![Preface
Lattice dynamical systems (LDS) are basically infinite dimensional systems of ordi-
nary differential equations, either autonomous or nonautonomous, and are formu-
lated as ordinary differential equations on Hilbert spaces of bi-infinite sequences.
There have been many generalisations to include delayed, random and stochastic
terms as well as multi-valued terms. LDS arise in a wide range of applications with
intrinsic discrete structures such as chemical reaction, pattern recognition, image
processing, living cell systems, cellular neural networks, etc. Sometimes they are
derived as spatial discretisations of models based on partial differential equations,
but they need not arise in this way.
There is an extensive literature on lattice dynamical systems. During the 1990s
there was a strong emphasis on travelling waves in such systems and in recent
decades on attractors. This book focuses on dissipative lattice dynamical sys-
tems and their attractors of various forms such as autonomous, nonautonomous
and random. The existence of such attractors is established by showing that the
corresponding dynamical system has an appropriate kind of absorbing set and is
asymptotically compact in some way.
Asymptotic compactness is usually established by showing that the system sat-
isfies an asymptotic tails property inside the absorbing set, which essentially leads
to a total boundedness property. This approach is based on a seminal paper of
Bates, Lu and Wang [Bates et al. (2001)], which has since been used and extended
many times in a broad variety of situations. In each case the technical details are
different, but the basic idea is similar.
There is now also a very large literature on dissipative lattice dynamical systems,
especially on attractors of all kinds in such systems. We cannot hope to do justice
to all of these papers here. Instead we have focused on key papers of representative
types of lattice systems and various types of attractors. Our selection is biased by
our own interests, in particular to those dealing with biological applications. Nev-
ertheless, we believe that this book will provide the reader with a solid introduction
to field, its main results and the methods that are used to obtain them.
vii
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![4 Dissipative Lattice Dynamical Systems
where ui(t), gi and f(ui(t)) correspond to u(xi, t), g(xi) and f(u(xi, t)) for each
i ∈ Z. When the function f in (1.1) depends also on x, then the corresponding
term in (1.2) becomes fi(ui) = f(xi, u(xi, t)).
Similarly, the spatial discretisation of a wave-like equation
∂2
u
∂t2
= ν
∂2
u
∂x2
− λu + f
x, u,
∂u
∂t
+ g(x),
leads to an LDS consisting of an infinite dimensional system of second order ordinary
differential equations such as
d2
ui
dt2
= ν (ui−1 − 2ui + ui+1) − λui + fi
ui,
dui
dt
+ gi, i ∈ Z.
This can be reformulated as an infinite dimensional system of a pair of first order
ordinary differential equations
dui
dt
= vi
dvi
dt
= ν (ui−1 − 2ui + ui+1) − λui + fi (ui, vi) + gi.
The appearance of switching effects and recovery delays in systems of excitable
cells leads to reaction-diffusion systems which are technically very difficult to analyse
[Kloeden and Lorenz (2017)]. This motivated [Han and Kloeden (2016)] to study
the following lattice system with a reaction term which is switched off when a certain
threshold is exceeded and restored after a suitable recovery time:
dui
dt
= ν(ui−1 − 2ui + ui+1) + fi(t, ui)H[ςi − max
−θ≤s≤0
ui(t + s)], i ∈ Z, (1.3)
ui(t) = ϕi(t − t0), ∀ t ∈ [t0 − θ, t0], i ∈ Z, t0 ∈ R.
Here ν = 1/κ 0 is the coupling coefficient where κ is the intercellular resistance,
while ςi ∈ R is the threshold triggering the switch-off at the i-th site and ui(t + ·) ∈
C([−θ, 0], R) is the segment of ui on time interval [t − θ, t] where θ is a positive
constant.
In addition, H is the Heaviside operator
H(x) =
(
1, x ≥ 0,
0, x 0,
x ∈ R. (1.4)
To facilitate the mathematical analysis, the Heaviside function is often replaced by
a set-valued mapping χ defined on R by
χ(s) =
{0}, s 0,
[0, 1], s = 0,
{1}, s 0,
s ∈ R. (1.5)
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![Lattice dynamical systems: a preview 5
Then the lattice system (1.3) can be reformulated as the lattice differential
inclusion
d
dt
ui(t) ∈ ν(ui−1 − 2ui + ui+1) + fi(t, ui)χ ςi − max
−θ≤s≤0
ui(t + s)
.
1.2.2 Neural field models
Lattice dynamical systems need not originate by discretising an underlying PDE
as above, but could arise from an integro-differential equation such as the Amari
neural field model [Amari (1977)] (see also Chapter 3 of [Coombes et al. (2014)] by
Amari):
∂u(t, x)
∂t
= −u(t, x) +
Z
Ω
K(x − y)H (u(t, y) − ς) dy, x ∈ Ω ⊂ R,
where ς 0 is a given threshold and H is the Heaviside function defined as in (1.4).
Such continuum neural models may lose their validity in capturing detailed
dynamics at discrete sites when the discrete structures of neural systems become
dominant, so a lattice model may be more appropriate. The following lattice version
of the Amari model was introduced in [Han and Kloeden (2019a)],
d
dt
ui(t) = fi(ui(t)) +
X
j∈Zd
κi,jH(uj(t) − ς) + gi(t), i ∈ Zd
. (1.6)
When the Heaviside function is replaced by the set-valued mapping χ defined in
(1.5), the lattice system (1.6) can be reformulated as the lattice differential inclusion
d
dt
ui(t) ∈ fi(ui(t)) +
X
j∈Zd
κi,jχ(uj(t) − ς) + gi(t), i ∈ Zd
.
The Heaviside function can also be approximated by a simplifying sigmoidal
function such as
σε(x) =
1
1 + e−x/ε
, x ∈ R, 0 ε 1.
This avoids the need to introduce a differential inclusion as above. This sigmoidal
function is globally Lipschitz with the Lipschitz constant Lσ = 1
ε and does not
lead to an inclusion equation. For example, Wang, Kloeden Yang [Wang et al.
(2020a)] considered the autonomous neural field lattice system with delays
d
dt
ui(t) = fi(ui(t)) +
X
j∈Zd
κi,jσε(uj(t − θj) − ς) + gi, i ∈ Zd
.
Delays are often included in neural field models to account for the finite transmission
time of signals between neurons.
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![6 Dissipative Lattice Dynamical Systems
1.2.3 Intrinsically discrete models
Some LDSs arise naturally without involving discretisation. Instead, they may be
developed as the infinite dimensional counterparts to a finite dimensional ODE sys-
tem. For example, based on the Hopfield neural network [Hopfield (1984)] modeled
by an n-dimensional system of ODEs
µi
dui(t)
dt
= −
ui(t)
κi
+
n
X
j=1
λi,jfj(uj(t)) + gi, i = 1, ..., n,
where ui is the mean soma potential of neuron i, µi and κi are the input capacitance
of the cell membrane and transmembrane resistance, respectively. Han, Usman
Kloeden [Han et al. (2019)] considered the random Hopfield neural lattice model:
µi
dui(t)
dt
= −
ui(t)
κi
+
i+n
X
j=i−n
λi,jfj(uj(t)) + gi(ϑt(ω)), i ∈ Z,
where ϑt(ω) is a sample path of a noise process.
1.3 Sequence spaces
An LDS can be formulated as an ordinary differential equation on an appropriate
space of infinite sequences.
Let ℓ2
be the Hilbert space of real-valued square summable bi-infinite sequences
u = (ui)i∈Z with norm and inner product
∥u∥ :=
X
i∈Z
u2
i
!1/2
, ⟨u, v⟩ :=
X
i∈Z
uivi for u = (ui)i∈Z, v = (vi)i∈Z ∈ ℓ2
.
For p ≥ 1, ℓp
denotes the Banach space of real-valued p-summable bi-infinite
sequences u = (ui)i∈Z with norm
∥u∥p :=
X
i∈Z
|ui|p
!1/p
, for u = (ui)i∈Z ∈ ℓp
.
Its dual space is ℓq
, where 1
p + 1
q = 1, with the dual coupling
Ju, vK :=
X
i∈Z
uivi for u = (ui)i∈Z ∈ ℓp
, v = (vi)i∈Z ∈ ℓq
.
Similarly, ℓ∞
is the Banach space of real-valued bounded bi-infinite sequences with
norm ∥u∥∞ := supi∈Z |ui|.
One can show that ℓ2
⊂ ℓp
⊂ ℓ∞
for p ≥ 2. (Note that these inclusions are in
the opposite direction to the Lebesgue integral spaces Lp
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![Lattice dynamical systems: a preview 7
Weighted norm sequence spaces
Since ui → 0 as i → ±∞ for u = (ui)i∈Z ∈ ℓ2
, the Hilbert space ℓ2
does not include
traveling wave solutions or solutions with non-zero constant components. Similarly,
in neural models the values at distant neurons need not vanish.
Weighted sequence spaces are used to handle such dynamical behaviour. For
greater applicability these will be defined for weighted space of bi-infinite sequences
with vectorial integer indices i = (i1, · · · , id) ∈ Zd
and any p ≥ 1. In particular,
given a positive sequence of weights (ρi)i∈Zd , ℓp
ρ denotes the Banach space
ℓp
ρ :=
n
u = (ui)i∈Zd :
X
i∈Zd
ρi|ui|p
∞, ui ∈ R
o
with the norm
∥u∥p,ρ :=
X
i∈Zd
ρi|ui|p
1/p
ρ.
for u = (ui)i∈Zd ∈ ℓp
For the special case with p = 2, ∥u∥2,ρ is written as ∥u∥ρ in short.
The weights ρi are often assumed to satisfy the following assumption.
Assumption 1.1. ρi 0 for all i ∈ Zd
and ρΣ
:=
P
i∈Zd
ρi ∞.
Lemma 1.1. Let Assumption 1.1 hold. Then ℓ2
⊂ ℓ2
ρ and ∥u∥ρ ≤
√
ρΣ
∥u∥ for u
∈ ℓ2
.
Proof. Let u ∈ ℓ2
. By Assumption 1.1, 0 ρi ρΣ
for each i ∈ Zd
. Hence
ρ =
∥u∥2
X
i∈Zd
ρiu2
i ≤
X
i∈Zd
ρΣ
u2
i = ρΣ
X
i∈Zd
u2
i = ρΣ
∥u∥2
.
Lemma 1.2 ([Han et al. (2011)]). Let Assumption 1.1 hold. Then ℓp
ρ is sepa-
rable. In particular, ℓ2
ρ is a separable Hilbert space.
Proof. Separability holds because
S
N≥1 ℓN is a countable dense subset of ℓp
ρ, where
ℓN = {u = (ui)i∈Zd : ui ∈ Q for i ∈ Zd
and ui = 0 for |i| N}.
First, it is clear that
S
N≥1 ℓN is a countable subset of ℓp
ρ. Then, given any
element u = (ui)i∈Zd ∈ ℓp
ρ and any ε 0, there exists a positive integer I(ε) ∈ N
such that
X
|i|I(ε)
ρi|ui|p
εp
/2.
¯
Choose ū = (ui)i∈Zd such that ūi ∈ Q for |i| ≤ I(ε) and ūi = 0 for |i| I(ε) with
X
|i|≤I(ε)
ρi|ui − ūi|p
εp
/2.
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![8 Dissipative Lattice Dynamical Systems
Then ū ∈
S
N≥1 ℓN and
∥u − ū∥p,ρ ε.
This implies that
S
N≥1 ℓN is dense in ℓp
ρ and hence ℓp
ρ is separable.
The following additional assumption on the weights with indices i ∈ Zd
will
often also be used.
Assumption 1.2. There exist positive constants γ0 and γ1 such that
ρi±1j ≤ γ0ρi, ρi±1j − ρi ≤ γ1ρi for all i = (i1, . . . , id) ∈ Zd
, j = 1, · · · , d,
where 1j represents the vector in Zd
with the jth element equals 1, and all other
elements equal 0. For example when d = 1, [Wang (2006)] considered the weights
ρi = (1 + i2
)−c
with c 1
2 for i ∈ Z.
1.4 An illustrative lattice reaction-diffusion model
The paper of Bates, Lu Wang [Bates et al. (2001)], has had a seminal influence
on the investigation of attractors in lattice dynamical systems. The main ideas will
be briefly outlined here in simplified form.
The authors assumed that the nonlinear function f : R → R in the LDS (1.2) is
continuously differentiable, hence locally Lipschitz, with f(0) = 0 and satisfies the
dissipativity condition
xf(x) ≤ 0, x ∈ R. (1.7)
In addition, it was assumed that g = (gi)i∈Z ∈ ℓ2
.
To simplify the exposition we will assume here that f is globally Lipschitz with
Lipschitz constant Lf . Then the function F defined component wise by Fi(u) :=
f(ui) for i ∈ Z is globally Lipschitz with
∥F(u) − F(v)∥2
=
X
i∈Z
|f(ui) − f(vi)|2
≤ L2
f
X
i∈Z
|ui − vi|2
and takes values in ℓ2
since
∥F(u)∥2
=
X
i∈Z
|f(ui) − f(0)|2
≤ L2
f
X
i∈Z
|ui|2
= L2
f ∥u∥2
.
Moreover, ⟨F(u), u⟩ ≤ 0 due to (1.7).
Define the operator Λ : ℓ2
→ ℓ2
by
(Λu)i = ui−1 − 2ui + ui+1, i ∈ Z
and the operators D+
, D−
: ℓ2
→ ℓ2
by
(D+
u)i = ui+1 − ui, (D−
u)i = ui−1 − ui, i ∈ Z.
It is straightforward to check that
−Λ = D+
D−
= D−
D+
and ⟨D−
u, v⟩ = ⟨u, D+
v⟩ ∀ u, v ∈ ℓ2
,
and hence ⟨Λu, u⟩ = −∥D+
u∥2
≤ 0 for any u ∈ ℓ2
.
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![Lattice dynamical systems: a preview 9
In ℓ2
this means Λ is negative definite since ∥D+
u∥ = 0 implies that all compo-
nents ui are identical and hence u is zero in ℓ2
. Moreover, Λ is a bounded linear
operator and generates a uniformly continuous semi-group. Λ is often called the
discrete Laplace operator.
The lattice system (1.1) can be written as an ODE
du(t)
dt
= νΛu − λu + F(u) + g (1.8)
on ℓ2
, where g = (gi)i∈Z, F : ℓ2
→ ℓ2
is given component wise by Fi(u) := f(ui)
for some continuously differentiable globally Lipschitz function f : R → R with
f(0) = 0. It follows that the function on the RHS of the infinite dimensional ODE
(1.8) maps ℓ2
into itself and is globally Lipschitz on ℓ2
.
Existence and uniqueness theorems for ODEs on Banach spaces (see e.g., [Deim-
ling (1977)]) ensure the global existence and uniqueness of a solution u(t) = u(t; uo)
in ℓ2
given initial datum u(0) = uo. Moreover, u(t; uo) generates a semi-group
{φ(t)}t≥0, i.e., an autonomous semi-dynamical system, on ℓ2
.
Existence of an absorbing set
It is easy to show that the semi-group {φ(t)}t≥0 has a positive invariant absorbing
set. In fact, taking the inner product in ℓ2
of (1.8) with u = u(t; uo) gives
d
dt
∥u∥2
+ 2ν∥D+
u∥2
+ 2λ∥u∥2
= 2⟨F(u), u⟩ + 2⟨g, u⟩ ≤ −λ∥u∥2
+
1
λ
∥g∥2
,
and hence
d
dt
∥u∥2
≤ −λ∥u∥2
+
1
λ
∥g∥2
.
The Gronwall inequality then gives
∥u(t)∥2
≤ ∥uo∥2
e−λt
+
1
λ
∥g∥2
,
and hence the closed and bounded subset of ℓ2
Q :=
u ∈ ℓ2
: ∥u∥2
≤ 1 +
1
λ
∥g∥2
is a positively invariant absorbing set for the semi-group {φ(t)}t≥0 on ℓ2
.
When the function f is assumed to be locally rather than globally Lipschitz
the above inequality shows that the solutions cannot blow up and hence can be
extended without restriction into the future.
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![10 Dissipative Lattice Dynamical Systems
Asymptotic tails and asymptotic compactness
A very significant contribution of the paper [Bates et al. (2001)] was to show that
the semi-group generated by the LDS (1.8) is asymptotically compact, from which it
follows that it has a global attractor A in ℓ2
. Their method of proof has since been
adapted and used repeatedly in a large number of other papers including almost all
of those discussed in this chapter.
The first step of the proof is to derive an asymptotic tails estimate for the
solutions u(t; uo) of the LDS in the absorbing set Q.
Lemma 1.3. For every ε 0 there exist T(ε) 0 and I(ε) ∈ N such that
X
|i|I(ε)
|ui(t; uo)|
2
≤ ε
for all uo ∈ Q and t ≥ T(ε).
The proof requires a smooth cut-off function ξ : R+
→ [0, 1] with ξ(s) = 0 for
0 ≤ s ≤ 1, ξ(s) ∈ [0, 1] for 1 ≤ s ≤ 2 and ξ(s) = 1 for s ≥ 1. For a large positive
fixed integer k (to be determined in the proof) the proof uses
vi(t) = ξk(|i|)ui(t) with ξk(|i|) = ξ
|i|
k
, i ∈ Z.
Multiplying equation (1.8) by vi(t) = ξk(|i|)ui(t) and summing over i ∈ Z gives
1
2
d
dt
X
i∈Z
ξk(|i|)|ui(t)|2
+ ν⟨D+
u, D+
v⟩ + λ
X
i∈Z
ξk(|i|)u2
i (t)
=
X
i∈Z
ξk(|i|)ui(t)f(ui(t))ui(t) +
X
i∈Z
ξk(|i|)gi.
After some skillful estimates this leads to
d
dt
X
i∈Z
ξk(|i|)|ui(t)|2
+ λ
X
i∈Z
ξk(|i|)u2
i (t) ≤
C
k
+
1
λ
X
|i|≥k
g2
i ≤
1
2
ε
for k ≥ I(ε) since g = (gi)i∈Z ∈ ℓ2
. Finally, by the Gronwall inequality,
X
|i|≥2k
|ui(t)|2
≤
X
i∈Z
ξk(|i|)|ui(t)|2
≤ ε
for t ≥ T(ε) (to handle the initial condition) and k ≥ I(ε).
To obtain asymptotic compactness a sequence u(tn; u
(n)
o ) with u
(n)
o ∈ Q and tn
→ ∞ is considered. Since Q is closed and bounded subset of the Hilbert space ℓ2
it
is weakly compact. This gives a weakly convergent subsequence with a limit in Q.
The asymptotic tail estimate is then used to separate a finite number of terms from
the small tail to show that the weak limit is in fact a strong limit. The existence of
a global attractor then follows by standard results in dynamical systems theory.
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![Lattice dynamical systems: a preview 11
1.5 Outline of this book
This monograph consists of 21 chapters divided into 6 parts: Background, Laplacian
LDS, A selection of lattice models, Stochastic and random LDS, Hopfield lattice
models and LDS in biology. The main emphasis is on establishing the existence of
attractors in such systems.
The Background part consists of two chapters, including this introductory chap-
ter and another chapter on dynamical systems which provide background material
on various kinds of dynamical systems and their attractors.
The Laplacian LDS part contains 3 chapters. In Chapter 3 we investigate the
existence of global attractors in the autonomous case of the basic Laplacian lattice
model of [Bates et al. (2001)] in some detail, in particular the asymptotic tails and
asymptotic compactness arguments. Chapter 4 concerns the approximation of such
attractors, first by finite dimensional versions of the lattice model and secondly by
Euler numerical approximations. In Chapter 5 a non-autonomous Laplacian lattice
model and its pullback attractor are considered on weighted sequence spaces.
Part III collects a selection of different lattice models to provide the reader with
an overview of broad range of different kinds of lattice models as well as to provide a
technical background for later applications that involve these types of models. There
are 5 chapters. Lattice models based on delay differential equations are considered
in Chapter 6 and on set-valued differential equations in Chapter 7, while Chapter 8
deals with lattice models based on second order differential equations. In Chapter
9 discrete time lattice models, i.e., described by difference equations rather than
differential equations, are investigated, which are based on models motivated by
spatial ecology. The resulting systems involve compact rather than asymptotically
compact operators, as elsewhere in the book. The final Chapter 10 briefly presents
and states without proofs results from the literature on the finite dimension of
attractors, exponential attractors and travelling waves. The aim is to provide the
reader with a quick overview of some important topics which are tangential to our
main interests and the methods used in the book.
Stochastic and random LDS are the focus of Part IV. Chapter 11 introduces
random dynamical systems and random ordinary differential equations which gen-
erate them. Random lattice models are then considered in detail in Chapters 12
and 13. These are generated by stochastic differential equations with additive or
linear multiplicative noise which can be transformed to random ordinary differential
equations by using Ornstein-Uhlenbeck processes. Finally, in Chapter 14 an LDS
driven by fractional Brownian motion is considered.
Part V on Hopfield lattice models has 3 chapters. Chapter 15 and Chapter
16 consider deterministic Hopfield models on unweighted and weighted sequence
spaces, respectively. Approximations of attractors are investigated depending on
the number of connections of each neuron going to infinity. Chapter 17 examines
the effects of noise on lattice Hopfield models.
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![12 Dissipative Lattice Dynamical Systems
Finally in Part VI we consider LDS in biology, which is of personal interest for us,
so the models there are biased to those we ourselves have investigated. Chapter 18
considers the Fitzhugh-Nagumo lattice model in weighted sequence spaces. Then
in Chapter19 we look at Amari lattice models, where the Heaviside function is
formulated as a set-valued mapping or approximated by a sigmoidal function, while
Chapter 20 deals with a neural lattice model with nonlinear state dependent noise
coefficients. Finally, Chapter 21 focuses on lattice systems with switching effects
and delayed recovery.
1.6 Endnotes
The proof of Lemma 1.2 was taken from [Han et al. (2011)], where sequence spaces
with weighted norms are considered, see also Chapter 5 and elsewhere in this book.
Further details of the asymptotic tails argument of [Bates et al. (2001)] sketched
above will be given in later chapters. An alternative compactness argument via
the quasi-stability concept [Chueshov (2015)] was used by [Czaja (2022)] in the
sequence space ℓ2
. See [Diestel et al. (1993)] for weak compactness in the space L2
,
and [Kisielewicz (1992)] for weak compactness in spaces C.
Other applications of lattice models are given in [Afraimovich and Nekorkin
(1994); Amigo et al. (2010); Bates and Chmaj (2003); Chow and Mallet-Paret
(1995); Han and Kloeden (2019b); Kapral (1991); McBride et al. (2010)] and refer-
enced therein.
1.7 Problems
Problem 1.1. Prove that ℓ2
is dense in ℓp
for p 2 or give a counter example
otherwise.
Problem 1.2. Determine a lattice version of the scalar porous media operator
∂
∂x u∂u
∂x
media
. Does the corresponding lattice system (1.2) with the discretised porous
operator instead of the discretised Laplacian operator have a global attractor?
Problem 1.3. Prove the existence and uniqueness of a global solution to the ODE
(1.8), given f is only locally Lipschitz with appropriate growth conditions.
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![Dynamical systems 15
and Treccani (1969)]. Such systems are often generated by differential inclusions
or differential equations without uniqueness [Aubin and Cellina (1984); Smirnov
(2002)].
Definition 2.3. A set-valued autonomous dynamical system on a metric space
(X, dX) is defined in terms of an attainability set mapping (t, x) 7→ Φ(t, x) on R+
×X
satisfying
(i) compactness: Φ(t, x0) is a non-empty compact subset of X for all (t, x0) ∈
R+
× X,
(ii) initial condition: Φ(0, x0) = {x0} for all x0 ∈ X,
(iii) semi-group: Φ(s + t, x0) = Φ (s, Φ(t, x0)) for all t, s ∈ R+
and all x0 ∈ X,
(iv) upper semi-continuity in initial conditions: (t, x) 7→ Φ(t, x) is upper semi-
continuous in (t, x) ∈ R+
× X with respect to the Hausdorff semi-distance
distX, i.e.,
distX (Φ(t, x), Φ(t0, x0)) → 0 as (t, x) → (t0, x0) in R+
× X,
(v) t 7→ Φ(t, x0) is continuous in t ∈ R+
with respect to the Hausdorff metric HX
uniformly in x0 in compact subsets B ∈ Pcc(X), i.e.,
sup
x0∈B
HX (Φ(t, x0), Φ(t0, x0)) → 0 as t → t0 in R+
.
2.1.2 Two-parameter non-autonomous dynamical systems
Two abstract formulations of non-autonomous dynamical systems will be consid-
ered in this book, presented in this section and Sect. 2.1.3, respectively. The
first is a more direct generalisation of the definition of an abstract autonomous
semi-dynamical system and is based on the properties of the solution mappings
of non-autonomous differential equations. It is called a process or two-parameter
semi-group. First define
R2
≥ := {(t, t0) ∈ R × R : t ≥ t0} .
Definition 2.4. (Process) A process on a metric space (X, dX) is a mapping ψ :
R2
≥ × X → X with the following properties:
(i) initial condition: ψ(t0, t0, x0) = x0 for all x0 ∈ X and t0 ∈ R.
(ii) two-parameter semi-group property:
ψ(t2, t0, x0) = ψ(t2, t1, ψ(t1, t0, x0))
for all (t1, t0), (t2, t1) ∈ R2
≥ and x0 ∈ X.
(iii) continuity: the mapping (t, t0, x0) 7→ ψ(t, t0, x0) is continuous.
Remark 2.3. We can consider a process ψ as a two-parameter family of mappings
ψt,t0 (·) on X that forms a two-parameter semi-group under composition, i.e.,
ψt2,t0 (x) = ψt2,t1 ◦ ψt1,t0 (x), ∀ t0 ≤ t1 ≤ t2 in R.
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![Dynamical systems 17
Remark 2.5. The base system ϑ serves as a driving system which makes the cocycle
mapping non-autonomous. Skew product flows often have very nice properties, in
particular, when the base space P is compact. This occurs when the driving system
is, for example, periodic or almost periodic. It provides more detailed information
about the dynamical behaviour of the system. George Sell, a pioneering researcher
in the area, described the effect of a compact base space as being equivalent to
compactifying time, see e.g., [Sell (1971)].
Remark 2.6. The skew product flow can also be used to define a random dy-
namical system, in which the driving system ϑ is an ergodic dynamical sys-
tem (Ω, F, P, {ϑt}t∈R), i.e., the base space (Ω, F, P) is a probability space and
(t, ω) 7→ ϑt(ω) is a measurable flow which is ergodic under P, and the cocycle
mapping π : (t, ω, x) 7→ π(t, ω, x) is measurable. More details on random dynamical
systems will be given in Chapter 11.
Definition 2.8. A set-valued skew product flow (ϑ, Π) on P×X consists of a driving
dynamical system ϑ and a cocycle attainability set mapping Π : R+
× P × X →
Pcc(X) satisfying
(i) compactness: Π(t, p, x) is a non-empty compact subset of X for all t ≥ 0, p ∈ P
and x ∈ X,
(ii) initial condition: Π(0, p, x) = {x} for all p ∈ P and x ∈ X,
(iii) cocycle property: for all s, t ≥ 0, p ∈ P and x ∈ X,
Π(s + t, p, x) = Π(s, ϑt(p), Π(t, p, x)),
(iv) continuity in time: limt→s HX(Π(t, p, x), Π(s, p, x)) = 0 for all t, s ≥ 0, p ∈ P
and x ∈ X,
(v) upper semi-continuity in parameter and initial conditions
lim
p→p0,x→x0
distX (Π(t, p, x), Π(t, p0, x0)) = 0
uniformly in t ∈ [T0, T1] for any 0 ≤ T0 T1 ∞ for all (p0, x0) ∈ P × X.
2.2 Invariant sets and attractors of dynamical systems
We are interested in the long term, i.e., asymptotic, behaviour of an underlying
dynamical system. The invariant sets of a dynamical system provide us with a lot
of useful information about the dynamical behaviour of the system, in particular its
asymptotic behaviour. In this section we first provide the definitions of invariant
sets for autonomous and non-autonomous dynamical systems, and then introduce
concepts and the theory of attractors for autonomous dynamical systems, processes,
and skew product flows, respectively.
Definition 2.9. Let φ : R × X → X be an autonomous dynamical system on X.
A non-empty subset D of X is said to be invariant (positively invariant, negatively
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![22 Dissipative Lattice Dynamical Systems
Remark 2.7. Theorem 2.4 characterises and gives the existence of a pullback
attractor. Notice that the actual construction assumes nothing about the dynamics
outside the absorbing sets, i.e., in particular that it is pullback absorbing. Thus
forward attractors can be constructed by a similar pullback argument within a for-
ward absorbing set, but this provides only a necessary condition for the family of
sets obtained so to be a forward attractor. Moreover, when they exist, forward
attractors need not be unique.
To define pullback attractors for set-valued processes, denote by P(X) the collec-
tion of all families of non-empty subsets of X and let D = {Dt : Dt ⊂ X, Dt ̸ =∅}t∈R.
For any D, D̃ ∈ 2X
, the notation D̃ ⊂ D means D̃t ⊂ Dt for every t ∈ R.
Definition 2.23. A subset D of P(X) is inclusion closed if for D ∈ D and D̃ ∈
P(X), then D̃ ⊂ D implies that D̃ ∈ P(X).
Such a collection D defined in Def. 2.23 is called a universe.
Definition 2.24. Let {Ψ(t, t0)}(t,t0)∈R2
≥
be a set-valued process on X. A family of
non-empty bounded sets Q := (Qt)t∈R is said to be D-pullback absorbing for the
set-valued process Ψ, if for any D = (Dt)t∈R ∈ D and each t ∈ R, there exists some
time TD(t) 0 such that
Ψ(t, t − τ, Dt−τ ) ⊂ Qt, for all τ ≥ TD.
A family of non-empty bounded sets Q := (Qt)t∈R is said to be D-pullback attract-
ing for the set-valued process Ψ, if every D = (Dt)t∈R ∈ D satisfies
lim
τ→∞
distX(Ψ(t, t − τ, Dt−τ ), Qt) = 0.
Definition 2.25. Let {Ψ(t, t0)}(t,t0)∈R2
≥
be a set-valued process on X and let D be
a universe. A family A = (At)t∈R is said to be a global D-pullback attractor for
Ψ if
(i) At ⊂ X is compact for any t ∈ R;
(ii) A is invariant;
(iii) A is D-pullback attracting.
The existence of a pullback attractor usually relies on some asymptotically com-
pactness. In this work we will use the following definition.
Definition 2.26. A set-valued process {Ψ(t, t0)}(t,t0)∈R2
≥
is said to be D-pullback
asymptotically upper semi-compact in X if for any fixed time t ∈ R, any sequence
yn ∈ Ψ(t, t−τn, xn) has a convergent subsequence in X whenever τn → ∞ as n → ∞
and xn ∈ Dt−τn with D = (Dt)t∈R ∈ D.
The following proposition from [Caraballo and Kloeden (2009)] gives the exis-
tence of pullback attractors.
Proposition 2.2. Let {Ψ(t, t0)}(t,t0)∈R2
≥
be a set-valued process on X and let D be
a universe. Assume that
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![Dynamical systems 23
(i) Ψ(t, t0, x) is upper semi-continuous in x for any (t, t0) ∈ R2
≥,
(ii) Ψ(t, t0) is D-pullback asymptotically upper semi-compact in X,
(iii) Ψ(t, t0) has a D-pullback absorbing set Q = (Qt)t∈R ∈ D.
Then the set-valued process {Ψ(t, t0)}(t,t0)∈R2
≥
has a unique D-pullback attractor
A = (At)t∈R with its components given by
At =
s≥0
[
τ≥s
Ψ(t, t − τ, Qt−τ ).
When investigating set-valued processes it is often convenient to consider their
single-valued trajectories.
Definition 2.27. A trajectory of a set-valued process {Ψ(t, t0)}(t,t0)∈R2
≥
is a single-
valued function ψ : [t0, t1] ∩ R → X for some (t1, t0) ∈ R2
≥ such that
ψ(t) ∈ Ψ(t, s, ψ(s)) for all t0 ≤ s ≤ t ≤ t1 in R.
A trajectory is called an entire trajectory if it is a trajectory on the whole time
set R.
In the discrete time case, trajectories are simply parts of sequences. Note that
in the continuous time case trajectories are not assumed to be continuous but
this follows from the next theorem, which is a generalisation of a theorem by
Barbashin.
Theorem 2.5. (Barbashin’s Theorem) Let {Ψ(t, t0)}(t,t0)∈R2
≥
be a set-valued pro-
cess on a complete metric space (X, dX). Then
(1) there exists a trajectory from x0 to x1 ∈ Ψ(t1, t0, x0) for each (t1, t0) ∈ R2
≥ and
x0 ∈ X;
(2) trajectories of a set-valued processes are continuous functions;
(3) the set J (t1, t0, K) of all trajectories joining x0 to an arbitrary x1 ∈ Ψ(t1, t0, x0)
with x0 ∈ K is compact in C([t0, t1]; X) for all (t1, t0) ∈ R2
≥ and any non-empty
compact subset K of X.
Definition 2.28. A family D = (Dt)t∈R of non-empty sets of X is said to be
invariant for a set-valued process Ψ if Ψ(t, t0, Dt0
) = Dt for all (t, t0) ∈ R2
≥; pos-
itively invariant if Ψ(t, t0, Dt0 ) ⊂ Dt for all (t, t0) ∈ R2
≥; and strongly negatively
invariant if Dt ⊂ Ψ(t, t0, Dt0
) for all (t, t0) ∈ R2
≥.
Theorem 2.6. Let {Ψ(t, t0)}(t,t0)∈R2
≥
be a set-valued process on a complete metric
space (X, dX) and let K = (Kt)t∈R be a family of non-empty compact subsets of X,
which is Ψ-positively invariant.
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![26 Dissipative Lattice Dynamical Systems
The compact sets are the closed subsets D of E with κ(D) = 0.
Basic properties of the Kuratowski measure of noncompactness on a Banach
space include:
(i) D is bounded if and only if κ(D) ∞.
(ii) κ(D̄) = κ(D), where D̄ denotes the closure of D.
(iii) D is compact if and only if κ(D) = 0.
(iv) κ(D1 ∪ D2) = max(κ(D1), κ(D2)) for any subsets D1 and D2.
(v) κ is continuous with respect to the Hausdorff distance of sets.
(vi) κ(aD) = |a|κ(D) for any scalar a.
(vii) κ(D1 + D2) ≤ κ(D1) + κ(D2) for any subsets D1 and D2.
(viii) κ(convD) = κ(D), where convD denotes the convex hull of D.
(ix) if D1 ⊇ D2 ⊇ D3 ⊇ · · · are non-empty closed subsets of E such that κ(Dn) → 0
as n → ∞, then n≥1 Dn is non-empty and compact.
T
2.3.2 Weak convergence and weak compactness
Let (H, ∥·∥H, ⟨·, ·⟩H) be a Hilbert space, which will typically be ℓ2
or ℓ2
ρ in this book.
Convergence with respect to the norm ∥ · ∥H is often called strong convergence,
i.e., un → u∗
strongly if and only if ∥un − u∗
∥H → 0 as n → ∞. Another useful
convergence is weak convergence. A sequence {un}n∈N converges weakly to u∗
in H
if and only if
⟨h, un − u∗
⟩H → 0 as n → ∞ for all h ∈ H.
Weak convergence is often written as un ⇀ u∗
.
Essentially, weak convergence is with respect to all linear functionals on H.
In general, weak convergence does not imply strong convergence, but the following
result holds. See [Banach and Saks (1930); Okada (1984); Partington (1977); Szlenik
(1965)].
Theorem 2.8. (Banach-Saks Theorem) A bounded sequence {un}n∈N in a
Hilbert H contains a subsequence {unk
}k∈N and a point u∗
such that
1
N
N
X
k=1
unk
−→ u∗
strongly as N → ∞.
Definition 2.33. A subset K of a Hilbert space H is said to be weakly compact if
it is weakly sequentially compact, i.e., if every sequence {un}n∈N in K has a weakly
convergent subsequence unk
⇀ u∗
in K.
The following theorem is a special case of a more general result of Kakutani, see
Theorem 3.17 in [Brezis (2011)].
Theorem 2.9. A closed and bounded (in norm) subset D of a Hilbert space H is
weakly compact.
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![Dynamical systems 27
¨
¨
A special case of the Banach-Alaoglu theorem is the sequential version of the
original theorem.
Theorem 2.10. (Banach-Alaoglu Theorem) The closed unit ball of the dual
space of a separable normed vector space is sequentially compact in the weak∗
-
topology.
The following result is from [Ulger (1991), Proposition 7], see also [Diestel
(1977)].
Lemma 2.3. (Ulger’s Lemma) Let (Ω, Σ, µ) be a probabilistic space, and E be
an arbitrary Banach space. For any weakly compact subset K ⊂ E, the set
¨
f ∈ L1
(µ, E) : f(ω) ∈ K for µ-almost every ω ∈ Ω
is relatively weakly compact.
The next result is due to [Ulger (1991), Corollary 5].
Lemma 2.4. Let (Ω, Σ, µ) be a probabilistic space and E be a Banach space. Set
U :=
f ∈ L1
(µ, E) : ∥f(ω)∥E ≤ 1 for µ − a.e. ω ∈ Ω .
A sequence
fk(·) k∈N
in U ⊂ L1
(µ, E) converges weakly to f ∈ L1
(µ, E) if and
only if for any sub-sequence
fkn
(·) n∈N
given, there exists a sequence
gn(·) n∈N
with gn ∈ co
fkn , fk(n+1)
, . . . such that for µ-a.e. ω ∈ Ω,
gn(ω) −→ f(ω) (n −→ ∞) weakly in E.
2.3.3 Ascoli-Arzelà Theorem
The Ascoli-Arzelà Theorem [Green and Valentine (1960/1961)] is a crucial tool in
the study of lattice dynamical systems. Let (E, ∥ · ∥E) be a Banach space, let I be
a closed and bounded interval in R and let C(I, E) be the space of all continuous
functions f : I → E with uniform norm ∥f∥∞ = maxt∈I ∥f(t)∥E.
Definition 2.34. A subset S of C(I, E) is said to be equi-continuous if for every ε
0 there exists δ = δ(ε) 0 which is independent of f ∈ S such that ∥f(s)−f(t)∥E
ε for all s, t ∈ I with |s − t| δ and all f ∈ S.
Theorem 2.11. (Ascoli-Arzelà Theorem). A subset S of C(I, E) is relatively
compact if and only if S is equi-continuous and S(t) := {f(t) : f ∈ S} is relatively
compact in E for every t ∈ I.
The following consequence of this theorem in a Hilbert space H, which will
typically be ℓ2
or ℓ2
ρ in this book, will be used in the sequel. See, e.g. [Lebl (2016)].
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![28 Dissipative Lattice Dynamical Systems
Corollary 2.2. Let {fn(·)}n∈N be a sequence in C([0, T], H), which is uniformly
bounded and equi-Lipschitz continuous on [0, T]. Then there is an f∗
(·) ∈
C([0, T], H) and a convergent subsequence {fnk
(·)}k∈N of {fn(·)}n∈N such that
fnk
(·) → f∗
(·) strongly in C([0, T], H) as nk → ∞
d
dt
fnk
(·) ⇀
d
dt
f∗
(·) weakly in L1
([0, T], H) as nk → ∞.
2.3.4 Asymptotic compactness properties
Some kind of compactness condition is required to ensure that the omega limit sets
defining an attractor are non-empty. For a dynamical system on the finite dimen-
sional state space this is easy since the compact subsets are the closed and bounded
subsets. Then, e.g., for an autonomous dynamical system φ with a positively in-
variant, closed and bounded (hence compact) absorbing set Q, the attractor
A =
t≥0
φ(t, Q)
is the non-empty intersection of the nested compact subsets φ(s, Q) ⊂ φ(t, Q) ⊂ Q
for s t, since continuous functions map compact subsets onto compact subsets.
In infinitely dimensional state spaces, closed and bounded subsets need not be
compact, so some compactness property must come from the dynamics. A simple
property is that the mappings φ(t, ·) are compact for t 0, i.e., map closed and
bounded subsets of X onto pre-compact subsets of X. This is usually too strong for
most applications, so a weaker asymptotic compactness property is often used.
For specific examples of lattice systems, to show that the system is asymptotic
compact, one usually first shows that the lattice dynamical system satisfies an
asymptotic tails property inside an absorbing set which is positively invariant closed
and bounded convex set (such as a ball). In particular, when the state space X is
a space of bi-infinite real-valued sequences such as ℓ2
and the set Q is also convex,
then it follows that φ is asymptotically compact in Q.
Similar proofs also hold in weighted Hilbert spaces of bi-infinite real-valued
sequences such as ℓ2
ρ.
Assumption 2.1. (Asymptotic tails property: autonomous systems) Let
φ = (φi)i∈Z be an autonomous semi-dynamical system on the Hilbert space (ℓ2
, ∥·∥)
and let B be a positively invariant, closed and bounded subset of ℓ2
, which is φ-
positive invariant. Then φ is said to satisfy an asymptotic tails property in B if for
every ε 0 there exist T(ε) 0 and I(ε) ∈ N such that
X
|i|I(ε)
|φi(t, x0)|2
≤ ε ∀ x0 ∈ B and t ≥ T(ε).
Lemma 2.5. Let Assumption 2.1 hold. Then the semi-dynamical system φ is
asymptotically compact in B.
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![30 Dissipative Lattice Dynamical Systems
Set I(t, ε) := max{I1(t, ε), I2(t, ε)}. Then, using (2.1) − (2.3), for n ≥ I(t, ε) it
follows that
ψ(t, t − tn, u(n)
) − v∗
2
=
X
|i|≤I(t,ε)
ψi(t, t − tn, u(n)
) − v∗
i
2
+
X
|i|I(t,ε)
ψi(t, t − tn, u(n)
) − v∗
i
2
≤
1
2
ε + 2
X
|i|I(t,ϵ)
ψi(t, t − tn, u(n)
)
2
+ |v∗
i |2
≤ ε.
Hence v(n)
(the subsequence) is strongly convergent in ℓ2
, so ψ is pullback asymp-
totic compact in B.
2.4 End notes
There are many classical monographs on autonomous dynamical systems, see, e.g.,
[Teschl (2012)]. See Mallet-Paret, Wu, Yi Zhu, [Mallet-Paret et al. (2012)] and
[Robinson (2001)] for infinite dimensional dynamical systems. For non-autonomous
dynamical systems see [Sell (1971)], [Kloeden and Rasmussen (2011)], [Caraballo
and Han (2016)], and [Kloeden and Yang (2021)].
Proofs of most of the results on non-autonomous systems stated in this chapter
can be found in [Kloeden and Rasmussen (2011)] and [Kloeden and Yang (2021)],
with the autonomous counterparts holding as special cases.
See [Ambrosio and Tilli (2004)] for general topics on analysis in metric spaces.
2.5 Problems
Problem 2.1. Consider the attractor Ap of the autonomous scalar ODE
dx
dt
= −x x4
− 2x2
+ 1 − p
with a parameter p ∈ P = [−2, 2]. Determine the attractor Ap for each p. Then
show that the attractors Ap converge upper semi-continuously to Ap0
as p → p0,
but need not converge continuously (in the Hausdorff metric). What properties will
ensure that the attractors converge continuously?
Problem 2.2. What is the exact relationship between the asymptotic tails property
and total boundedness?
Problem 2.3. Describe the major differences between the process and skew-
product flow formulations of non-autonomous dynamical systems. In what scenarios
is one more convenient than the other?
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![Chapter 3
Lattice Laplacian models
A lattice reaction-diffusion model is a lattice dynamical system obtained by spa-
tially discretising the Laplacian operator in a parabolic partial differential equation
modelling a reaction-diffusion equation such as equation (1.1) in Chapter 1. Bates,
Lu Wang [Bates et al. (2001)] investigated dynamical behaviour of the lattice dy-
namical system (LDS) based on this reaction-diffusion equation and their results,
which have profoundly influenced the development of the theory of dissipative LDS,
will be presented here.
Consider the autonomous LDS
dui
dt
= ν (ui−1 − 2ui + ui+1) + f(ui) + gi, i ∈ Z, (3.1)
in the space ℓ2
(which was defined in Section 1.3), which will be investigated here
under the following assumptions.
Assumption 3.1. The function f : R → R is a continuously differentiable function
satisfying
f(s)s ≤ −αs2
∀ s ∈ R,
for some α 0.
Assumption 3.2. The function g = (gi)i∈Z ∈ ℓ2
.
Remark 3.1. Since f is smooth, the Assumption 3.1 implies that f(0) = 0.
3.1 The discrete Laplace operator
The Laplacian operator on one-dimensional spatial domain is just the second
derivative. Using central difference quotient to approximate it leads to the one-
dimensional discrete Laplace operator on an appropriate sequence space.
For any u = (ui)i∈Z ∈ ℓ2
, the discrete Laplace operator Λ is defined from ℓ2
to
ℓ2
component wise by
(Λu)i = ui−1 − 2ui + ui+1, i ∈ Z. (3.2)
33
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Dissipative Lattice Dynamical Systems Peter Kloeden Xiaoying Han
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- 7. INTERDISCIPLINARY MATHEMATICAL SCIENCES* Series Editor: Jinqiao Duan (Illinois Institute of Technology, Chicago, USA) Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin, Charles Doering, Paul Fischer, Andrei V. Fursikov, Xiaofan Li, Sergey V. Lototsky, Fred R. McMorris, Daniel Schertzer, Bjorn Schmalfuss, Yuefei Wang, Xiangdong Ye, and Jerzy Zabczyk Published Vol. 22 Dissipative Lattice Dynamical Systems by Xiaoying Han & Peter Kloeden Vol. 21 An Introduction to Nonautonomous Dynamical Systems and their Attractors by Peter Kloeden & Meihua Yang Vol. 20 Stochastic PDEs and Modelling of Multiscale Complex System eds. Xiaopeng Chen, Yan Lv & Wei Wang Vol. 19 Kernel-based Approximation Methods using MATLAB by Gregory Fasshauer & Michael McCourt Vol. 18 Global Attractors of Non-Autonomous Dynamical and Control Systems (Second Edition) by David N Cheban Vol. 17 Festschrift Masatoshi Fukushima: In Honor of Masatoshi Fukushima’s Sanju eds. Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda & Toshihiro Uemura Vol. 16 Hilbert–Huang Transform and Its Applications (Second Edition) eds. Norden E Huang & Samuel S P Shen Vol. 15 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis: Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with Their Numerical Simulations eds. Changpin Li, Yujiang Wu & Ruisong Ye Vol. 14 Recent Developments in Computational Finance: Foundations, Algorithms and Applications eds. Thomas Gerstner & Peter Kloeden Vol. 13 Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan eds. Tusheng Zhang & Xunyu Zhou Vol. 12 New Trends in Stochastic Analysis and Related Topics: A Volume in Honour of Professor K D Elworthy eds. Huaizhong Zhao & Aubrey Truman *For the complete list of titles in this series, please go to http://www.worldscientific.com/series/ims Downloaded from www.worldscientific.com
- 9. 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 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Interdisciplinary Mathematical Sciences — Vol. 22 DISSIPATIVE LATTICE DYNAMICAL SYSTEMS Copyright © 2023 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. ISBN 978-981-126-775-8 (hardcover) ISBN 978-981-126-776-5 (ebook for institutions) ISBN 978-981-126-777-2 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13182#t=suppl Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore Downloaded from www.worldscientific.com
- 10. Dedicated to my parents (XH) Dedicated to the memory of Karin Wahl-Kloeden (PEK) Downloaded from www.worldscientific.com
- 11. T T Th h hi i is s s p p pa a ag g ge e e i i in n nt t te e en n nt t ti i io o on n na a al l ll l ly y y l l le e ef f ft t t b b bl l la a an n nk k k Downloaded from www.worldscientific.com
- 12. Preface Lattice dynamical systems (LDS) are basically infinite dimensional systems of ordi- nary differential equations, either autonomous or nonautonomous, and are formu- lated as ordinary differential equations on Hilbert spaces of bi-infinite sequences. There have been many generalisations to include delayed, random and stochastic terms as well as multi-valued terms. LDS arise in a wide range of applications with intrinsic discrete structures such as chemical reaction, pattern recognition, image processing, living cell systems, cellular neural networks, etc. Sometimes they are derived as spatial discretisations of models based on partial differential equations, but they need not arise in this way. There is an extensive literature on lattice dynamical systems. During the 1990s there was a strong emphasis on travelling waves in such systems and in recent decades on attractors. This book focuses on dissipative lattice dynamical sys- tems and their attractors of various forms such as autonomous, nonautonomous and random. The existence of such attractors is established by showing that the corresponding dynamical system has an appropriate kind of absorbing set and is asymptotically compact in some way. Asymptotic compactness is usually established by showing that the system sat- isfies an asymptotic tails property inside the absorbing set, which essentially leads to a total boundedness property. This approach is based on a seminal paper of Bates, Lu and Wang [Bates et al. (2001)], which has since been used and extended many times in a broad variety of situations. In each case the technical details are different, but the basic idea is similar. There is now also a very large literature on dissipative lattice dynamical systems, especially on attractors of all kinds in such systems. We cannot hope to do justice to all of these papers here. Instead we have focused on key papers of representative types of lattice systems and various types of attractors. Our selection is biased by our own interests, in particular to those dealing with biological applications. Nev- ertheless, we believe that this book will provide the reader with a solid introduction to field, its main results and the methods that are used to obtain them. vii Downloaded from www.worldscientific.com
- 13. viii Dissipative Lattice Dynamical Systems At the end of each chapter we have included a section with some problems. These are not meant to be exercises for students, although some could serve that purpose. Their main goal is to draw the reader’s attention to important issues for clarification and extension of the material and proofs in the book. Some are fairly straightforward, but others are serious research problems, in some cases very difficult ones. Auburn, Xiaoying Han Tübingen Peter Kloeden June 2022 Downloaded from www.worldscientific.com
- 14. Contents Preface vii Background 1 1. Lattice dynamical systems: a preview 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Examples of lattice dynamical systems . . . . . . . . . . . . . . . . 3 1.2.1 PDE based models . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Neural field models . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Intrinsically discrete models . . . . . . . . . . . . . . . . . 6 1.3 Sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 An illustrative lattice reaction-diffusion model . . . . . . . . . . . 8 1.5 Outline of this book . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. Dynamical systems 13 2.1 Abstract dynamical systems . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Autonomous dynamical systems . . . . . . . . . . . . . . . 14 2.1.2 Two-parameter non-autonomous dynamical systems . . . . 15 2.1.3 Skew product flows . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Invariant sets and attractors of dynamical systems . . . . . . . . . 17 2.2.1 Attractors of autonomous semi-dynamical systems . . . . 18 2.2.2 Attractors of processes . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Attractors of skew product flows . . . . . . . . . . . . . . 24 2.3 Compactness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Kuratowski measure of non-compactness . . . . . . . . . . 25 2.3.2 Weak convergence and weak compactness . . . . . . . . . 26 ix Downloaded from www.worldscientific.com
- 15. x Dissipative Lattice Dynamical Systems 2.3.3 Ascoli-Arzelà Theorem . . . . . . . . . . . . . . . . . . . . 27 2.3.4 Asymptotic compactness properties . . . . . . . . . . . . . 28 2.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Laplacian LDS 31 3. Lattice Laplacian models 33 3.1 The discrete Laplace operator . . . . . . . . . . . . . . . . . . . . 33 3.2 The autonomous reaction-diffusion LDS . . . . . . . . . . . . . . 34 3.2.1 Existence of an absorbing set . . . . . . . . . . . . . . . . 35 3.2.2 Asymptotic tails property . . . . . . . . . . . . . . . . . . 35 3.3 Nonautonomous lattice reaction-diffusion LDS . . . . . . . . . . . 37 3.4 p-Laplacian reaction-diffusion LDS . . . . . . . . . . . . . . . . . . 39 3.4.1 Discretised p-Laplacian . . . . . . . . . . . . . . . . . . . . 40 3.4.2 Existence and uniqueness of solutions . . . . . . . . . . . . 41 3.4.3 Existence of a global attractor . . . . . . . . . . . . . . . . 41 3.5 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4. Approximation of attractors of LDS 45 4.1 Finite dimensional approximations . . . . . . . . . . . . . . . . . . 45 4.2 Upper semi-continuous convergence of the finite dimensional attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Numerical approximation of lattice attractors . . . . . . . . . . . . 50 4.4 Finite dimensional approximations of the IES . . . . . . . . . . . . 57 4.4.1 Finite dimensional numerical attractors A (h) N . . . . . . . . 57 4.4.2 Upper semi continuous convergence . . . . . . . . . . . . 58 4.4.3 Convergence of numerical attractors . . . . . . . . . . . . 61 4.5 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5. Non-autonomous Laplacian lattice systems in weighted sequence spaces 63 5.1 The discrete Laplacian on weighted sequence spaces . . . . . . . . 64 5.2 Generation of a non-autonomous dynamical system on ℓ2 ρ . . . . . 66 5.2.1 Existence and uniqueness of solutions in ℓ2 . . . . . . . . . 67 5.2.2 Lipschitz continuity of solutions in initial data in the ℓ2 ρ norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.3 Generation of semi-group on ℓ2 p . . . . . . . . . . . . . . . 70 5.3 Existence of pullback attractors . . . . . . . . . . . . . . . . . . . 70 5.3.1 Existence of an absorbing set . . . . . . . . . . . . . . . . 70 5.3.2 Asymptotic tails and asymptotic compactness . . . . . . . 71 Downloaded from www.worldscientific.com
- 16. Contents xi 5.4 Uniformly strictly contracting Laplacian lattice systems . . . . . . 74 5.5 Forward dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.6 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A selection of lattice models 79 6. Lattice dynamical systems with delays 81 6.1 The coefficient terms . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . 82 6.2.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . 83 6.2.2 An a prior estimate of solutions . . . . . . . . . . . . . . . 84 6.2.3 Uniqueness of solutions . . . . . . . . . . . . . . . . . . . . 87 6.3 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.3.1 Tails estimate . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.2 Existence of the global attractor . . . . . . . . . . . . . . . 91 6.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7. Set-valued lattice models 93 7.1 Set-valued lattice system on ℓ2 . . . . . . . . . . . . . . . . . . . . 93 7.2 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.3 Set-valued semi-dynamical systems with compact values . . . . . . 99 7.4 Existence of a global attractor . . . . . . . . . . . . . . . . . . . . 102 7.5 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8. Second order lattice dynamical systems 105 8.1 Existence and uniqueness of solution . . . . . . . . . . . . . . . . . 107 8.2 Existence of a bounded absorbing set . . . . . . . . . . . . . . . . 108 8.3 Existence of a global attractor . . . . . . . . . . . . . . . . . . . . 111 8.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9. Discrete time lattice systems 117 9.1 Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.1.2 Existence of a global attractor . . . . . . . . . . . . . . . . 121 9.1.3 Finite dimensional approximations of the global attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.2 Convergent sequences of interconnection weights . . . . . . . . . . 125 9.3 Lattice systems with finitely many interconnections . . . . . . . . 127 Downloaded from www.worldscientific.com
- 17. xii Dissipative Lattice Dynamical Systems 9.4 Nonautonomous systems . . . . . . . . . . . . . . . . . . . . . . . 128 9.4.1 Existence of a pullback attractor . . . . . . . . . . . . . . 129 9.4.2 Existence of a forward ω-limit sets . . . . . . . . . . . . . 130 9.5 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10. Three topics in brief 133 10.1 Finite dimension of lattice attractors . . . . . . . . . . . . . . . . . 133 10.2 Exponential attractors . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.2.1 Application to general lattice systems . . . . . . . . . . . . 136 10.2.2 First order lattice systems . . . . . . . . . . . . . . . . . . 138 10.2.3 Partly dissipative lattice systems . . . . . . . . . . . . . . 139 10.2.4 Second order lattice systems . . . . . . . . . . . . . . . . . 140 10.3 Traveling waves for lattice neural field equations . . . . . . . . . . 141 10.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Stochastic and Random LDS 147 11. Random dynamical systems 149 11.1 Random ordinary differential equations . . . . . . . . . . . . . . . 149 11.1.1 RODEs with canonical noise . . . . . . . . . . . . . . . . . 150 11.1.2 Existence und uniqueness results for RODEs . . . . . . . . 150 11.2 Random dynamical systems . . . . . . . . . . . . . . . . . . . . . . 151 11.3 Random attractors for general RDS in weighted spaces . . . . . . 154 11.4 Stochastic differential equations as RODEs . . . . . . . . . . . . . 156 11.5 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12. Stochastic LDS with additive noise 159 12.1 Random dynamical systems generated by stochastic LDS . . . . . 159 12.1.1 Ornstein-Uhlenbeck process . . . . . . . . . . . . . . . . . 160 12.1.2 Transformation to a random ordinary differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.1.3 Existence and uniqueness of solutions . . . . . . . . . . . . 163 12.1.4 Random dynamical systems generated by random LDS . . . . . . . . . . . . . . . . . . . . . . . . . . 167 12.2 Existence of global random attractors in weighted space . . . . . . 168 12.2.1 Existence of tempered random bounded absorbing sets . . . . . . . . . . . . . . . . . . . . . . . . . 169 12.2.2 Existence of global random attractors . . . . . . . . . . . . 171 Downloaded from www.worldscientific.com
- 18. Contents xiii 12.3 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 13. Stochastic LDS with multiplicative noise 175 13.1 Random dynamical systems generated by stochastic LDS . . . . . 175 13.1.1 Transformation to a random LDS . . . . . . . . . . . . . . 176 13.1.2 Existence and uniqueness of solutions to the random LDS . . . . . . . . . . . . . . . . . . . . . . . . . . 177 13.1.3 Random dynamical systems generated by random LDS . . . . . . . . . . . . . . . . . . . . . . . . . . 183 13.2 Existence of global random attractors in weighted space . . . . . . 184 13.2.1 Existence of tempered random bounded absorbing sets . . . . . . . . . . . . . . . . . . . . . . . . . 185 13.2.2 Existence of global random attractors . . . . . . . . . . . . 187 13.3 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 13.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 14. Stochastic lattice models with fractional Brownian motions 193 14.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 14.2 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 197 14.2.1 Standing assumptions . . . . . . . . . . . . . . . . . . . . . 198 14.2.2 Properties of operators . . . . . . . . . . . . . . . . . . . . 198 14.2.3 Existence of mild solutions . . . . . . . . . . . . . . . . . . 200 14.3 Generation of an RDS . . . . . . . . . . . . . . . . . . . . . . . . . 203 14.4 Exponential stability of the trivial solution . . . . . . . . . . . . . 204 14.4.1 Existence of trivial solutions . . . . . . . . . . . . . . . . . 205 14.4.2 The cut–off strategy . . . . . . . . . . . . . . . . . . . . . 206 14.4.3 Preliminary estimates . . . . . . . . . . . . . . . . . . . . . 207 14.4.4 Exponential stability . . . . . . . . . . . . . . . . . . . . . 208 14.5 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 14.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Hopfield Lattice Models 215 15. Hopfield neural network lattice model 217 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 15.2 Formulation as an ODE . . . . . . . . . . . . . . . . . . . . . . . . 218 15.3 Existence of attractors . . . . . . . . . . . . . . . . . . . . . . . . . 220 15.4 Finite dimensional approximations . . . . . . . . . . . . . . . . . . 225 15.5 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 15.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Downloaded from www.worldscientific.com
- 19. xiv Dissipative Lattice Dynamical Systems 16. The Hopfield lattice model in weighted spaces 237 16.1 Reformulation as an ODE on ℓ2 ρ . . . . . . . . . . . . . . . . . . . 238 16.2 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . 239 16.3 Existence of attractors . . . . . . . . . . . . . . . . . . . . . . . . . 241 16.3.1 Existence of absorbing sets . . . . . . . . . . . . . . . . . . 241 16.3.2 Asymptotic compactness . . . . . . . . . . . . . . . . . . . 242 16.4 Upper semi-continuity of attractors in λi,j . . . . . . . . . . . . . . 245 16.5 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 16.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 17. A random Hopfield lattice model 253 17.1 Basic properties of solutions . . . . . . . . . . . . . . . . . . . . . 253 17.2 Existence of random attractors . . . . . . . . . . . . . . . . . . . . 259 17.3 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 17.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 LDS in Biology 267 18. FitzHugh-Nagumo lattice model 269 18.1 Generation of a semi-dynamical system on ℓ2 ρ × ℓ2 ρ . . . . . . . . . 270 18.1.1 Existence and uniqueness of solutions in ℓ2 × ℓ2 . . . . . . 270 18.1.2 Lipschitz ℓ2 ρ-continuity of solutions in initial data . . . . . 271 18.1.3 Existence and uniqueness of solutions in ℓ2 ρ × ℓ2 ρ . . . . . . 273 18.2 Existence of a global attractor . . . . . . . . . . . . . . . . . . . . 273 18.2.1 Existence of an absorbing set . . . . . . . . . . . . . . . . 273 18.2.2 Asymptotic tails and asymptotic compactness . . . . . . . 274 18.3 Limit of the global attractors as δ → 0 . . . . . . . . . . . . . . . 276 18.3.1 Uniform bound on the global attractors . . . . . . . . . . 277 18.3.2 Pre-compactness of the union of the global attractors . . . 279 18.3.3 Upper semi-continuity of the global attractors . . . . . . . 281 18.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 18.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 19. The Amari lattice neural field model 285 19.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 19.1.1 Standing assumptions . . . . . . . . . . . . . . . . . . . . . 287 19.1.2 Basic estimates . . . . . . . . . . . . . . . . . . . . . . . . 287 19.2 Set-valued lattice systems . . . . . . . . . . . . . . . . . . . . . . . 289 19.2.1 Inflated lattice systems . . . . . . . . . . . . . . . . . . . . 290 19.2.2 Relations between Heaviside, sigmoid, and inflated . . . . 291 Downloaded from www.worldscientific.com
- 20. Contents xv 19.3 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 291 19.3.1 The sigmoidal lattice system . . . . . . . . . . . . . . . . . 291 19.3.2 The inflated system . . . . . . . . . . . . . . . . . . . . . . 292 19.3.3 The set-valued lattice system . . . . . . . . . . . . . . . . 292 19.4 Convergence of sigmoidal solutions . . . . . . . . . . . . . . . . . . 292 19.4.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . 292 19.4.2 The convergence theorem . . . . . . . . . . . . . . . . . . 297 19.5 Set-valued dynamical systems with compact values . . . . . . . . . 301 19.6 Attractors of the sigmoidal and lattice systems . . . . . . . . . . . 307 19.6.1 Comparison of the attractors . . . . . . . . . . . . . . . . 308 19.7 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 19.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 20. Stochastic neural field models with nonlinear noise 311 20.1 Well-posedness of the LDS in ℓ2 ρ . . . . . . . . . . . . . . . . . . . 311 20.2 Existence of mean-square solutions . . . . . . . . . . . . . . . . . . 314 20.2.1 Solutions of the truncated system . . . . . . . . . . . . . . 315 20.2.2 Existence of a global mean-square solution . . . . . . . . . 316 20.3 Weak pullback mean random attractors . . . . . . . . . . . . . . . 325 20.3.1 Preliminaries on mean random dynamical systems . . . . . 325 20.3.2 Existence of absorbing sets . . . . . . . . . . . . . . . . . . 326 20.3.3 Existence of a mean random attractor . . . . . . . . . . . 328 20.3.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 20.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 21. Lattice systems with switching effects and delayed recovery 331 21.1 Set-valued delay differential inclusions . . . . . . . . . . . . . . . . 332 21.2 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 336 21.3 Long term behavior of lattice system . . . . . . . . . . . . . . . . . 339 21.3.1 Generation of set-valued process . . . . . . . . . . . . . . . 340 21.3.2 Existence of an absorbing set . . . . . . . . . . . . . . . . 341 21.3.3 Tail estimations . . . . . . . . . . . . . . . . . . . . . . . . 342 21.3.4 Existence of a nonautonomous attractor . . . . . . . . . . 344 21.4 End notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 21.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Bibliography 349 Index 359 Downloaded from www.worldscientific.com
- 21. T T Th h hi i is s s p p pa a ag g ge e e i i in n nt t te e en n nt t ti i io o on n na a al l ll l ly y y l l le e ef f ft t t b b bl l la a an n nk k k Downloaded from www.worldscientific.com
- 23. T T Th h hi i is s s p p pa a ag g ge e e i i in n nt t te e en n nt t ti i io o on n na a al l ll l ly y y l l le e ef f ft t t b b bl l la a an n nk k k Downloaded from www.worldscientific.com
- 24. Chapter 1 Lattice dynamical systems: a preview 1.1 Introduction Lattice dynamical systems (LDS), as considered in this book, are essentially infinite dimensional systems of ordinary differential equations (ODEs). In particular, they can be formulated as ordinary differential equations on a Hilbert or Banach space of bi-infinite sequences. The infinite dimensionality of this state space takes their investigation beyond the usual qualitative theory of ODEs, but its special nature often means that such an investigation is not as technically complicated as for the corresponding partial differential equation (PDE) from which an LDS may have been derived. This allows a greater focus on the dynamical behaviour of such systems. Not all lattice dynamical systems originate by discretising an underlying PDE. Some may arise by discretising integral equations, others are intrinsically discrete. 1.2 Examples of lattice dynamical systems Lattice dynamical systems may arise from discretisation of continuum models or as infinite dimensional counterparts of finite ODE models. 1.2.1 PDE based models A classical lattice dynamical system is based on a reaction-diffusion equation ∂u ∂t = ν ∂2 u ∂x2 − λu + f(u) + g(x), (1.1) where λ and ν are positive constants, on a one-dimensional domain R. It is obtained by using a central difference quotient to discretise the Laplacian. Setting the stepsize scaled to equal 1 leads to the infinite dimensional system of ordinary differential equations dui dt = ν (ui−1 − 2ui + ui+1) − λui + f(ui) + gi, i ∈ Z, (1.2) 3 Downloaded from www.worldscientific.com
- 25. 4 Dissipative Lattice Dynamical Systems where ui(t), gi and f(ui(t)) correspond to u(xi, t), g(xi) and f(u(xi, t)) for each i ∈ Z. When the function f in (1.1) depends also on x, then the corresponding term in (1.2) becomes fi(ui) = f(xi, u(xi, t)). Similarly, the spatial discretisation of a wave-like equation ∂2 u ∂t2 = ν ∂2 u ∂x2 − λu + f x, u, ∂u ∂t + g(x), leads to an LDS consisting of an infinite dimensional system of second order ordinary differential equations such as d2 ui dt2 = ν (ui−1 − 2ui + ui+1) − λui + fi ui, dui dt + gi, i ∈ Z. This can be reformulated as an infinite dimensional system of a pair of first order ordinary differential equations dui dt = vi dvi dt = ν (ui−1 − 2ui + ui+1) − λui + fi (ui, vi) + gi. The appearance of switching effects and recovery delays in systems of excitable cells leads to reaction-diffusion systems which are technically very difficult to analyse [Kloeden and Lorenz (2017)]. This motivated [Han and Kloeden (2016)] to study the following lattice system with a reaction term which is switched off when a certain threshold is exceeded and restored after a suitable recovery time: dui dt = ν(ui−1 − 2ui + ui+1) + fi(t, ui)H[ςi − max −θ≤s≤0 ui(t + s)], i ∈ Z, (1.3) ui(t) = ϕi(t − t0), ∀ t ∈ [t0 − θ, t0], i ∈ Z, t0 ∈ R. Here ν = 1/κ 0 is the coupling coefficient where κ is the intercellular resistance, while ςi ∈ R is the threshold triggering the switch-off at the i-th site and ui(t + ·) ∈ C([−θ, 0], R) is the segment of ui on time interval [t − θ, t] where θ is a positive constant. In addition, H is the Heaviside operator H(x) = ( 1, x ≥ 0, 0, x 0, x ∈ R. (1.4) To facilitate the mathematical analysis, the Heaviside function is often replaced by a set-valued mapping χ defined on R by χ(s) = {0}, s 0, [0, 1], s = 0, {1}, s 0, s ∈ R. (1.5) Downloaded from www.worldscientific.com
- 26. Lattice dynamical systems: a preview 5 Then the lattice system (1.3) can be reformulated as the lattice differential inclusion d dt ui(t) ∈ ν(ui−1 − 2ui + ui+1) + fi(t, ui)χ ςi − max −θ≤s≤0 ui(t + s) . 1.2.2 Neural field models Lattice dynamical systems need not originate by discretising an underlying PDE as above, but could arise from an integro-differential equation such as the Amari neural field model [Amari (1977)] (see also Chapter 3 of [Coombes et al. (2014)] by Amari): ∂u(t, x) ∂t = −u(t, x) + Z Ω K(x − y)H (u(t, y) − ς) dy, x ∈ Ω ⊂ R, where ς 0 is a given threshold and H is the Heaviside function defined as in (1.4). Such continuum neural models may lose their validity in capturing detailed dynamics at discrete sites when the discrete structures of neural systems become dominant, so a lattice model may be more appropriate. The following lattice version of the Amari model was introduced in [Han and Kloeden (2019a)], d dt ui(t) = fi(ui(t)) + X j∈Zd κi,jH(uj(t) − ς) + gi(t), i ∈ Zd . (1.6) When the Heaviside function is replaced by the set-valued mapping χ defined in (1.5), the lattice system (1.6) can be reformulated as the lattice differential inclusion d dt ui(t) ∈ fi(ui(t)) + X j∈Zd κi,jχ(uj(t) − ς) + gi(t), i ∈ Zd . The Heaviside function can also be approximated by a simplifying sigmoidal function such as σε(x) = 1 1 + e−x/ε , x ∈ R, 0 ε 1. This avoids the need to introduce a differential inclusion as above. This sigmoidal function is globally Lipschitz with the Lipschitz constant Lσ = 1 ε and does not lead to an inclusion equation. For example, Wang, Kloeden Yang [Wang et al. (2020a)] considered the autonomous neural field lattice system with delays d dt ui(t) = fi(ui(t)) + X j∈Zd κi,jσε(uj(t − θj) − ς) + gi, i ∈ Zd . Delays are often included in neural field models to account for the finite transmission time of signals between neurons. Downloaded from www.worldscientific.com
- 27. 6 Dissipative Lattice Dynamical Systems 1.2.3 Intrinsically discrete models Some LDSs arise naturally without involving discretisation. Instead, they may be developed as the infinite dimensional counterparts to a finite dimensional ODE sys- tem. For example, based on the Hopfield neural network [Hopfield (1984)] modeled by an n-dimensional system of ODEs µi dui(t) dt = − ui(t) κi + n X j=1 λi,jfj(uj(t)) + gi, i = 1, ..., n, where ui is the mean soma potential of neuron i, µi and κi are the input capacitance of the cell membrane and transmembrane resistance, respectively. Han, Usman Kloeden [Han et al. (2019)] considered the random Hopfield neural lattice model: µi dui(t) dt = − ui(t) κi + i+n X j=i−n λi,jfj(uj(t)) + gi(ϑt(ω)), i ∈ Z, where ϑt(ω) is a sample path of a noise process. 1.3 Sequence spaces An LDS can be formulated as an ordinary differential equation on an appropriate space of infinite sequences. Let ℓ2 be the Hilbert space of real-valued square summable bi-infinite sequences u = (ui)i∈Z with norm and inner product ∥u∥ := X i∈Z u2 i !1/2 , ⟨u, v⟩ := X i∈Z uivi for u = (ui)i∈Z, v = (vi)i∈Z ∈ ℓ2 . For p ≥ 1, ℓp denotes the Banach space of real-valued p-summable bi-infinite sequences u = (ui)i∈Z with norm ∥u∥p := X i∈Z |ui|p !1/p , for u = (ui)i∈Z ∈ ℓp . Its dual space is ℓq , where 1 p + 1 q = 1, with the dual coupling Ju, vK := X i∈Z uivi for u = (ui)i∈Z ∈ ℓp , v = (vi)i∈Z ∈ ℓq . Similarly, ℓ∞ is the Banach space of real-valued bounded bi-infinite sequences with norm ∥u∥∞ := supi∈Z |ui|. One can show that ℓ2 ⊂ ℓp ⊂ ℓ∞ for p ≥ 2. (Note that these inclusions are in the opposite direction to the Lebesgue integral spaces Lp ). Downloaded from www.worldscientific.com
- 28. Lattice dynamical systems: a preview 7 Weighted norm sequence spaces Since ui → 0 as i → ±∞ for u = (ui)i∈Z ∈ ℓ2 , the Hilbert space ℓ2 does not include traveling wave solutions or solutions with non-zero constant components. Similarly, in neural models the values at distant neurons need not vanish. Weighted sequence spaces are used to handle such dynamical behaviour. For greater applicability these will be defined for weighted space of bi-infinite sequences with vectorial integer indices i = (i1, · · · , id) ∈ Zd and any p ≥ 1. In particular, given a positive sequence of weights (ρi)i∈Zd , ℓp ρ denotes the Banach space ℓp ρ := n u = (ui)i∈Zd : X i∈Zd ρi|ui|p ∞, ui ∈ R o with the norm ∥u∥p,ρ := X i∈Zd ρi|ui|p 1/p ρ. for u = (ui)i∈Zd ∈ ℓp For the special case with p = 2, ∥u∥2,ρ is written as ∥u∥ρ in short. The weights ρi are often assumed to satisfy the following assumption. Assumption 1.1. ρi 0 for all i ∈ Zd and ρΣ := P i∈Zd ρi ∞. Lemma 1.1. Let Assumption 1.1 hold. Then ℓ2 ⊂ ℓ2 ρ and ∥u∥ρ ≤ √ ρΣ ∥u∥ for u ∈ ℓ2 . Proof. Let u ∈ ℓ2 . By Assumption 1.1, 0 ρi ρΣ for each i ∈ Zd . Hence ρ = ∥u∥2 X i∈Zd ρiu2 i ≤ X i∈Zd ρΣ u2 i = ρΣ X i∈Zd u2 i = ρΣ ∥u∥2 . Lemma 1.2 ([Han et al. (2011)]). Let Assumption 1.1 hold. Then ℓp ρ is sepa- rable. In particular, ℓ2 ρ is a separable Hilbert space. Proof. Separability holds because S N≥1 ℓN is a countable dense subset of ℓp ρ, where ℓN = {u = (ui)i∈Zd : ui ∈ Q for i ∈ Zd and ui = 0 for |i| N}. First, it is clear that S N≥1 ℓN is a countable subset of ℓp ρ. Then, given any element u = (ui)i∈Zd ∈ ℓp ρ and any ε 0, there exists a positive integer I(ε) ∈ N such that X |i|I(ε) ρi|ui|p εp /2. ¯ Choose ū = (ui)i∈Zd such that ūi ∈ Q for |i| ≤ I(ε) and ūi = 0 for |i| I(ε) with X |i|≤I(ε) ρi|ui − ūi|p εp /2. Downloaded from www.worldscientific.com
- 29. 8 Dissipative Lattice Dynamical Systems Then ū ∈ S N≥1 ℓN and ∥u − ū∥p,ρ ε. This implies that S N≥1 ℓN is dense in ℓp ρ and hence ℓp ρ is separable. The following additional assumption on the weights with indices i ∈ Zd will often also be used. Assumption 1.2. There exist positive constants γ0 and γ1 such that ρi±1j ≤ γ0ρi, ρi±1j − ρi ≤ γ1ρi for all i = (i1, . . . , id) ∈ Zd , j = 1, · · · , d, where 1j represents the vector in Zd with the jth element equals 1, and all other elements equal 0. For example when d = 1, [Wang (2006)] considered the weights ρi = (1 + i2 )−c with c 1 2 for i ∈ Z. 1.4 An illustrative lattice reaction-diffusion model The paper of Bates, Lu Wang [Bates et al. (2001)], has had a seminal influence on the investigation of attractors in lattice dynamical systems. The main ideas will be briefly outlined here in simplified form. The authors assumed that the nonlinear function f : R → R in the LDS (1.2) is continuously differentiable, hence locally Lipschitz, with f(0) = 0 and satisfies the dissipativity condition xf(x) ≤ 0, x ∈ R. (1.7) In addition, it was assumed that g = (gi)i∈Z ∈ ℓ2 . To simplify the exposition we will assume here that f is globally Lipschitz with Lipschitz constant Lf . Then the function F defined component wise by Fi(u) := f(ui) for i ∈ Z is globally Lipschitz with ∥F(u) − F(v)∥2 = X i∈Z |f(ui) − f(vi)|2 ≤ L2 f X i∈Z |ui − vi|2 and takes values in ℓ2 since ∥F(u)∥2 = X i∈Z |f(ui) − f(0)|2 ≤ L2 f X i∈Z |ui|2 = L2 f ∥u∥2 . Moreover, ⟨F(u), u⟩ ≤ 0 due to (1.7). Define the operator Λ : ℓ2 → ℓ2 by (Λu)i = ui−1 − 2ui + ui+1, i ∈ Z and the operators D+ , D− : ℓ2 → ℓ2 by (D+ u)i = ui+1 − ui, (D− u)i = ui−1 − ui, i ∈ Z. It is straightforward to check that −Λ = D+ D− = D− D+ and ⟨D− u, v⟩ = ⟨u, D+ v⟩ ∀ u, v ∈ ℓ2 , and hence ⟨Λu, u⟩ = −∥D+ u∥2 ≤ 0 for any u ∈ ℓ2 . Downloaded from www.worldscientific.com
- 30. Lattice dynamical systems: a preview 9 In ℓ2 this means Λ is negative definite since ∥D+ u∥ = 0 implies that all compo- nents ui are identical and hence u is zero in ℓ2 . Moreover, Λ is a bounded linear operator and generates a uniformly continuous semi-group. Λ is often called the discrete Laplace operator. The lattice system (1.1) can be written as an ODE du(t) dt = νΛu − λu + F(u) + g (1.8) on ℓ2 , where g = (gi)i∈Z, F : ℓ2 → ℓ2 is given component wise by Fi(u) := f(ui) for some continuously differentiable globally Lipschitz function f : R → R with f(0) = 0. It follows that the function on the RHS of the infinite dimensional ODE (1.8) maps ℓ2 into itself and is globally Lipschitz on ℓ2 . Existence and uniqueness theorems for ODEs on Banach spaces (see e.g., [Deim- ling (1977)]) ensure the global existence and uniqueness of a solution u(t) = u(t; uo) in ℓ2 given initial datum u(0) = uo. Moreover, u(t; uo) generates a semi-group {φ(t)}t≥0, i.e., an autonomous semi-dynamical system, on ℓ2 . Existence of an absorbing set It is easy to show that the semi-group {φ(t)}t≥0 has a positive invariant absorbing set. In fact, taking the inner product in ℓ2 of (1.8) with u = u(t; uo) gives d dt ∥u∥2 + 2ν∥D+ u∥2 + 2λ∥u∥2 = 2⟨F(u), u⟩ + 2⟨g, u⟩ ≤ −λ∥u∥2 + 1 λ ∥g∥2 , and hence d dt ∥u∥2 ≤ −λ∥u∥2 + 1 λ ∥g∥2 . The Gronwall inequality then gives ∥u(t)∥2 ≤ ∥uo∥2 e−λt + 1 λ ∥g∥2 , and hence the closed and bounded subset of ℓ2 Q := u ∈ ℓ2 : ∥u∥2 ≤ 1 + 1 λ ∥g∥2 is a positively invariant absorbing set for the semi-group {φ(t)}t≥0 on ℓ2 . When the function f is assumed to be locally rather than globally Lipschitz the above inequality shows that the solutions cannot blow up and hence can be extended without restriction into the future. Downloaded from www.worldscientific.com
- 31. 10 Dissipative Lattice Dynamical Systems Asymptotic tails and asymptotic compactness A very significant contribution of the paper [Bates et al. (2001)] was to show that the semi-group generated by the LDS (1.8) is asymptotically compact, from which it follows that it has a global attractor A in ℓ2 . Their method of proof has since been adapted and used repeatedly in a large number of other papers including almost all of those discussed in this chapter. The first step of the proof is to derive an asymptotic tails estimate for the solutions u(t; uo) of the LDS in the absorbing set Q. Lemma 1.3. For every ε 0 there exist T(ε) 0 and I(ε) ∈ N such that X |i|I(ε) |ui(t; uo)| 2 ≤ ε for all uo ∈ Q and t ≥ T(ε). The proof requires a smooth cut-off function ξ : R+ → [0, 1] with ξ(s) = 0 for 0 ≤ s ≤ 1, ξ(s) ∈ [0, 1] for 1 ≤ s ≤ 2 and ξ(s) = 1 for s ≥ 1. For a large positive fixed integer k (to be determined in the proof) the proof uses vi(t) = ξk(|i|)ui(t) with ξk(|i|) = ξ |i| k , i ∈ Z. Multiplying equation (1.8) by vi(t) = ξk(|i|)ui(t) and summing over i ∈ Z gives 1 2 d dt X i∈Z ξk(|i|)|ui(t)|2 + ν⟨D+ u, D+ v⟩ + λ X i∈Z ξk(|i|)u2 i (t) = X i∈Z ξk(|i|)ui(t)f(ui(t))ui(t) + X i∈Z ξk(|i|)gi. After some skillful estimates this leads to d dt X i∈Z ξk(|i|)|ui(t)|2 + λ X i∈Z ξk(|i|)u2 i (t) ≤ C k + 1 λ X |i|≥k g2 i ≤ 1 2 ε for k ≥ I(ε) since g = (gi)i∈Z ∈ ℓ2 . Finally, by the Gronwall inequality, X |i|≥2k |ui(t)|2 ≤ X i∈Z ξk(|i|)|ui(t)|2 ≤ ε for t ≥ T(ε) (to handle the initial condition) and k ≥ I(ε). To obtain asymptotic compactness a sequence u(tn; u (n) o ) with u (n) o ∈ Q and tn → ∞ is considered. Since Q is closed and bounded subset of the Hilbert space ℓ2 it is weakly compact. This gives a weakly convergent subsequence with a limit in Q. The asymptotic tail estimate is then used to separate a finite number of terms from the small tail to show that the weak limit is in fact a strong limit. The existence of a global attractor then follows by standard results in dynamical systems theory. Downloaded from www.worldscientific.com
- 32. Lattice dynamical systems: a preview 11 1.5 Outline of this book This monograph consists of 21 chapters divided into 6 parts: Background, Laplacian LDS, A selection of lattice models, Stochastic and random LDS, Hopfield lattice models and LDS in biology. The main emphasis is on establishing the existence of attractors in such systems. The Background part consists of two chapters, including this introductory chap- ter and another chapter on dynamical systems which provide background material on various kinds of dynamical systems and their attractors. The Laplacian LDS part contains 3 chapters. In Chapter 3 we investigate the existence of global attractors in the autonomous case of the basic Laplacian lattice model of [Bates et al. (2001)] in some detail, in particular the asymptotic tails and asymptotic compactness arguments. Chapter 4 concerns the approximation of such attractors, first by finite dimensional versions of the lattice model and secondly by Euler numerical approximations. In Chapter 5 a non-autonomous Laplacian lattice model and its pullback attractor are considered on weighted sequence spaces. Part III collects a selection of different lattice models to provide the reader with an overview of broad range of different kinds of lattice models as well as to provide a technical background for later applications that involve these types of models. There are 5 chapters. Lattice models based on delay differential equations are considered in Chapter 6 and on set-valued differential equations in Chapter 7, while Chapter 8 deals with lattice models based on second order differential equations. In Chapter 9 discrete time lattice models, i.e., described by difference equations rather than differential equations, are investigated, which are based on models motivated by spatial ecology. The resulting systems involve compact rather than asymptotically compact operators, as elsewhere in the book. The final Chapter 10 briefly presents and states without proofs results from the literature on the finite dimension of attractors, exponential attractors and travelling waves. The aim is to provide the reader with a quick overview of some important topics which are tangential to our main interests and the methods used in the book. Stochastic and random LDS are the focus of Part IV. Chapter 11 introduces random dynamical systems and random ordinary differential equations which gen- erate them. Random lattice models are then considered in detail in Chapters 12 and 13. These are generated by stochastic differential equations with additive or linear multiplicative noise which can be transformed to random ordinary differential equations by using Ornstein-Uhlenbeck processes. Finally, in Chapter 14 an LDS driven by fractional Brownian motion is considered. Part V on Hopfield lattice models has 3 chapters. Chapter 15 and Chapter 16 consider deterministic Hopfield models on unweighted and weighted sequence spaces, respectively. Approximations of attractors are investigated depending on the number of connections of each neuron going to infinity. Chapter 17 examines the effects of noise on lattice Hopfield models. Downloaded from www.worldscientific.com
- 33. 12 Dissipative Lattice Dynamical Systems Finally in Part VI we consider LDS in biology, which is of personal interest for us, so the models there are biased to those we ourselves have investigated. Chapter 18 considers the Fitzhugh-Nagumo lattice model in weighted sequence spaces. Then in Chapter19 we look at Amari lattice models, where the Heaviside function is formulated as a set-valued mapping or approximated by a sigmoidal function, while Chapter 20 deals with a neural lattice model with nonlinear state dependent noise coefficients. Finally, Chapter 21 focuses on lattice systems with switching effects and delayed recovery. 1.6 Endnotes The proof of Lemma 1.2 was taken from [Han et al. (2011)], where sequence spaces with weighted norms are considered, see also Chapter 5 and elsewhere in this book. Further details of the asymptotic tails argument of [Bates et al. (2001)] sketched above will be given in later chapters. An alternative compactness argument via the quasi-stability concept [Chueshov (2015)] was used by [Czaja (2022)] in the sequence space ℓ2 . See [Diestel et al. (1993)] for weak compactness in the space L2 , and [Kisielewicz (1992)] for weak compactness in spaces C. Other applications of lattice models are given in [Afraimovich and Nekorkin (1994); Amigo et al. (2010); Bates and Chmaj (2003); Chow and Mallet-Paret (1995); Han and Kloeden (2019b); Kapral (1991); McBride et al. (2010)] and refer- enced therein. 1.7 Problems Problem 1.1. Prove that ℓ2 is dense in ℓp for p 2 or give a counter example otherwise. Problem 1.2. Determine a lattice version of the scalar porous media operator ∂ ∂x u∂u ∂x media . Does the corresponding lattice system (1.2) with the discretised porous operator instead of the discretised Laplacian operator have a global attractor? Problem 1.3. Prove the existence and uniqueness of a global solution to the ODE (1.8), given f is only locally Lipschitz with appropriate growth conditions. Downloaded from www.worldscientific.com
- 34. Chapter 2 Dynamical systems Background material on autonomous and non-autonomous dynamical systems is summarised in this Chapter for the reader’s convenience. More details and proofs can be found in the literature mentioned in the Endnotes. Throughout this chapter, let (X, dX) be a complete metric space, and let Pcc(X) denote the collection of all non-empty compact subsets of X. The distance between two points x, y ∈ X is given by dX(x, y) = dX(y, x) (symmetric). We define the distance between a point x ∈ X and a non-empty compact subset B in X by distX(x, B) := inf b∈B dX(x, b). Remark 2.1. The mapping b 7→ dX(x, b) is continuous for x fixed, in fact |dX(x, b) − dX(x, c)| ≤ dX(b, c), and the subset B is non-empty and compact, so the inf can be replaced by min here, i.e., it is actually attained. Then we define the distance of a compact subset A from a compact subset B by distX(A, B) := sup a∈A distX(a, B) = sup a∈A inf b∈B dX(a, b), which is sometimes written as H∗ X(A, B) and called the Hausdorff separation or semi-distance of A from B. Remark 2.2. The function a 7→ distX(a, B) is continuous for fixed B and the set A is compact, so the sup here can be replaced by max. The Hausdorff separation, distX(A, B) satisfies the triangle inequality distX(A, B) ≤ distX(A, C) + distX(C, B). 13 Downloaded from www.worldscientific.com
- 35. 14 Dissipative Lattice Dynamical Systems However, distX(A, B) is not a metric, since it can be equal to zero without the sets being equal, i.e., distX(A, B) = 0 if A ⊂ B. Define HX(A, B) := max {distX(A, B), distX(B, A)} . This is a metric on Pcc(X) called the Hausdorff metric. Theorem 2.1. (Pcc(X), HX) is a complete metric space. 2.1 Abstract dynamical systems In this section we introduce the concepts of autonomous and non-autonomous dy- namical systems, respectively. In particular, definitions of single-valued and set- valued autonomous dynamical systems are given in Sect. 2.1.1, the process formu- lation of single-valued and set-valued non-autonomous dynamical systems are given in Sect. 2.1.2, and the skew product formulation of single-valued and set-valued non-autonomous dynamical systems are given in Sect. 2.1.3. 2.1.1 Autonomous dynamical systems Definition 2.1. An autonomous dynamical system on a metric space (X, dX) is given by mapping φ : R × X → X, which satisfies the properties: (i) initial condition: φ(0, x0) = x0 for all x0 ∈ X, (ii) group under composition: φ(s + t, x0) = φ(s, φ(t, x0)) for all s, t ∈ R, x0 ∈ X, (iii) continuity: the mapping (t, x) 7→ φ(t, x) is continuous at all points (t0, x0) ∈ R × X. Throughout this book, define R+ := {t ∈ R : t ≥ 0}. Definition 2.2. An autonomous semi-dynamical dynamical system on a metric space (X, dX) is given by mapping φ : R+ × X → X, which satisfies the properties: (i) initial condition: φ(0, x0) = x0 for all x0 ∈ X, (ii) semi-group under composition: φ(s + t, x0) = φ(s, φ(t, x0)) for all s, t ∈ R+ , x0 ∈ X, (iii) continuity: the mapping (t, x) 7→ φ(t, x) is continuous at all points (t0, x0) ∈ R+ × X. Next we provide the definition of set-valued autonomous dynamical systems. There is a large literature for autonomous set-valued dynamical systems, which are often called set-valued semi-groups or general dynamical systems, see e.g., [Szegö Downloaded from www.worldscientific.com
- 36. Dynamical systems 15 and Treccani (1969)]. Such systems are often generated by differential inclusions or differential equations without uniqueness [Aubin and Cellina (1984); Smirnov (2002)]. Definition 2.3. A set-valued autonomous dynamical system on a metric space (X, dX) is defined in terms of an attainability set mapping (t, x) 7→ Φ(t, x) on R+ ×X satisfying (i) compactness: Φ(t, x0) is a non-empty compact subset of X for all (t, x0) ∈ R+ × X, (ii) initial condition: Φ(0, x0) = {x0} for all x0 ∈ X, (iii) semi-group: Φ(s + t, x0) = Φ (s, Φ(t, x0)) for all t, s ∈ R+ and all x0 ∈ X, (iv) upper semi-continuity in initial conditions: (t, x) 7→ Φ(t, x) is upper semi- continuous in (t, x) ∈ R+ × X with respect to the Hausdorff semi-distance distX, i.e., distX (Φ(t, x), Φ(t0, x0)) → 0 as (t, x) → (t0, x0) in R+ × X, (v) t 7→ Φ(t, x0) is continuous in t ∈ R+ with respect to the Hausdorff metric HX uniformly in x0 in compact subsets B ∈ Pcc(X), i.e., sup x0∈B HX (Φ(t, x0), Φ(t0, x0)) → 0 as t → t0 in R+ . 2.1.2 Two-parameter non-autonomous dynamical systems Two abstract formulations of non-autonomous dynamical systems will be consid- ered in this book, presented in this section and Sect. 2.1.3, respectively. The first is a more direct generalisation of the definition of an abstract autonomous semi-dynamical system and is based on the properties of the solution mappings of non-autonomous differential equations. It is called a process or two-parameter semi-group. First define R2 ≥ := {(t, t0) ∈ R × R : t ≥ t0} . Definition 2.4. (Process) A process on a metric space (X, dX) is a mapping ψ : R2 ≥ × X → X with the following properties: (i) initial condition: ψ(t0, t0, x0) = x0 for all x0 ∈ X and t0 ∈ R. (ii) two-parameter semi-group property: ψ(t2, t0, x0) = ψ(t2, t1, ψ(t1, t0, x0)) for all (t1, t0), (t2, t1) ∈ R2 ≥ and x0 ∈ X. (iii) continuity: the mapping (t, t0, x0) 7→ ψ(t, t0, x0) is continuous. Remark 2.3. We can consider a process ψ as a two-parameter family of mappings ψt,t0 (·) on X that forms a two-parameter semi-group under composition, i.e., ψt2,t0 (x) = ψt2,t1 ◦ ψt1,t0 (x), ∀ t0 ≤ t1 ≤ t2 in R. Downloaded from www.worldscientific.com
- 37. 16 Dissipative Lattice Dynamical Systems Remark 2.4. For an autonomous system, a process reduces to ψ(t, t0, x0) = φ(t − t0, x0), since the solutions depend only on the elapsed time t − t0, i.e., just one parameter instead of independently on the actual time t and the initial time t0, i.e., two parameters. Definition 2.5. (Set-valued process) A set-valued process on metric space (X, dX) is given by a mapping R2 ≥ × X ∋ (t, t0, x) 7→ Ψ(t, t0, x0) ∈ Pcc(X) such that (i) Ψ(t0, t0, x0) = {x0} for all x0 ∈ X and all t0 ∈ R, (ii) Ψ(t2, t0, x0) = Ψ (t2, t1, Ψ(t1, t0, x0)) for all t0 ≤ t1 ≤ t2 in R and all x0 ∈ X, (iii) (t, t0, x0) 7→ Ψ(t, t0, x0) is upper semi-continuous in (t, t0, x0) ∈ R2 ≥ × X with respect to the Hausdorff semi-distance distX, i.e., distX (Ψ(s, s0, y0), Ψ(t, t0, x0)) → 0 as (s, s0, y0) → (t, t0, x0) in R2 ≥ × X, (iv) t 7→ Ψ(t, t0, x0) is continuous in t ∈ R with respect to the Hausdorff metric uniformly in (t0, x0) in compact subsets of R × X, i.e., sup (t0,x0)∈B HX (Ψ(s, t0, x0), Ψ(t, t0, x0)) → 0 as s → t in R for each B ∈ Pcc(R × X). 2.1.3 Skew product flows A skew product flow consists of an autonomous dynamical system (full group) on a base space P, which is the source of the non-autonomity in a cocycle mapping acting on a state space X. The autonomous dynamical system is often called the driving system. Throughout this section, suppose that (P, dP) is a complete metric space and consider the time set R. Definition 2.6. An autonomous dynamical system ϑ = (ϑt)t∈R acting on the base space (P, dP) is a driving dynamical system if (i) ϑ0(p) = p all p ∈ P, (ii) ϑs+t(p) = ϑs ◦ ϑt(p) for all p ∈ P and s, t ∈ R, (iii) (t, p) 7→ ϑt(p) is continuous for all p ∈ P and s, t ∈ R. Definition 2.7. A skew product flow (ϑ, π) on P×X consists of a driving dynamical system ϑ = {ϑt}t∈R acting on the base space (P, dP) and a cocycle mapping π : R+ × P × X → X acting on the state space (X, dX) satisfying (i) initial condition: π(0, p, x) = x for all p ∈ P and x ∈ X, (ii) cocycle property: for all s, t ∈ R+ , p ∈ P and x ∈ X, π(s + t, p, x) = π(s, ϑt(p), π(t, p, x)), (iii) continuity: (t, p, x) 7→ π(t, p, x) is continuous. Downloaded from www.worldscientific.com
- 38. Dynamical systems 17 Remark 2.5. The base system ϑ serves as a driving system which makes the cocycle mapping non-autonomous. Skew product flows often have very nice properties, in particular, when the base space P is compact. This occurs when the driving system is, for example, periodic or almost periodic. It provides more detailed information about the dynamical behaviour of the system. George Sell, a pioneering researcher in the area, described the effect of a compact base space as being equivalent to compactifying time, see e.g., [Sell (1971)]. Remark 2.6. The skew product flow can also be used to define a random dy- namical system, in which the driving system ϑ is an ergodic dynamical sys- tem (Ω, F, P, {ϑt}t∈R), i.e., the base space (Ω, F, P) is a probability space and (t, ω) 7→ ϑt(ω) is a measurable flow which is ergodic under P, and the cocycle mapping π : (t, ω, x) 7→ π(t, ω, x) is measurable. More details on random dynamical systems will be given in Chapter 11. Definition 2.8. A set-valued skew product flow (ϑ, Π) on P×X consists of a driving dynamical system ϑ and a cocycle attainability set mapping Π : R+ × P × X → Pcc(X) satisfying (i) compactness: Π(t, p, x) is a non-empty compact subset of X for all t ≥ 0, p ∈ P and x ∈ X, (ii) initial condition: Π(0, p, x) = {x} for all p ∈ P and x ∈ X, (iii) cocycle property: for all s, t ≥ 0, p ∈ P and x ∈ X, Π(s + t, p, x) = Π(s, ϑt(p), Π(t, p, x)), (iv) continuity in time: limt→s HX(Π(t, p, x), Π(s, p, x)) = 0 for all t, s ≥ 0, p ∈ P and x ∈ X, (v) upper semi-continuity in parameter and initial conditions lim p→p0,x→x0 distX (Π(t, p, x), Π(t, p0, x0)) = 0 uniformly in t ∈ [T0, T1] for any 0 ≤ T0 T1 ∞ for all (p0, x0) ∈ P × X. 2.2 Invariant sets and attractors of dynamical systems We are interested in the long term, i.e., asymptotic, behaviour of an underlying dynamical system. The invariant sets of a dynamical system provide us with a lot of useful information about the dynamical behaviour of the system, in particular its asymptotic behaviour. In this section we first provide the definitions of invariant sets for autonomous and non-autonomous dynamical systems, and then introduce concepts and the theory of attractors for autonomous dynamical systems, processes, and skew product flows, respectively. Definition 2.9. Let φ : R × X → X be an autonomous dynamical system on X. A non-empty subset D of X is said to be invariant (positively invariant, negatively Downloaded from www.worldscientific.com
- 39. 18 Dissipative Lattice Dynamical Systems invariant (resp.)) under φ if φ(t, D) = (⊂, ⊃ (resp.))D for all t ∈ R, where φ(t, D) := [ x∈D {φ(t, x)}. Definition 2.10. Let ψ : R2 ≥ × X → X be a process on X. A family of non-empty subsets D = (Dt)t∈R of X is said to be invariant (positively invariant, negatively invariant (resp.)) under ψ if ψ(t, t0, Dt0 ) = (⊂, ⊃ (resp.))Dt for all (t, t0) ∈ R2 ≥. Definition 2.11. Let (ϑ, π) be a skew product flow on P × X. A family D = (Dp)p∈P of non-empty subsets Dp of X is said to be invariant (positively invariant, negatively invariant (resp.)) under π if π (t, p, Dp) = (⊂, ⊃ (resp.))Dϑt(p) for all p ∈ P and t ∈ R+ . There are two types of invariance concepts for set-valued dynamical systems, one depending on the full sets, and the other involving only certain trajectories, referred to as strong and weak invariance, respectively. In this book we only consider the strong invariance. In fact, replacing the dynamical system φ, process ψ and skew product flow π by set-valued dynamical system Φ, set-valued process Ψ and set-valued skew product flow Π, respectively, in the above definitions, give the corresponding definitions of strongly invariance, strongly positive invariance and strongly negative invariance under set-valued dynamical systems, processes, and skew product flows, respectively. 2.2.1 Attractors of autonomous semi-dynamical systems Definition 2.12. An entire path of a semi-dynamical system {φ(t)}t≥0 on a com- plete metric state space (X, dX) is a mapping e : R → X with the property that e(t) = φ(t − s, e(s)) ∀ (t, s) ∈ R2 ≥. Note that t−s ∈ R+ , the time set on which the semi-dynamical system φ is defined. However, the entire solution e is defined for all t ∈ R, not just in R+ . Lemma 2.1. Let K be a compact invariant set w.r.t. a semi-dynamical system {φ(t)}t≥0. Then for every x ∈ K there exists an entire solution ex : R → K with ex(0) = x. The ω-limit sets of a semi-dynamical system characterise its asymptotic be- haviour as t → ∞. Definition 2.13. (Omega-limit sets) The ω-limit set of a bounded set B ⊂ X is defined by ω(B) = {x ∈ X : ∃ tk → ∞, yk ∈ B with φ(tk, yk) → x} . Downloaded from www.worldscientific.com
- 40. Dynamical systems 19 The ω-limit sets have the following properties. Theorem 2.2. For a non-empty bounded subset B of X, ω(B) = t≥0 [ s≥t φ(s, B) . An attractor is an invariant set of special interest since it contains all the long term dynamics of a dissipative dynamical system, i.e., it is where every thing ends up. In particular, it contains the omega-limit set ω(B) of every non-empty bounded subset B of the state space X. In addition, an attractor is the omega-limit set of a neighbourhood of itself, i.e., it attracts a neighbourhood of itself. This additional stability property distinguishes an attractor from omega-limit sets in general. Definition 2.14. A global attractor of a semi-dynamical system {φ(t)}t≥0 is a non- empty compact invariant set A of X which attracts all non-empty bounded subsets B of X, i.e., distX (φ(t, B), A) → 0 as t → ∞. An attractor may have a very complicated geometrical shape, e.g., the fractal dimensional set in the Lorenz ODE system. It is often easier to determine a closed and bounded absorbing set with a simpler geometrical shape such as a ball, in particular in infinite dimensional spaces, where closed and bounded subsets are much more common and easily determined than compact subsets. Definition 2.15. A non-empty subset Q of X is called an absorbing set of φ if for every non-empty bounded subset B of X there exists a TB ≥ 0 such that φ(t, B) ⊂ Q ∀ t ≥ TB. All of the future dynamics is in Q, which need not be invariant, but often it is positively invariant, i.e., φ(t, Q) ⊂ Q for all t ∈ R+ . Some additional compactness property of the semi-group φ in Q such as its asymptotic compactness is then needed to ensure the non-emptiness of the attractor. Definition 2.16. A semi-dynamical system {φ(t)}t≥0 on a complete metric space (X, dX) is said to be asymptotically compact if, for every sequence {tk}k∈N in R+ with tk → ∞ as k → ∞ and every bounded sequence {xk}k∈N in X, the sequence {φ(tk, xk)}k∈N has a convergent subsequence. Theorem 2.3. Let {φ(t)}t≥0 be an autonomous semi-dynamical system on a com- plete metric space (X, dX) which is asymptotically compact and has a closed and bounded absorbing set Q ⊂ X. Then φ has an attractor A, which is contained in Q and is given by A = t≥0 [ s≥t φ(s, Q). Downloaded from www.worldscientific.com
- 41. 20 Dissipative Lattice Dynamical Systems Moreover, if Q is positively invariant then A = t≥0 φ(t, Q). In particular, A = ω(Q). An attractor, when it exists, is characterised by the bounded entire paths of the systems. Corollary 2.1. Let A be an attractor of a semi-dynamical system {φ(t)}t≥0. Then for every a ∈ A there exists an entire solution ea : R → A with ea(0) = a. Definition 2.17. A compact subset A is said to be a (strong) global attractor for a set-valued dynamical system Φ, if it satisfies (i) Φ(t, A) = A for all t. (ii) A attracts every bounded subset of X. 2.2.2 Attractors of processes Now consider a two-parameter semi-group or process {ψ(t, t0)}(t,t0)∈R2 ≥ on the met- ric state space X and time set R. An entire path of a two-parameter semi-group or process ψ is defined analogously to an entire solution of an autonomous semi-dynamical system. Definition 2.18. An entire solution of a process {ψ(t, t0)}(t,t0)∈R2 ≥ on a com- plete metric space (X, dX) is a mapping e : R → X with the property that e(t) = ψ(t, t0, e(t0)) for all (t, t0) ∈ R2 ≥. Steady state solutions are entire solutions, but there may be other interesting bounded entire solutions. Lemma 2.2. Let D = (Dt)t∈R be a ψ-invariant family of subsets of X. Then for any t0 ∈ R and any x0 ∈ Dt0 , there exists an entire solution ex0,t0 : R → X of ψ such that ex0,t0 (t0) = x0 and ex0,t0 (t) ∈ Dt for all t ∈ R. An attractor for a process ψ should thus be a family A = (At)t∈R of non-empty compact subsets At of X, which is ψ-invariant, i.e., ψ(t, t0, At0 ) = At for all (t, t0) ∈ R2 ≥. There is, however, a problem with convergence. There are two possibilities, one with pullback convergence and one with forward convergence. Definition 2.19. A family A = (At)t∈R of non-empty compact subsets of X, which is ψ-invariant, is called a Downloaded from www.worldscientific.com
- 42. Dynamical systems 21 • pullback attractor if it pullback attracts all bounded subsets B of X, i.e., lim t0→−∞ distX (ψ(t, t0, B), At) = 0, (fixed t) • forward attractor if it forward attracts all bounded subsets B of X, i.e., lim t→∞ distX (ψ(t, t0, B), At) = 0, (fixed t0). We say that a pullback attractor A = (At)t∈R is uniformly bounded if S t∈R At is bounded or, equivalently, if there is a common bounded subset B of X such that At ⊆ B for all t ∈ R. We have the following characterisation of a uniformly bounded pullback attractor. Proposition 2.1. A uniformly bounded pullback attractor A = (At)t∈R of a process ψ is uniquely determined by the bounded entire solutions of the process, i.e., a0 ∈ At0 ⇐⇒ ∃ a bounded entire solution e(·) with e(t0) = a0. To handle non-uniformities in the dynamics, which are typical in non- autonomous behaviour, we consider a pullback absorbing family of sets instead of a single set and assume that it absorbs a family of non-empty bounded subsets, which do not grow too quickly. The following definition is needed to ensure that the component sets in the non-autonomous family do not grow too quickly. Definition 2.20. A family B = (Bt)t∈R of non-empty bounded subsets Bt of X is said to have subexponential growth if lim sup |t|→∞ ∥Bt∥ec|t| = 0 ∀c 0, where ∥Bt∥ = sup b∈Bt ∥b∥. In this case it is called a tempered family. Definition 2.21. A family Q = (Qt)t∈R of non-empty subsets of X is called a pullback absorbing family for a process ψ on X if for each t ∈ R and every tempered family B = (Bt)t∈R of non-empty bounded subsets of X there exists a Tt,B ∈ R+ such that ψ (t, t0, Bt0 ) ⊆ Qt for all t0 ≤ t − Tt,B. Definition 2.22. A process ψ on a Banach space X is said to be pullback asymp- totically compact if, for each t ∈ R, each sequence {tk}k∈N in R with tk ≤ t and tk → −∞ as k → ∞, and each bounded sequence {xk}k∈N in X, the sequence {ψ(t, tk, xk)}k∈N has a convergent subsequence. Theorem 2.4. (Existence of a pullback attractor) Suppose that a process ψ on a complete metric space (X, dX) is pullback asymptotically compact and has a ψ- positive invariant pullback absorbing family Q = (Qt)t∈R of compact sets. Then ψ has a global pullback attractor A = (At)t∈R with component subsets determined by At = t0≤t ψ (t, t0, Qt0 ) for each t ∈ R. Moreover, if A is uniformly bounded then it is unique. Downloaded from www.worldscientific.com
- 43. 22 Dissipative Lattice Dynamical Systems Remark 2.7. Theorem 2.4 characterises and gives the existence of a pullback attractor. Notice that the actual construction assumes nothing about the dynamics outside the absorbing sets, i.e., in particular that it is pullback absorbing. Thus forward attractors can be constructed by a similar pullback argument within a for- ward absorbing set, but this provides only a necessary condition for the family of sets obtained so to be a forward attractor. Moreover, when they exist, forward attractors need not be unique. To define pullback attractors for set-valued processes, denote by P(X) the collec- tion of all families of non-empty subsets of X and let D = {Dt : Dt ⊂ X, Dt ̸ =∅}t∈R. For any D, D̃ ∈ 2X , the notation D̃ ⊂ D means D̃t ⊂ Dt for every t ∈ R. Definition 2.23. A subset D of P(X) is inclusion closed if for D ∈ D and D̃ ∈ P(X), then D̃ ⊂ D implies that D̃ ∈ P(X). Such a collection D defined in Def. 2.23 is called a universe. Definition 2.24. Let {Ψ(t, t0)}(t,t0)∈R2 ≥ be a set-valued process on X. A family of non-empty bounded sets Q := (Qt)t∈R is said to be D-pullback absorbing for the set-valued process Ψ, if for any D = (Dt)t∈R ∈ D and each t ∈ R, there exists some time TD(t) 0 such that Ψ(t, t − τ, Dt−τ ) ⊂ Qt, for all τ ≥ TD. A family of non-empty bounded sets Q := (Qt)t∈R is said to be D-pullback attract- ing for the set-valued process Ψ, if every D = (Dt)t∈R ∈ D satisfies lim τ→∞ distX(Ψ(t, t − τ, Dt−τ ), Qt) = 0. Definition 2.25. Let {Ψ(t, t0)}(t,t0)∈R2 ≥ be a set-valued process on X and let D be a universe. A family A = (At)t∈R is said to be a global D-pullback attractor for Ψ if (i) At ⊂ X is compact for any t ∈ R; (ii) A is invariant; (iii) A is D-pullback attracting. The existence of a pullback attractor usually relies on some asymptotically com- pactness. In this work we will use the following definition. Definition 2.26. A set-valued process {Ψ(t, t0)}(t,t0)∈R2 ≥ is said to be D-pullback asymptotically upper semi-compact in X if for any fixed time t ∈ R, any sequence yn ∈ Ψ(t, t−τn, xn) has a convergent subsequence in X whenever τn → ∞ as n → ∞ and xn ∈ Dt−τn with D = (Dt)t∈R ∈ D. The following proposition from [Caraballo and Kloeden (2009)] gives the exis- tence of pullback attractors. Proposition 2.2. Let {Ψ(t, t0)}(t,t0)∈R2 ≥ be a set-valued process on X and let D be a universe. Assume that Downloaded from www.worldscientific.com
- 44. Dynamical systems 23 (i) Ψ(t, t0, x) is upper semi-continuous in x for any (t, t0) ∈ R2 ≥, (ii) Ψ(t, t0) is D-pullback asymptotically upper semi-compact in X, (iii) Ψ(t, t0) has a D-pullback absorbing set Q = (Qt)t∈R ∈ D. Then the set-valued process {Ψ(t, t0)}(t,t0)∈R2 ≥ has a unique D-pullback attractor A = (At)t∈R with its components given by At = s≥0 [ τ≥s Ψ(t, t − τ, Qt−τ ). When investigating set-valued processes it is often convenient to consider their single-valued trajectories. Definition 2.27. A trajectory of a set-valued process {Ψ(t, t0)}(t,t0)∈R2 ≥ is a single- valued function ψ : [t0, t1] ∩ R → X for some (t1, t0) ∈ R2 ≥ such that ψ(t) ∈ Ψ(t, s, ψ(s)) for all t0 ≤ s ≤ t ≤ t1 in R. A trajectory is called an entire trajectory if it is a trajectory on the whole time set R. In the discrete time case, trajectories are simply parts of sequences. Note that in the continuous time case trajectories are not assumed to be continuous but this follows from the next theorem, which is a generalisation of a theorem by Barbashin. Theorem 2.5. (Barbashin’s Theorem) Let {Ψ(t, t0)}(t,t0)∈R2 ≥ be a set-valued pro- cess on a complete metric space (X, dX). Then (1) there exists a trajectory from x0 to x1 ∈ Ψ(t1, t0, x0) for each (t1, t0) ∈ R2 ≥ and x0 ∈ X; (2) trajectories of a set-valued processes are continuous functions; (3) the set J (t1, t0, K) of all trajectories joining x0 to an arbitrary x1 ∈ Ψ(t1, t0, x0) with x0 ∈ K is compact in C([t0, t1]; X) for all (t1, t0) ∈ R2 ≥ and any non-empty compact subset K of X. Definition 2.28. A family D = (Dt)t∈R of non-empty sets of X is said to be invariant for a set-valued process Ψ if Ψ(t, t0, Dt0 ) = Dt for all (t, t0) ∈ R2 ≥; pos- itively invariant if Ψ(t, t0, Dt0 ) ⊂ Dt for all (t, t0) ∈ R2 ≥; and strongly negatively invariant if Dt ⊂ Ψ(t, t0, Dt0 ) for all (t, t0) ∈ R2 ≥. Theorem 2.6. Let {Ψ(t, t0)}(t,t0)∈R2 ≥ be a set-valued process on a complete metric space (X, dX) and let K = (Kt)t∈R be a family of non-empty compact subsets of X, which is Ψ-positively invariant. Downloaded from www.worldscientific.com
- 45. 24 Dissipative Lattice Dynamical Systems Then there exists a family of non-empty compact subsets K∞ = (K∞ t )t∈R con- tained in K in the sense that K∞ t ⊂ Kt for each t ∈ R, which is Ψ-strongly invariant. The component sets K∞ t are given by K∞ t = t0≤t Ψ(t, t0, Kt0 ), t ∈ R. 2.2.3 Attractors of skew product flows For complete metric spaces (P, dP) and (X, dX), let (ϑ, π) be a skew product flow on P × X. Similarly to processes we have two types of attractors for skew product flows, pullback and forward attractors. Definition 2.29. A family A = (Ap)p∈P of π-invariant non-empty compact subsets of X is called a pullback attractor if it pullback attracts families B = (Bp)p∈P of non-empty bounded subsets of X, i.e., lim t→∞ distX π(t, ϑ−t(p), Bϑ−t(p)), Ap = 0 for each p ∈ P. It is called a forward attractor if it forward attracts families of non-empty bounded subsets B = (Bp)p∈P of X, i.e., lim t→∞ distX π(t, p, Bp), Aϑt(p) = 0 for each p ∈ P. Also, as for a process, the existence of a pullback attractor for skew product flow is ensured by that of a pullback absorbing family. To handle nonuniformities, as for processes, the following definition similar to Definition 2.20 is needed to ensure that the component sets in the non-autonomous family should not do too quickly. Definition 2.30. A family B = (Bp)p∈P of non-empty bounded subsets Bp of X is said to have sub-exponential growth if lim sup |t|→∞ ∥Bϑ−t(p0)∥ec|t| = 0 ∀c 0 where ∥Bϑ−t(p0)∥ = sup b∈Bϑ−t(p0) ∥b∥. In this case it is called a tempered family. Definition 2.31. A family Q = (Qp)p∈P of non-empty subsets of X is called a pullback absorbing family for a skew product flow (ϑ, π) on P × X if for each p ∈ P and every tempered family B = (Bp)p∈P of non-empty bounded subsets of X there exists a Tp,B ∈ R+ such that π t, ϑ−t(p), Bϑ−t(p) ⊆ Qp for all t ≥ Tp,B. Definition 2.32. A skew product flow (ϑ, π) on P × X is said to be D-pullback asymptotically compact if for any p ∈ P and D = (Dt)t∈R ∈ D, the sequence π(tn, ϑ−tn (p), xn) has a convergence subsequence for any sequences tn → +∞ and xn ∈ Dϑ−tn (p). Downloaded from www.worldscientific.com
- 46. Dynamical systems 25 The proof of the following theorem here is similar to that of Theorem 2.4. Theorem 2.7. (Existence of a pullback attractor) Let (P, dP) and (X, dX) be complete metric spaces and suppose that a skew product flow (ϑ, π) on P × X is pullback asymptotic compact and has a pullback tempered absorbing family Q = (Qp)p∈P of non-empty closed and bounded sets. Then the skew product flow (ϑ, π) has a pullback attractor A = (Ap)p∈P with component subsets determined by Ap = t≤0 [ s≥t π t, ϑ−t(p), Qϑ−t(p) for each p ∈ P. If Q is π-positively invariant then Ap = t≤0 π t, ϑ−t(p), Qϑ−t(p) for each p ∈ P. Moreover, A is unique if the components sets are uniformly bounded. Note that if the pullback attractor is uniformly pullback attracting, i.e., if lim t→∞ sup p∈P distX π(t, ϑ−t(p), Qϑ−t(p)), Ap = 0 for each p ∈ P, then it is uniformly forward attracting, since writing a = ϑ−t(p), sup p∈P distX π(t, ϑ−t(p), Qϑ−t(p)), Ap = sup a∈P distX π(t, a, Qa), Aϑt(a) . In this case this uniform pullback/forward attractor is called a uniform (non- autonomous) attractor. 2.3 Compactness criteria In a finite dimensional space such as Rd the compact subsets are the closed and bounded subsets. In an infinite dimensional Banach space (E, ∥ · ∥E) the compact subsets are the closed and totally bounded subsets, i.e., they can be covered by the union of a finite number of balls of arbitrarily small radius. Equivalently, a subset D of (E, ∥ · ∥E) is compact if it is sequentially compact, i.e., if every sequence in D has a convergent subsequence in D. 2.3.1 Kuratowski measure of non-compactness Let (E, ∥ · ∥E) be a Banach space. A mapping S is called a κ-contraction on E when it is a contraction with respect the Kuratowski measure of noncompactness of subsets of E, i.e., if there is a positive number q 1 such that κ(S(D)) qκ(D) for every subset D of E. The Kuratowski measure of noncompactness of a subset D of Banach space (E, ∥ · ∥E) is defined by κ(D) = inf{d 0 : there exists an open cover of D with sets of diameter ≤ d}. Downloaded from www.worldscientific.com
- 47. 26 Dissipative Lattice Dynamical Systems The compact sets are the closed subsets D of E with κ(D) = 0. Basic properties of the Kuratowski measure of noncompactness on a Banach space include: (i) D is bounded if and only if κ(D) ∞. (ii) κ(D̄) = κ(D), where D̄ denotes the closure of D. (iii) D is compact if and only if κ(D) = 0. (iv) κ(D1 ∪ D2) = max(κ(D1), κ(D2)) for any subsets D1 and D2. (v) κ is continuous with respect to the Hausdorff distance of sets. (vi) κ(aD) = |a|κ(D) for any scalar a. (vii) κ(D1 + D2) ≤ κ(D1) + κ(D2) for any subsets D1 and D2. (viii) κ(convD) = κ(D), where convD denotes the convex hull of D. (ix) if D1 ⊇ D2 ⊇ D3 ⊇ · · · are non-empty closed subsets of E such that κ(Dn) → 0 as n → ∞, then n≥1 Dn is non-empty and compact. T 2.3.2 Weak convergence and weak compactness Let (H, ∥·∥H, ⟨·, ·⟩H) be a Hilbert space, which will typically be ℓ2 or ℓ2 ρ in this book. Convergence with respect to the norm ∥ · ∥H is often called strong convergence, i.e., un → u∗ strongly if and only if ∥un − u∗ ∥H → 0 as n → ∞. Another useful convergence is weak convergence. A sequence {un}n∈N converges weakly to u∗ in H if and only if ⟨h, un − u∗ ⟩H → 0 as n → ∞ for all h ∈ H. Weak convergence is often written as un ⇀ u∗ . Essentially, weak convergence is with respect to all linear functionals on H. In general, weak convergence does not imply strong convergence, but the following result holds. See [Banach and Saks (1930); Okada (1984); Partington (1977); Szlenik (1965)]. Theorem 2.8. (Banach-Saks Theorem) A bounded sequence {un}n∈N in a Hilbert H contains a subsequence {unk }k∈N and a point u∗ such that 1 N N X k=1 unk −→ u∗ strongly as N → ∞. Definition 2.33. A subset K of a Hilbert space H is said to be weakly compact if it is weakly sequentially compact, i.e., if every sequence {un}n∈N in K has a weakly convergent subsequence unk ⇀ u∗ in K. The following theorem is a special case of a more general result of Kakutani, see Theorem 3.17 in [Brezis (2011)]. Theorem 2.9. A closed and bounded (in norm) subset D of a Hilbert space H is weakly compact. Downloaded from www.worldscientific.com
- 48. Dynamical systems 27 ¨ ¨ A special case of the Banach-Alaoglu theorem is the sequential version of the original theorem. Theorem 2.10. (Banach-Alaoglu Theorem) The closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak∗ - topology. The following result is from [Ulger (1991), Proposition 7], see also [Diestel (1977)]. Lemma 2.3. (Ulger’s Lemma) Let (Ω, Σ, µ) be a probabilistic space, and E be an arbitrary Banach space. For any weakly compact subset K ⊂ E, the set ¨ f ∈ L1 (µ, E) : f(ω) ∈ K for µ-almost every ω ∈ Ω is relatively weakly compact. The next result is due to [Ulger (1991), Corollary 5]. Lemma 2.4. Let (Ω, Σ, µ) be a probabilistic space and E be a Banach space. Set U := f ∈ L1 (µ, E) : ∥f(ω)∥E ≤ 1 for µ − a.e. ω ∈ Ω . A sequence fk(·) k∈N in U ⊂ L1 (µ, E) converges weakly to f ∈ L1 (µ, E) if and only if for any sub-sequence fkn (·) n∈N given, there exists a sequence gn(·) n∈N with gn ∈ co fkn , fk(n+1) , . . . such that for µ-a.e. ω ∈ Ω, gn(ω) −→ f(ω) (n −→ ∞) weakly in E. 2.3.3 Ascoli-Arzelà Theorem The Ascoli-Arzelà Theorem [Green and Valentine (1960/1961)] is a crucial tool in the study of lattice dynamical systems. Let (E, ∥ · ∥E) be a Banach space, let I be a closed and bounded interval in R and let C(I, E) be the space of all continuous functions f : I → E with uniform norm ∥f∥∞ = maxt∈I ∥f(t)∥E. Definition 2.34. A subset S of C(I, E) is said to be equi-continuous if for every ε 0 there exists δ = δ(ε) 0 which is independent of f ∈ S such that ∥f(s)−f(t)∥E ε for all s, t ∈ I with |s − t| δ and all f ∈ S. Theorem 2.11. (Ascoli-Arzelà Theorem). A subset S of C(I, E) is relatively compact if and only if S is equi-continuous and S(t) := {f(t) : f ∈ S} is relatively compact in E for every t ∈ I. The following consequence of this theorem in a Hilbert space H, which will typically be ℓ2 or ℓ2 ρ in this book, will be used in the sequel. See, e.g. [Lebl (2016)]. Downloaded from www.worldscientific.com
- 49. 28 Dissipative Lattice Dynamical Systems Corollary 2.2. Let {fn(·)}n∈N be a sequence in C([0, T], H), which is uniformly bounded and equi-Lipschitz continuous on [0, T]. Then there is an f∗ (·) ∈ C([0, T], H) and a convergent subsequence {fnk (·)}k∈N of {fn(·)}n∈N such that fnk (·) → f∗ (·) strongly in C([0, T], H) as nk → ∞ d dt fnk (·) ⇀ d dt f∗ (·) weakly in L1 ([0, T], H) as nk → ∞. 2.3.4 Asymptotic compactness properties Some kind of compactness condition is required to ensure that the omega limit sets defining an attractor are non-empty. For a dynamical system on the finite dimen- sional state space this is easy since the compact subsets are the closed and bounded subsets. Then, e.g., for an autonomous dynamical system φ with a positively in- variant, closed and bounded (hence compact) absorbing set Q, the attractor A = t≥0 φ(t, Q) is the non-empty intersection of the nested compact subsets φ(s, Q) ⊂ φ(t, Q) ⊂ Q for s t, since continuous functions map compact subsets onto compact subsets. In infinitely dimensional state spaces, closed and bounded subsets need not be compact, so some compactness property must come from the dynamics. A simple property is that the mappings φ(t, ·) are compact for t 0, i.e., map closed and bounded subsets of X onto pre-compact subsets of X. This is usually too strong for most applications, so a weaker asymptotic compactness property is often used. For specific examples of lattice systems, to show that the system is asymptotic compact, one usually first shows that the lattice dynamical system satisfies an asymptotic tails property inside an absorbing set which is positively invariant closed and bounded convex set (such as a ball). In particular, when the state space X is a space of bi-infinite real-valued sequences such as ℓ2 and the set Q is also convex, then it follows that φ is asymptotically compact in Q. Similar proofs also hold in weighted Hilbert spaces of bi-infinite real-valued sequences such as ℓ2 ρ. Assumption 2.1. (Asymptotic tails property: autonomous systems) Let φ = (φi)i∈Z be an autonomous semi-dynamical system on the Hilbert space (ℓ2 , ∥·∥) and let B be a positively invariant, closed and bounded subset of ℓ2 , which is φ- positive invariant. Then φ is said to satisfy an asymptotic tails property in B if for every ε 0 there exist T(ε) 0 and I(ε) ∈ N such that X |i|I(ε) |φi(t, x0)|2 ≤ ε ∀ x0 ∈ B and t ≥ T(ε). Lemma 2.5. Let Assumption 2.1 hold. Then the semi-dynamical system φ is asymptotically compact in B. Downloaded from www.worldscientific.com
- 50. Dynamical systems 29 An analogous result also holds for pullback asymptotic compactness of processes and skew product flows. The proof of Lemma 2.5 follows as a simpler case of the proof for Lemma 2.6 below for processes. Assumption 2.2. (Pullback asymptotic tails property for process) Let ψ = (ψi)i∈Z be a process on the Hilbert space (ℓ2 , ∥ · ∥) and let B = {Bt}t∈R be ψ-positively invariant and consist of closed and bounded subsets of ℓ2 . Then ψ is said to satisfy a pullback asymptotic tails property in B if for every t ∈ R and ε 0 there exist T(t, ε) 0 and I(t, ε) ∈ N such that X |i|I(t,ε) |ψi(t, t0, x0)|2 ≤ ε, ∀ x0 ∈ Bt0 and t0 ≤ t − T(t, ε). Lemma 2.6. Let Assumption 2.2 hold. Then the process ψ is pullback asymptoti- cally compact in B. Proof. We only need to show that every sequence v(n) ∈ ψ(t, t − tn, Bt−tn ) ⊂ Bt with tn → ∞ as n → ∞ has a converging subsequence in ℓ2 . For a sequence {tn} with tn → ∞ as n → ∞, let u(n) ∈ Bt−tn and v(n) = ψ(t, t − tn, u(n) ) ∈ Bt, n = 1, 2, · · · . Since Bt is non-empty, closed, and bounded in ℓ2 , it is weakly compact so there is a subsequence of {v(n) } (still denoted by {v(n) }), and v∗ ∈ Bt such that v(n) = ψ(t, t − tn, u(n) ) ⇀ v∗ (i.e., weakly in ℓ2 ). We now show that this weak convergence is actually strong. Given any ε 0, by the Assumption 2.2, there exists I1(t, ε) 0 and N1(t, ε) 0 such that X |i|≥I1(t,ε) |ψi(t, t − tn, uo)| 2 ≤ 1 8 ε, ∀ n ≥ N1(t, ε), (2.1) for every uo ∈ Bt−tn . Moreover, since v∗ = (v∗ i )i∈Z ∈ ℓ2 , there exists an I2(ε) 0 such that X |i|≥I2(ε) |v∗ i |2 ≤ ε 8 . (2.2) Set I(t, ε) := max{I1(t, ε), I2(ε)}. Since ψ(t, t − tn, u(n) ) ⇀ v∗ in ℓ2 , it follows component wise that ψi(t, t − tn, u(n) ) −→ v∗ i for |i| ≤ I(t, ε), as n → ∞. Therefore there exists N2(t, ε) 0 such that X |i|≤I(t,ε) ψi(t, t − tn, u(n) ) − v∗ i 2 ≤ 1 2 ε, ∀ n ≥ N2(t, ϵ). (2.3) Downloaded from www.worldscientific.com
- 51. 30 Dissipative Lattice Dynamical Systems Set I(t, ε) := max{I1(t, ε), I2(t, ε)}. Then, using (2.1) − (2.3), for n ≥ I(t, ε) it follows that ψ(t, t − tn, u(n) ) − v∗ 2 = X |i|≤I(t,ε) ψi(t, t − tn, u(n) ) − v∗ i 2 + X |i|I(t,ε) ψi(t, t − tn, u(n) ) − v∗ i 2 ≤ 1 2 ε + 2 X |i|I(t,ϵ) ψi(t, t − tn, u(n) ) 2 + |v∗ i |2 ≤ ε. Hence v(n) (the subsequence) is strongly convergent in ℓ2 , so ψ is pullback asymp- totic compact in B. 2.4 End notes There are many classical monographs on autonomous dynamical systems, see, e.g., [Teschl (2012)]. See Mallet-Paret, Wu, Yi Zhu, [Mallet-Paret et al. (2012)] and [Robinson (2001)] for infinite dimensional dynamical systems. For non-autonomous dynamical systems see [Sell (1971)], [Kloeden and Rasmussen (2011)], [Caraballo and Han (2016)], and [Kloeden and Yang (2021)]. Proofs of most of the results on non-autonomous systems stated in this chapter can be found in [Kloeden and Rasmussen (2011)] and [Kloeden and Yang (2021)], with the autonomous counterparts holding as special cases. See [Ambrosio and Tilli (2004)] for general topics on analysis in metric spaces. 2.5 Problems Problem 2.1. Consider the attractor Ap of the autonomous scalar ODE dx dt = −x x4 − 2x2 + 1 − p with a parameter p ∈ P = [−2, 2]. Determine the attractor Ap for each p. Then show that the attractors Ap converge upper semi-continuously to Ap0 as p → p0, but need not converge continuously (in the Hausdorff metric). What properties will ensure that the attractors converge continuously? Problem 2.2. What is the exact relationship between the asymptotic tails property and total boundedness? Problem 2.3. Describe the major differences between the process and skew- product flow formulations of non-autonomous dynamical systems. In what scenarios is one more convenient than the other? Downloaded from www.worldscientific.com
- 53. T T Th h hi i is s s p p pa a ag g ge e e i i in n nt t te e en n nt t ti i io o on n na a al l ll l ly y y l l le e ef f ft t t b b bl l la a an n nk k k Downloaded from www.worldscientific.com
- 54. Chapter 3 Lattice Laplacian models A lattice reaction-diffusion model is a lattice dynamical system obtained by spa- tially discretising the Laplacian operator in a parabolic partial differential equation modelling a reaction-diffusion equation such as equation (1.1) in Chapter 1. Bates, Lu Wang [Bates et al. (2001)] investigated dynamical behaviour of the lattice dy- namical system (LDS) based on this reaction-diffusion equation and their results, which have profoundly influenced the development of the theory of dissipative LDS, will be presented here. Consider the autonomous LDS dui dt = ν (ui−1 − 2ui + ui+1) + f(ui) + gi, i ∈ Z, (3.1) in the space ℓ2 (which was defined in Section 1.3), which will be investigated here under the following assumptions. Assumption 3.1. The function f : R → R is a continuously differentiable function satisfying f(s)s ≤ −αs2 ∀ s ∈ R, for some α 0. Assumption 3.2. The function g = (gi)i∈Z ∈ ℓ2 . Remark 3.1. Since f is smooth, the Assumption 3.1 implies that f(0) = 0. 3.1 The discrete Laplace operator The Laplacian operator on one-dimensional spatial domain is just the second derivative. Using central difference quotient to approximate it leads to the one- dimensional discrete Laplace operator on an appropriate sequence space. For any u = (ui)i∈Z ∈ ℓ2 , the discrete Laplace operator Λ is defined from ℓ2 to ℓ2 component wise by (Λu)i = ui−1 − 2ui + ui+1, i ∈ Z. (3.2) 33 Downloaded from www.worldscientific.com
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- 62. No. 1 A, Patent Reflex Iron Oven. “NEW PROCESS” OVENS. No. 1 A, “New Process” Top Oven. The accompanying cuts represent our patent top oven, No. 1 A. The construction of the flues is such that the distribution of heat is equalized in all parts of the oven, consequently this oven will bake equally well upon the top or bottom shelf. The outer wall is made of iron, heavily embossed, which gives it strength. The inner wall is corrugated tin. The “New Process” is without doubt the best baking oven on the market. It is made to cover two burners, the inside dimensions of the oven being 19½ × 13¼ × 12 inches. The door of this oven is hinged at the bottom, so that when opened it forms a very convenient shelf, supported by chains as
- 63. No. 1 A, Patent Reflex Iron Oven, With Door Lowered. shown in the cut. The oven is knocked down and packed in very compact form for convenience in shipping. No. 1 B, “New Process” Has the same general construction, except that the outside is made of tin. No. 1 A, Iron Oven, price $4.00. Code Word “Granite.” No. 1 B, Tin Oven, price $4.00. Code Word “Marble.” Weight 28 lbs.
- 64. “Standard” Perfect Bakers. A Good Oven Is one in which the heat flues are constructed in such a manner that the heat is distributed evenly through the baking compartment. The “Standard” Perfect Bakers Will bake quickly and evenly on the top and bottom alike. A poor oven often causes a good stove to be condemned, and is dear at any price. Standard Ovens are properly constructed of good materials, and are sold at the right price. We have the best facilities for the manufacture of ovens, and solicit your orders. Special Prices Quoted on Large Orders. Size 13″ × 21″ × 18″, Outside Measurement.
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- 69. Operation. amount of oil in reservoir. The heater is provided with a bail for convenience in handling. Has improved smokeless device. The “New Process” Oil Heater produces a white flame, smokeless and odorless, of great intensity and heating power. It is light and portable, and can be put just where it is needed most. It will comfortably heat in cold weather a room of ordinary size. Height, 28¾″. Diameter, 9¼″. Weight of Heater, 13¾ lbs. Boxed, 23 lbs.
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- 72. No. 25. The Mammoth Globe Incandescent Lamp. 320 Candle Power. No. 25, Complete. Price in Brass, $3.25; Nickel, $3.60. Packages Extra. The accompanying cut shows the No. 25 Mammoth Globe Incandescent lamp with 20-inch tin reflector and chimney complete. We furnish it in either brass or nickel-plated finish. The reservoir holds enough oil for eight hours’ burning, and is provided with an indicator showing at all times the amount of oil in reservoir. The reservoir never becomes heated. The lamp is fitted with a duplex
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- 74. No. 250. No. 5000. The Mammoth Globe Incandescent Lamp 320 Candle Power. No. 250, Price Complete in Brass, $3.60; Nickel, $3.90. No. 5000, Complete in Brass, $5.00; Nickel, $5.75. Packages Extra. The above cuts represent two additional styles of the Globe Incandescent Lamp. The No. 250 Lamp is provided with 14-inch
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