There is an extensive literature in the form of papers (but no books) on lattice dynamical systems. The book focuses on dissipative lattice dynamical systems 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.
There is now a very large literature on lattice dynamical systems, especially on attractors of all kinds in such systems. We cannot hope to do justice to all of them here. Instead, we have focused on key areas 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. One of the important results is the approximation of Heaviside switching functions in LDS by sigmoidal functions.
Nevertheless, we believe that this book will provide the reader with a solid introduction to the field, its main results and the methods that are used to obtain them.
Author(s): Xiaoying Han, Peter Kloeden
Series: Interdisciplinary Mathematical Sciences 22
Edition: 1
Publisher: World Scientific
Year: 2023
Language: English
Pages: 380
Tags: Lattice Dynamical Systems, LDS, Lattice Laplacian Models, Attractors, Set-Valued Lattice Models, Stochastic LDS, Hopfield Lattice Models, FitzHugh-Nagumo Lattice Model
Contents
Preface
Background
1. Lattice dynamical systems: a preview
1.1 Introduction
1.2 Examples of lattice dynamical systems
1.2.1 PDE based models
1.2.2 Neural field models
1.2.3 Intrinsically discrete models
1.3 Sequence spaces
1.4 An illustrative lattice reaction-diffusion model
1.5 Outline of this book
1.6 Endnotes
1.7 Problems
2. Dynamical systems
2.1 Abstract dynamical systems
2.1.1 Autonomous dynamical systems
2.1.2 Two-parameter non-autonomous dynamical systems
2.1.3 Skew product flows
2.2 Invariant sets and attractors of dynamical systems
2.2.1 Attractors of autonomous semi-dynamical systems
2.2.2 Attractors of processes
2.2.3 Attractors of skew product flows
2.3 Compactness criteria
2.3.1 Kuratowski measure of non-compactness
2.3.2 Weak convergence and weak compactness
2.3.3 Ascoli-Arzel`a Theorem
2.3.4 Asymptotic compactness properties
2.4 End notes
2.5 Problems
Laplacian LDS
3. Lattice Laplacian models
3.1 The discrete Laplace operator
3.2 The autonomous reaction-diffusion LDS
3.2.1 Existence of an absorbing set
3.2.2 Asymptotic tails property
3.3 Nonautonomous lattice reaction-diffusion LDS
3.4 p-Laplacian reaction-diffusion LDS
3.4.1 Discretised p-Laplacian
3.4.2 Existence and uniqueness of solutions
3.4.3 Existence of a global attractor
3.4.3.1 Existence of an absorbing set
3.4.3.2 Asymptotic tails property
3.4.3.3 Variable exponent Laplace operators
3.5 End notes
3.6 Problems
4. Approximation of attractors of LDS
4.1 Finite dimensional approximations
4.2 Upper semi-continuous convergence of the finite dimensional attractors
4.3 Numerical approximation of lattice attractors
4.4 Finite dimensional approximations of the IES
4.4.1 Finite dimensional numerical attractors A(h)N
4.4.2 Upper semi continuous convergence
4.4.3 Convergence of numerical attractors
4.5 End notes
4.6 Problems
5. Non-autonomous Laplacian lattice systems in weighted sequence spaces
5.1 The discrete Laplacian on weighted sequence spaces
5.2 Generation of a non-autonomous dynamical system on ℓ2ρ
5.2.1 Existence and uniqueness of solutions in ℓ2
5.2.2 Lipschitz continuity of solutions in initial data in the ℓ2ρ norm
5.2.3 Generation of semi-group on ℓ2p
5.3 Existence of pullback attractors
5.3.1 Existence of an absorbing set
5.3.2 Asymptotic tails and asymptotic compactness
5.4 Uniformly strictly contracting Laplacian lattice systems
5.5 Forward dynamics
5.6 End notes
5.7 Problems
A selection of lattice models
6. Lattice dynamical systems with delays
6.1 The coefficient terms
6.2 Existence and uniqueness of solutions
6.2.1 Existence of solutions
6.2.2 An a prior estimate of solutions
6.2.3 Uniqueness of solutions
6.3 Asymptotic behaviour
6.3.1 Tails estimate
6.3.2 Existence of the global attractor
6.4 End notes
6.5 Problems
7. Set-valued lattice models
7.1 Set-valued lattice system on ℓ2
7.2 Existence of solutions
7.3 Set-valued semi-dynamical systems with compact values
7.4 Existence of a global attractor
7.5 Endnotes
7.6 Problems
8. Second order lattice dynamical systems
8.1 Existence and uniqueness of solution
8.2 Existence of a bounded absorbing set
8.3 Existence of a global attractor
8.4 End notes
8.5 Problems
9. Discrete time lattice systems
9.1 Autonomous systems
9.1.1 Preliminaries
9.1.2 Existence of a global attractor
9.1.3 Finite dimensional approximations of the global attractor
9.2 Convergent sequences of interconnection weights
9.3 Lattice systems with finitely many interconnections
9.4 Nonautonomous systems
9.4.1 Existence of a pullback attractor
9.4.2 Existence of a forward ω-limit sets
9.5 Endnotes
9.6 Problems
10. Three topics in brief
10.1 Finite dimension of lattice attractors
10.2 Exponential attractors
10.2.1 Application to general lattice systems
10.2.2 First order lattice systems
10.2.3 Partly dissipative lattice systems
10.2.4 Second order lattice systems
10.3 Traveling waves for lattice neural field equations
10.4 End notes
10.5 Problems
Stochastic and Random LDS
11. Random dynamical systems
11.1 Random ordinary differential equations
11.1.1 RODEs with canonical noise
11.1.2 Existence und uniqueness results for RODEs
11.2 Random dynamical systems
11.3 Random attractors for general RDS in weighted spaces
11.4 Stochastic differential equations as RODEs
11.5 End notes
11.6 Problems
12. Stochastic LDS with additive noise
12.1 Random dynamical systems generated by stochastic LDS
12.1.1 Ornstein-Uhlenbeck process
12.1.2 Transformation to a random ordinary differential equation
12.1.3 Existence and uniqueness of solutions
12.1.4 Random dynamical systems generated by random LDS
12.2 Existence of global random attractors in weighted space
12.2.1 Existence of tempered random bounded absorbing sets
12.2.2 Existence of global random attractors
12.3 End notes
12.4 Problems
13. Stochastic LDS with multiplicative noise
13.1 Random dynamical systems generated by stochastic LDS
13.1.1 Transformation to a random LDS
13.1.2 Existence and uniqueness of solutions to the random LDS
13.1.3 Random dynamical systems generated by random LDS
13.2 Existence of global random attractors in weighted space
13.2.1 Existence of tempered random bounded absorbing sets
13.2.2 Existence of global random attractors
13.3 End notes
13.4 Problems
14. Stochastic lattice models with fractional Brownian motions
14.1 Preliminaries
14.2 Existence of solutions
14.2.1 Standing assumptions
14.2.2 Properties of operators
14.2.3 Existence of mild solutions
14.3 Generation of an RDS
14.4 Exponential stability of the trivial solution
14.4.1 Existence of trivial solutions
14.4.2 The cut–off strategy
14.4.3 Preliminary estimates
14.4.4 Exponential stability
14.5 End notes
14.6 Problems
Hopfield Lattice Models
15. Hopfield neural network lattice model
15.1 Introduction
15.2 Formulation as an ODE
15.3 Existence of attractors
15.4 Finite dimensional approximations
15.5 End notes
15.6 Problems
16. The Hopfield lattice model in weighted spaces
16.1 Reformulation as an ODE on ℓ2p
16.2 Existence and uniqueness of solutions
16.3 Existence of attractors
16.3.1 Existence of absorbing sets
16.3.2 Asymptotic compactness
16.4 Upper semi-continuity of attractors in λi,j
16.5 End notes
16.6 Problems
17. A random Hopfield lattice model
17.1 Basic properties of solutions
17.2 Existence of random attractors
17.3 End notes
17.4 Problems
LDS in Biology
18. FitzHugh-Nagumo lattice model
18.1 Generation of a semi-dynamical system on ℓ2ρ × ℓ2ρ
18.1.1 Existence and uniqueness of solutions in ℓ2 × ℓ2
18.1.2 Lipschitz ℓ2ρ-continuity of solutions in initial data
18.1.3 Existence and uniqueness of solutions in ℓ2ρ × ℓ2ρ
18.2 Existence of a global attractor
18.2.1 Existence of an absorbing set
18.2.2 Asymptotic tails and asymptotic compactness
18.3 Limit of the global attractors as δ → 0
18.3.1 Uniform bound on the global attractors
18.3.2 Pre-compactness of the union of the global attractors
18.3.3 Upper semi-continuity of the global attractors
18.4 End notes
18.5 Problems
19. The Amari lattice neural field model
19.1 Preliminaries
19.1.1 Standing assumptions
19.1.2 Basic estimates
19.2 Set-valued lattice systems
19.2.1 Inflated lattice systems
19.2.2 Relations between Heaviside, sigmoid, and inflated
19.3 Existence of solutions
19.3.1 The sigmoidal lattice system
19.3.2 The inflated system
19.3.3 The set-valued lattice system
19.4 Convergence of sigmoidal solutions
19.4.1 A priori estimates
19.4.2 The convergence theorem
19.5 Set-valued dynamical systems with compact values
19.6 Attractors of the sigmoidal and lattice systems
19.6.1 Comparison of the attractors
19.7 End notes
19.8 Problems
20. Stochastic neural field models with nonlinear noise
20.1 Well-posedness of the LDS in ℓ2ρ
20.2 Existence of mean-square solutions
20.2.1 Solutions of the truncated system
20.2.2 Existence of a global mean-square solution
20.3 Weak pullback mean random attractors
20.3.1 Preliminaries on mean random dynamical systems
20.3.2 Existence of absorbing sets
20.3.3 Existence of a mean random attractor
20.3.4 End notes
20.4 Problems
21. Lattice systems with switching effects and delayed recovery
21.1 Set-valued delay differential inclusions
21.2 Existence of solutions
21.3 Long term behavior of lattice system
21.3.1 Generation of set-valued process
21.3.2 Existence of an absorbing set
21.3.3 Tail estimations
21.3.4 Existence of a nonautonomous attractor
21.4 End notes
21.5 Problems
Bibliography
Index
