Dissipative Lattice Dynamical Systems

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