kn}∞ n=1 → K. If we take a copy of K, reduced by three times (r = 1/3), then the whole set K can be composed of N = 4 such copies. One of the most important properties of the Koch’s snowflake is its infinite length, i.e. the length of the curve Kn = 4n/3n and its limit lim n→∞ (4n/3n) = ∞. Since the number of new triangles yielded by n iterations is tn = 3 · 4n−1 = 3 4 · 4n , (2.9) and the area of each new triangle is sn = sn−1 9 = s0 9n , (2.10) the total new area obtained in iteration n follows sntn = 3 4 4 9 n sn , (2.11) Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 37. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 19 Introduction to Fractal Dynamics 19 Fig. 2.3 Koch’s snowflake. and finally the total area of the snowflake is sn = s0 1 + 3 4 n k=1 4 9 k = s0 1 + 3 5 1 − 4 9 n = s0 5 8 − 3 4 9 n , (2.12) where s0 is the area of the starting triangle. The snowflake perimeter after n iterations follows Ln = 3 · l · 4 3 n , (2.13) where l denotes the length of each triangle sides. In the limiting case n → ∞, one gets lim n→∞ Ln = ∞, lim n→∞ sn = 8 5 s0. (2.14) Nowadays, various variants of the Koch curves have been proposed by taking into account various angles (quadratic types 1 and 2 Koch curve and Cesáro fractal), quadratic type 1 and type 2 Koch surfaces, sphere Koch flake and Koch cube, etc. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 38. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 20 20 Deterministic Chaos in One-Dimensional Continuous Systems 2.3 1D Maps 1D maps, also known as iterated maps or recursion relations or dif- ference equations, are mathematical systems governing the dynam- ics of a single variable x over discrete time values. Let us assume that a simple discrete dynamical system consists of an initial point x0 and the function f(x). Sequence {xn}∞ n=0 = f(n)(xn) ∞ n=0 is called an orbit f(n)(x) = f (f . . . (f(x))) of the initial point x0. We take x0 as a real, and f as an elementary function, for instance f (x) = x2 + c, f (x) = cx (1 − x), f (x) = cos x. Here, the map f is called a compressing one. Therefore, the theorem of a fixed point cannot be applied, and furthermore we cannot get any conclusion for the convergence of the series {xn}∞ n=0. Investigating a chaotic dynamics we consider a nonlinear (non-affine) function which cannot be presented in the form f (x) = ax+b, since in either linear or affine cases, chaos is not observed. Let ∃ |f (x)|. If x is a fixed point and |f (x)| 1, then x is an attracting point. If |f (x)| 1, then it is a repiller, and if |f(x)| = 1, then any conclusion regarding point x cannot be made. An orbit is called periodic of period p, if xn + p = xn for n = 0, 1, 2, . . . , and p is the smallest integer number. If the equation of periodicity xn + p = xn becomes true only after a certain finite number of steps, say, n ≥ n0, then the orbit is also periodic after a few iterative steps. The so- called cobweb diagram is often used to follow an orbit of the real function f. We consider function f (x) = x2 (Fig. 2.4). The shown diago- nal lines are plots of y = x (y/x denotes vertical/horizontal axis), whereas the curved line is a plot of the map f(x). The remaining series of lines with arrows are cobweb elements. How to construct a cobweb diagram? Let us take the map y = xn+1 = f(xn). The initial point x0 is mapped to a new point x1, which is found by drawing a vertical line from the axis x to the curve f(x). Then we move horizontally to the line y = x to find the next x-coordinate xn+1. A fixed point of the map is defined as an intersection of the graph y = f(x) and the line y = x. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 39. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 21 Introduction to Fractal Dynamics 21 Fig. 2.4 Cobweb diagram of the map f (x) = x2 for the following initial condi- tions: (a) x0 = 1.1, (b) x0 = −0.9. Another case of iterations is a limit-cycle. It represents a periodic dynamics, since the orbit tends to a series of points that repeats itself (a number of iterations needed to repeat itself is called a period of the orbit). In our case x0 = 0 and x0 = 1 are fixed points of the map y = x2. If x0 1, then the orbit tends to + ∞, if 0 x0 1 or −1 x0 0, then the orbit tends to a fixed point 0. If x0 = 1, then the orbit is [−1, 1 1 1 . . .], i.e. it is periodic, and finally if x0 −1, then xn → ∞ and the orbit diverges. In the given case, the fixed point 0 is attractive, whereas the fixed point 1 is repelling. The functions f (x) = x2 and f (x) = x2 −1 are particular cases of the map f (x) = x2 + c, and they are widely used in the theory of dynamical systems [Sharkovsky et al. (1997)]. Although f (x) = x2+c is only a quadratic function, it is widely applied. We consider a real case, i.e. when x and c are real numbers. We solve equation x2 +c = x, and we get ξ = 1+ √ 1−4c 2 ; η = 1− √ 1−4c 2 . We deal with a fixed point for 1 − 4c ≥ 0 (c ≤ 1 4 ), if −ξ η ξ, and besides, f (−ξ) = ξ. Orbits for x0 ξ and x0 −ξ tend to + ∞. For −3/4 c 1/4, the fixed point η is attractive, i.e. |f (x)| 1 and all orbits tend to η. For c −3/4 |f (x)| 1, i.e. η becomes a repeller. When c transits over the value −3/4, the system exhibits a period doubling bifurcation. A second period doubling bifurcation takes place for c = −5/4. If c −5/4, then an attractive periodic orbit of period 4 appears. Decreasing c, attractive orbits of length 8, 16 and 32 are observed, i.e. after period doubling bifurcations chaos Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 40. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 22 22 Deterministic Chaos in One-Dimensional Continuous Systems occurs. For c = −2, there exist periodic orbits of f (x) with periods 2, 3, 4, . . . . Feigenbaum studied intervals between period doubling bifurca- tion of the square function f (x) = cx (1 − x), also known as the logistic map [Feigenbaum (1990); Thunberg (2001)]. The obtained diagram of orbits is similar to that of the function f (x) = x2 + c. The fundamental meaning of the Feigenbaum analysis relies on the detection of the mechanism’s universality of a route to chaos via period doubling bifurcation, which is exhibited not only by the logis- tic map f (x) = cx (1 − x), but also by the maps f (x) = x2 + c, f (x) = c sin (πx) , f (x) = cx2 sin (πx) defined on appropriate inter- vals. The aforementioned class includes functions f (x), defined on the interval [0, 1] and achieving a maximum at point xm ∈ (0, 1) for f(xm) = 0, whereas f (x) on [0, xm] and [xm, 1], as well as its Schwartz derivative Sf (x) = f(x) f(x) − 3 2 f(x) f(x) 2 is negative ∀ x ∈ [0, 1]. We denote by c0, c1, c2, . . . , bifurcation points on the orbits’ diagram (Fig. 2.5), i.e. those points, where iterations f (x) = x2 + c change the attracting orbit of period 2n into attracting orbit 2n+1. One may check that c∞ = lim n→∞ = −1.401155 . . . , i.e. we deal with Fig. 2.5 Diagram exhibiting period doubling bifurcation of the map f (x) = x2 + c. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 41. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 23 Introduction to Fractal Dynamics 23 the Feigenbaum type process. For c = −1.7548777 . . . period-3 orbit exists, which corresponds to the more lighted part of the diagram. Further, we have d = lim n→∞ cn−cn−1 cn+1−cn = 4.669162 . . . , which defines the Feigenbaum constant. It should be noted that it is, in general, difficult to find analyti- cally bifurcation points cn for any given function, for instance x2 +c, and hence to define the Feigenbaum constant. However, there exists another approach. For each pair of bifurcation points cn and cn+1 there exists a point c∗ n, which corresponds to an orbit with period 2n. For this value of c, the critical point x0 of the function fc satisfies the equation f (n) c = x0. Then, the Feigenbaum constant d can be found through the following equation d = lim n→∞ c∗ n − c∗ n−1 c∗ n+1 − c∗ n , (2.15) where c∗ n is found numerically with the help of the Newton method. The stretching and folding exhibited by the logistic map produces sequences of iterates, which finally yield an exponential divergence validated by the Lyapunov exponent computation. The logistic map correlation dimension is 0.5, and its Hausdorff dimension is 0.538. The logistic map as well as other chaotic maps have found applica- tions in developing image encryption schemes [Pareek et al. (2006); Wong et al. (2008)]. 2.4 Sharkovsky’s Theorem Diagram of orbits shown in Fig. 2.5 presents attracting periodic orbits for fc(x). For 1/4 c −3/4, there is an attracting orbit of period 1. For −3/4 c −9/4, we deal with the attracting orbit of period 2, which bifurcates into attracting period-4 orbit, for c = −5/4. On some intervals, the diagram exhibits windows. For instance, for c ≈ −1, 75 we have a white zone with period-3 orbits. A nat- ural question appears: Do other periodic orbits exist? Those orbits should be repellers, since in the diagram we observe only attracting Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 42. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 24 24 Deterministic Chaos in One-Dimensional Continuous Systems orbits. It is shown that the occurrence of a period-3 orbit implies the occurrence of orbits with periods n = 1, 2, 3, . . . . In 1964, A. N. Sharkovsky published a general theorem, regarding a map onto itself valid for the real functions, which had an important impact on the development of the theory of nonlinear dynamical systems [Sharkovsky (1964)]. A brief description of Sharkovsky’s theorem follows. Let I be finite or infinite interval in R. We assume the map f : I → I is continuous. We define the number x as a periodic point of period m if fm(x) = x and fk(x) = x for all 0 k m. Let us construct the following ordering of the positive integers 3, 5, 7, 9, . . . 2 · 3, 2 · 5, 2 · 7, 2 · 9, . . . 22 · 3 , 22 · 5 , 22 · 7 , 22 · 9 , . . . 23 · 3 , 23 · 5 , 23 · 7 , 23 · 9 , . . . . . . . . . . . . . . . , 2n , . . . , 23 , 22 , 21 , 1. (2.16) We start with the construction of the odd numbers in the increas- ing order, followed by 2 times the odds, 22 times the odds, 23 times the odds, and so on, whereas at the end we take powers of two in the decreasing order. Then, if f has a periodic point of least period m and m precedes in the ordering (2.16), f also has a periodic point of least period n. Assuming now that f has only a finite number of periodic points, then those points have periods numbers of which are powers of 2. If there exists a periodic point of period-3, then there exist also periodic points of all other periods. It should be emphasized that the Sharkovsky theorem does not concern with the stability of periodic orbits but rather their existence. In other words, if one takes the logistic map, for a wide range of the control parameter only one period-3 orbit is detected, which is an attractor. The remaining orbits of all periods are not visible, since they are unstable and cannot be detected through the cobweb construction described so far. The Sharkovsky Theorem is valid only for a real function defined on a real interval. If, for instance, function f is given by a rotation of each point lying on a circle about the angle 2/n, then the orbits Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 43. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 25 Introduction to Fractal Dynamics 25 of each point have the same period n. In this case, there is lack of any additional periods, and Sharkovsky’s theorem is not applicable. On the other hand, if there exists an orbit of the odd period larger than one, then a number of different periods tends to infinity. 2.5 Julia Set We denote by C a set of complex numbers a + bi = z, where a and b are respectively real and imaginary parts of z, i.e. a = Re (z), b = Im(z), |z| = √ a2 + b2. Series of complex numbers {zn}∞ n=1 → lim n→∞ zn = ∞, which means that ∀ M 0, ∃ N 0, ∀ n N : |zn| M, i.e. all points zn lie inside a circle of radius M for suffi- ciently large n and it is necessary that absolute magnitudes increase to infinity. Let f(z) = anzn + an−1zn−1 + . . . + a1z + a0, an = 0, be a poly- nomial of order n ≥ 2, and its coefficients an, an−1, , . . . , a1, a0 are complex numbers (in particular case, they are real). The Julia set of a function f (known as J(f) set) is the limit of a set of points z, approaching infinity while iterating f(z). This set honors the French mathematician Gaston Julia [Julia (1918)] (1893–1978), who simultaneously with Pierre Fabout [Fatou (1917)] (1878–1929) in 1917–1919 wrote the fundamental papers devoted to the iteration of the functions of a complex variable. The factor set of the func- tion exhibits a property that all nearby values behave in a similar manner under the action of repeated iterations of functions. On the other hand, the Julia set exhibits values in which perturbations yield drastic changes in the sequence of iterated function values. A simple example of Julia’s set is: f(z) = z2. In what follows, we consider the following Julia set fc (z) = z2 + c, where c is constant in C. In Figs. 2.6 and 2.7, a few Julia sets are presented. In order to proceed with a more rigorous statement, we consider a complex rational function f(z) = p(z)/g(z) from the plane onto itself, where p(z), g(z) are complex polynominals. There exists a finite number of open sets F1, F2, . . . that are invariant owing to the action of f(z) having two properties: (i) a set of Fi is dense in the plane; (ii) f(z) behaves regularly on each of the sets Fi. One may Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 44. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 26 26 Deterministic Chaos in One-Dimensional Continuous Systems Fig. 2.6 The Julia set for f (z) = z2 − 1, where z is a complex number (adapted from [Douady (1986)]). Fig. 2.7 The Julia set for f (z) = z2 − 0.20 + 0.75i (adapted from [Sierpiński (1916)]). distinguish between attracting cycles and neutral cycles. The sets Fi are Fatou domains of f(z), and each of the domains has at least one critical point of f(z). The complement of F(f) (Fatou set) is the Julia set J(f). The latter one is invariant by f(z), and the iteration is repelling in a neighborhood of z, which yields chaotic deterministic iterations of f(z) on the Julia set. The Julia set may include a finite number of regular points, i.e. those whose sequence of operation is finite. There exist various equivalent descriptions of the Julia set. It can be also proved that the Julia set and the Fatou set of f are completely invariant under iterations of the holomorphic function f, Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 45. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 27 Introduction to Fractal Dynamics 27 i.e. f−1 (J(f)) = f(J(f)) = J(f), f−1 (F(f)) = f(F(f)) = F(f). (2.17) More detailed description of Fatou and Julia sets can be found in references [Barnsley (1988); Douady (1986); Lauwerier (1991); Pait- gen and Richter (1986)]. There are also a few proposals for the direct application of Julia/Fatou set theories to pure and applied sciences. The Newton’s-secants method for finding numerical solutions to the nonlinear equation F(z) = z2 − C = 0 has been proposed and then extended to solve Newton’s-secants and Tchebishev’s-secants imag- inary problems based on the Julia set theory by Tomova [Tomova (2001)]. Another application of Julia sets to switched dynamical pro- cesses has been presented by Lakhatakia [Lakhtakia (1991)]. The Julia set theory has been employed to transient chaos detection in process control systems [Russell and Alpigini (1996)]. The quaternion Julia set has been applied to generate real-time-based symmetric keys for cryptography in the reference [Rubesh Anand et al. (2009)]. 2.6 Mandelbrot’s Set Mandelbrot extended investigations of French mathematicians Fatou and Julia by studying the parameter space of quadratic polynomials in 1980 [Mandelbrot (1980)]. The Mandelbrot set is governed by the quadratic recurrence equation zn+1 = z2 n + z0, (2.18) where the orbit zn does not tend to infinity (remains bounded). Namely, a complex number z0 = c belongs to the Mandelbrot set M, when the absolute value of zn remains bounded. For example, for z0 = 1 we get the sequence {0, −1, 0, −1, 0, . . . } being bounded, whereas for z0 = 1 we obtain the sequence {0, 1, 2, 5, 26, . . . }, which is unbounded. The Mandelbrot set boundary also exhibits a smaller version of the main shape (see Fig. 2.8), and hence the fractal prop- erty of self-similarity is applied to the entire set. A zone of period-3 Mandelbrot’s set is shown in Fig. 2.9. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 46. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 28 28 Deterministic Chaos in One-Dimensional Continuous Systems Fig. 2.8 Mandelbrot’s set fc(z) = z2 +c (a) and the window (b) of Mandelbrot’s set M around point c = −1.75+0i (points belonging to set M are colored black); adapted from [Mandelbrot (1980)]. More precisely, the Mandelbrot set can be defined in the following way. Set M for fc(z) = z2+c is defined as {c} ∈ C, for which the orbit of point 0 is bounded, i.e.: (i) M = {c ∈ C : {f (n) c (0)}∞ n=0 bounded} or equivalently (ii) M = {c ∈ C : f (n) c (0) → ∞ for n → ∞}. Equivalence of definition 1 and definition 2 is yielded by the observation that lim z→∞ z2+c z = ∞, i.e. ∃ R 0 we have |z| R → |fc(c)| 2|z|. If ∀n0 ∃|f (n0) c (0)| R, ∀n n0 : |f (n0) c (0)| 2n−n0 R, i.e. f (n) c (0) → ∞. Point zero is the only one point where the deriva- tive is equal to zero. Problems regarding boundaries of the orbits should be justified using the following theorem. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 47. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 29 Introduction to Fractal Dynamics 29 Fig. 2.9 Area of period-3 of Mandelbrot’s set (it should be considered simulta- neously with Figs. 2.8 and 2.9); adapted from [Mandelbrot (1980)]. If |c| 2 and |z| ≥ c, then on orbit z → ∞. In particular, this implies that points c / ∈ M. Point c = −2 is the only one point of the circle |c| = 2, which belongs to Mandelbrot’s set. In Fig. 2.10, certain parts of the Mandelbrot’s set corresponding to the existence of the attractive periodic orbits of different orbits are presented. The presented diagram of the orbits shows what happens on the real axis of the Mandelbrot set. Each bifurcation corresponds to a new frame, which intersects the abscissa axis, and the period corresponds to a number of branches of the orbital diagram. A value of c, for which periodic attracting points of period-2 exist in the Julia set lie inside the circle |c + 1| = 1/4. It has been impossible so far to find an analytical dependence between a period and a frame for periods larger than two. The Hausdorff dimension of the Mandelbrot set boundary is equal to two [Shishikura (1998)]. There is a strong correspondence between the geometry of the Mandelbrot set at a given point and Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 48. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch02 page 30 30 Deterministic Chaos in One-Dimensional Continuous Systems Fig. 2.10 Periods of frames (Mandelbrot’s set); adapted from [Mandelbrot (1980)]. the associated Julia set. Properties and various peculiarities of the Mandelbrot set are described and illustrated in numerous publica- tions including [Lei (1990, 2000)]. The real-world applications of fractal and Mandelbrot sets can be found in physics, mechanics, eco- nomics, biomechanics and financial mathematics. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 49. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 31 Chapter 3 Introduction to Chaos and Wavelets This chapter is devoted to definitions of chaos and description of various routes to chaos including the Landau–Hopf scenario, the Ruelle–Takens–Newhouse scenario, the Feigenbaum scenario and the Pomeau–Manneville scenario. The synchronization of chaos is briefly addressed. Quantification of chaotic dynamics via Fourier analy- sis, Poincaré maps, Lyapunov exponents is described. The Melnikov method to detect strange chaotic attractors is presented. The remain- ing chapter part is focused on advantages of the wavelet analysis ver- sus the classical Fourier analysis. Properties of different wavelets are illustrated and discussed. 3.1 Routes to Chaos 3.1.1 Introduction There is no rigorous definition of chaos. Encyclopedia Britanica refers to the Greek word “χαoζ”. Poet Ovidius in his “Metamorphoses” writes: Before the seas, and this terrestrial ball, And Heav’n’s high canopy, that covers all, One was the face of Nature; if a face: Rather a rude and indigested mass: A lifeless lump, unfashion’d, and unfram’d, Of jarring seeds; and justly Chaos nam’d. No sun was lighted up, the world to view; No moon did yet her blunted horns renew: Nor yet was Earth suspended in the sky, Nor pois’d, did on her own foundations lye: Nor seas about the shores their arms had thrown. 31 Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 50. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 32 32 Deterministic Chaos in One-Dimensional Continuous Systems Therefore, Ovidius understood chaos as the infinite space existing before the appearance of all other matters. Romes understood chaos as the initial disordered matter, where a Creator introduced order and harmony, i.e. the same description for chaos holds for Romes and Greeks. Nothing has been changed up till now: G. Schuster, for instance, describes chaos as a state of disorder and irregularity [Schuster (1998)]. Nowadays, more attention is paid to the investigation of the struc- tural stochasticity (chaos is born from order), which can be exhib- ited by the deterministic dynamics of structural members in their pre-critical states. Stochasticity is implied by a complex intrinsic system dynamics and is not given as a result of the noise or fluc- tuation input. It can be treated, in some cases, as the occurrence of a turbulent behavior. It is well known that there exists relatively large amount of the scenarios of transitions into turbulent behavior being (in majority of cases) governed by the Navier–Stokes equa- tions, describing dynamics of the uncompressed fluid, which have the following form ∂u ∂ t + (u · ∇) u − ν∇2 u = − 1 ρ ∇p + f, (3.1) div u = 0, (3.2) u = 0 on D, (3.3) where u = u(xi, t), i = 1, 2, 3, p is pressure, D is boundary of the space containing a fluid, ρ is fluid density, f is external force, ν is kinematic viscosity. Energy dissipation is governed by the term ν∇2u. Equation (3.1) presents a 3D partial differential equation regarding u (velocities) with respect to a fixed system of coordinates (Euler’s approach), whereas Eq. (3.2) is responsible for the condition of the uncompressed fluid behavior, and Eq. (3.3) governs the boundary conditions. Note that there is lack of proof of the turbulent solution exis- tence to Eqs. (3.1)–(3.3), when time goes to infinity. However, there is a proof of the turbulent solutions existence for 2D equations. On the other hand, physical properties described by the Navier– Stokes equations are relatively well investigated and understood. The first work in this field was published by Reynolds [Reynolds Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 51. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 33 Introduction to Chaos and Wavelets 33 (1883)]. He introduced the following non-dimensional parameters: Reynold’s number R = UL ν , u/U — non-dimensional velocities, xi/L — non-dimensional coordinates, t = L/U — non-dimensional time, p = p/ρU2 — non-dimensional pressure, and he considered the following partial differential equation ∂u ∂t + (u · ∇) u − 1 R ∇2 u = −∇p. (3.4) Reynolds showed that an increase in R may change the fluid motion qualitatively from a regular (laminar) to disordered chaotic (turbulent) flow. In hydrodynamics, the word “turbulence” is used as a descrip- tion of the spatio-temporal chaos. It implies that chaos in a fluid is exhibited in all scales in space and time. However, the mathemati- cal description of this state belongs to one of the most tedious and complicated problems to be solved. Up to now, it is not clear how a stochastic attractor of the turbulent flow should be constructed. On the other hand, more important results have been recently obtained for the investigation of simple dynamical systems yielded, for exam- ple, by truncation of the original continuous system, and being gov- erned by ordinary differential equations as well as maps (difference equations). However, truncated systems include only timing chaos. It allows, in the first approximation, to analyze a turbulence birth, i.e. the case when the velocity field began to fluctuate in time in a disordered manner. One of the aims of the book is to illustrate and discuss various scenarios of transition of the space structural members as beams into spatio-temporal chaos for different mathematical models taken into account. In what follows, we address a few examples of transitions from regular to chaotic dynamics exhibited by simple dynamical sys- tems, in order to introduce the reader briefly to the topics covered by the book contents. 3.1.2 On chaos definitions In the beginning, we introduce a few definitions of chaos exhibited by dynamical systems. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 52. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 34 34 Deterministic Chaos in One-Dimensional Continuous Systems (i) Devaney’s definition [Devaney (1989); Eckmann and Ruelle (1985); Stewart (1989)]. The Devaney three features of chaos are formulated for a continuous map f : X → X, where X is a metric space. First (1) Devaney’s condition states that for all non-empty open subsets U and V of X there exists k (natural number) such that fk(U) ∩ V is non-empty, which defines f as transitive. Second (2) Devaney’s condition exhibits the so-called “element of regularity”, where there are periodic points of f, forming a dense subset of X. Third (3) Devaney’s condition states, that f satisfies the property of sensitive dependence on initial conditions, which means that if there is a positive real number δ such that for every x in X and every neighborhood N of X there exists a point y in N and n (nonnegative integer) such that the iterates fn(x), fn(y) are more than distance δ apart. Hence, the fundamental characteristics of chaos require three con- ditions: essential dependence on the initial conditions; mixing caused by transitivity; regularity condition implied by density of periodic points. There is a theorem saying that if f : X → X is transitive and has dense periodic points then f has sensitive dependence on ini- tial conditions, which has been proved in the reference [Banks et al. (1992)]. Let us present Devaney’s definition in a more rigorous way: let there be a metric space (x, d). Now we explain the meaning of the already introduced three con- ditions and the introduced metric space (X, d). 1. Let x ∈ X and U be an open set containing x. Map f depends essentially on the initial conditions if (∀δ 0), (∃n 0), (y ∈ U): d(f(n) (x), f(n) (y)) δ. (3.5) 2. Let f is transitive, then ∀(u, v) of open sets (∃n 0) f(n) (u)Πv = 0. (3.6) 3. Density of the periodic points means that at any arbitrary neigh- bourhood of any point in X there exists at least one periodic point. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 53. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 35 Introduction to Chaos and Wavelets 35 (ii) Baker’s map ([Driebe (1999); Fox (1997); Hasegawa and Saphir (1992)]). A map from the unit square onto itself is called the Baker map, which is a chaotic map. The name comes from the bakers’ operation. Namely, kneading is a process applied by bakers in the making of the dough. During kneading, the dough is folded in half and compressed. The horseshoe map is topologically conjugate to the baker’s map. It preserves the 2D Lebesgue measure, it is strong topological mixing and it can be understood as a two-sided operator of the symbolic dynamics of a 1D lattice. (iii) One may apply the following simple properties of chaos, i.e. there either exists the essential sensitivity on the initial conditions or one of the Lyapunov exponents measuring divergence of the neigh- borhood trajectories is positive. 3.1.3 Landau–Hopf (LH) scenario The first scenario was proposed by Landau [Landau (1944)] in 1944, and then by Hopf [Hopf (1948)] in 1948. When Reynold’s number R regarding parameters characterizing flow activity achieves the critical value R, the previously stationary flow loses its stability. For R → ∞, the velocity of occurrence of new frequencies increases (Pn0; 1), and the solution can be presented in the following form: u(x, t) = ∞ n=1 Am(x)i m (ω t+δ) , (3.7) where ω = {ω1, ω2, . . . , ωn}; n → ∞; R → ∞. Although the frequency ratio is irrational, the spectrum becomes a broad band and similar to chaotic one, i.e. an infinite quasi-periodic process of “turbulence” is observed, which has not been verified by experiments (the Couette flow and the Rayleigh–Bernard convection). The so far described LH scenario (Fig. 3.1) is associated with Hopf theory of bifurcations, which we will briefly present now. Let a motion be described by ordinary differential equations: dx dt = Fp(x), x = x1, . . . , xk, (3.8) Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
- 54. February 17, 2016 13:51 Deterministic Chaos in One-Dimensional Continuous Systems 9in x 6in b2304-ch03 page 36 36 Deterministic Chaos in One-Dimensional Continuous Systems Fig. 3.1 The LH scenario. Fig. 3.2 (a) Spiral type trajectories approaching a stable critical point; (b) Spiral type trajectories approaching a stable limit cycle. where p is a system parameter (for example, p can be an amplitude of excitation). Critical points of equation (3.8) are the points x = xc, where dxc dt = 0 , i.e. Fp(xc ) = 0. (3.9) Stability of the points defined by (3.9) is estimated via the asso- ciated linearized equations and their characteristic values λ = λ(p). Assuming that λ lies in the left half-plane, i.e. they have negative real parts, the investigated critical point is stable [Fig. 3.2(a)]. The Hopf bifurcation takes place under the condition that a com- plex conjugated pair of the eigenvalues moves from the left-hand side of the complex plane to its right-hand side. For the critical values of Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.
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