Monotone Nonautonomous Dynamical Systems

✍ Scribed by David N. Cheban

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 479
• Edition: 2024
• Category: Library

✦ Synopsis

The monograph present ideas and methods, developed by the author, to solve the problem of existence of Bohr/Levitan almost periodic (respectively, almost recurrent in the sense of Bebutov, almost authomorphic, Poisson stable) solutions and global attractors of monotone nonautonomous differential/difference equations. Namely, the text provides answers to the following problems:

1. Problem of existence of at least one Bohr/Levitan almost periodic solution for cooperative almost periodic differential/difference equations;

2. Problem of existence of at least one Bohr/Levitan almost periodic solution for uniformly stable and dissipative monotone differential equations (I. U. Bronshtein’s conjecture, 1975);

3. Problem of description of the structure of the global attractor for monotone nonautonomous dynamical systems;  

4. The structure of the invariant/minimal sets and global attractors for one-dimensional monotone nonautonomous dynamical systems;  

5. Asymptotic behavior of monotone nonautonomous dynamical systems with a first integral (Poisson stable motions, convergence, asymptotically Poisson stable motions and structure of the Levinson center (compact global attractor) of dissipative systems);

6. Existence and convergence to Poisson stable motions of monotone sub-linear nonautonomous dynamical systems.

This book will be interesting to the mathematical community working in the field of nonautonomous dynamical systems and their applications (population dynamics, oscillation theory, ecology, epidemiology, economics, biochemistry etc). The book should be accessible to graduate and PhD  students who took courses in real analysis (including the elements of functional analysis, general topology) and with general background in dynamical systems and qualitative theory of differential/difference equations.

✦ Table of Contents

Preface
Contents
Notation
1 Poisson Stable Motions of Dynamical Systems
1.1 Introduction
1.2 Some Notions, Notation, and Facts from the Theory of Dynamical Systems
1.3 Minimal Sets and Recurrent Motions
1.3.1 Minimal Sets
1.3.2 Almost Recurrent Motions
1.3.3 Recurrent Motions
1.4 Lyapunov Stable Sets
1.5 Almost Periodic Motions
1.6 Comparability by the Character of Recurrence of Motions for Dynamical Systems
1.6.1 B. A. Shcherbakov's Principle of Comparability of Motions by Their Character of Recurrence
1.6.2 Some Generalization of B. A. Shcherbakov's Results
1.6.3 Strongly Comparability of Motions by Their Character of Recurrence
1.7 Asymptotically Almost Periodic Motions
1.7.1 Asymptotically Poisson Stable Motions
1.7.2 Criterion of Asymptotical Almost Periodicity
1.7.3 Asymptotically Periodic Motions
1.7.4 Comparability by the Character of Recurrence in the Limit of Asymptotically Poisson Stable Motions
1.8 Almost Periodic and Asymptotically Almost Periodic Functions
1.8.1 Bohr Almost Periodic Functions
1.8.2 Levitan Almost Periodic Functions
1.8.3 Fréchet Asymptotically Almost Periodic Functions
2 Compact Global Attractors
2.1 Introduction
2.2 Limit Properties of Dynamical Systems
2.3 Levinson Center
2.4 Dissipative Systems on the Local Compact Spaces
2.5 Criterions of Compact Dissipativity
2.6 Global Attractor of Nonautonomous Dynamical Systems
2.6.1 Nonautonomous Sets
2.6.2 Maximal Compact Invariant Sets
2.7 Pullback Attractors of Cocycles
2.8 Global Attractors of Cocycles
2.9 Forward Attractors for Cocycles
3 V-Monotone Nonautonomous Dynamical Systems
3.1 Introduction
3.2 Global Attractors of V-Monotone NDS
3.3 On the Structure of Levinson Center of V-Monotone NDS
3.4 Almost Periodic Solutions of V-Monotone Systems
3.5 Pullback Attractors of V-Monotone NDS
3.6 Applications
3.6.1 Finite-Dimensional Systems
3.6.2 Caratheodory`s Differential Equations
3.6.3 ODEs with Impulse
3.6.4 Evolution Equations with Monotone Operators
3.7 Continuous Invariant Sections of NDS
3.7.1 Contraction Principle and Its Generalizations
3.7.2 Invariant Sections of NDS
3.7.3 Global Attractors of V-monotone NDS
3.7.4 Bohr/Levitan Almost Periodic Motions of V-Monotone NDS
3.7.5 Applications
Finite-Dimensional Systems
Caratheodory's Differential Equations
ODEs with Impulse
ODEs in Banach Spaces
Evolution Equations with Monotone Operators
3.8 Second Order Monotone Equations in the Hilbert Spaces
3.8.1 Some NDS Generated by Differential Equations in the Banach Spaces
3.8.2 Invariant Sections of Second Order Differential Equations
Invariant Sections
Linear Case
Quasiperiodic Solutions
Invariant Sections of Equation x''=V(x)+f(σ(t,y))
3.8.3 Almost Automorphic Solutions of Monotone Second Order Differential Equation
4 Poisson Stable Motions and Global Attractors of Monotone Nonautonomous Dynamical Systems
4.1 Introduction
4.2 Poisson Stable Motions of Monotone NDS
4.2.1 Structure of the ω-Limit Set
4.2.2 Comparable Motions by Character of their Recurrence for Monotone NDS
4.2.3 Applications
Ordinary Differential Equations
Functional Differential Equations with Finite Delay
Parabolic Systems
4.3 I. U. Bronshtein's Conjecture for Monotone NDS
4.3.1 I. U. Bronshtein's Conjecture for Cocycles
4.3.2 Applications
Dissipative Cocycles
Ordinary Differential Equations
Difference Equations
4.4 Structure of the Levinson Center for Monotone NDS
4.5 Applications
4.5.1 Ordinary Differential Equations
Scalar Differential Equations
4.5.2 Difference Equations
4.5.3 Functional Differential Equations with Finite Delay
4.5.4 Parabolic Systems
5 One-Dimensional Monotone Nonautonomous Dynamical Systems
5.1 Introduction
5.2 One-Dimensional Monotone Differential/Difference Equations
5.3 Invariant Sets
5.4 Structure of Levinson Center for One-Dimensional Monotone Cocycles
5.5 Pinched Sets
5.6 Minimal Sets of One-Dimensional Monotone Cocycles
5.7 Applications
5.7.1 Scalar Differential Equations
5.7.2 Scalar Difference Equations
5.8 Bohr/Levitan Almost Periodic Solutions of Scalar Differential Equations
5.8.1 Some Criterion of the Existence of Fixed Point for a Semigroup of Transformations
5.8.2 Distal NDS with the Minimal Base
5.8.3 Structure of the ω-Limit Set
5.8.4 Scalar Differential Equations
5.9 Bohr/Levitan Almost Periodic Solutions of the Second Order Differential Equation
5.9.1 Some Nonautonomous Dynamical Systems Generated by the Second Order Differential Equations
5.9.2 Levitan Almost Periodic and Almost Automorphic Solutions of the Second Order Differential Equations
5.9.3 Quasiperiodic, Bohr/Levitan Almost Periodic, and Recurrent Solutions of the Second Order Differential Equations
5.9.4 Some Generalizations
6 Monotone Nonautonomous Dynamical Systems with a First Integral
6.1 Introduction
6.2 Convergence in NDS with a Strictly Monotone First Integral
6.3 Asymptotically Almost Periodic Motions of NDS with a First Integral
6.4 Applications
6.4.1 Ordinary Differential Equations
6.4.2 Linear Differential Equations
6.4.3 Linear Differential Equations with Constant Matrix
6.4.4 Difference Equations
6.4.5 Linear Difference Equations
6.4.6 Perron–Frobenius Dynamics
6.5 Levinson Center for Monotone NDS with a Strictly Monotone First Integral
6.6 Applications
6.6.1 Nonlinear Differential Equations
6.6.2 Linear Differential Equations
6.6.3 Difference Equations
6.6.4 Linear Difference Equations
7 Monotone Sub-Linear Nonautonomous Dynamical Systems
7.1 Introduction
7.2 Existence and Convergence to Poisson Stable Motions of Monotone NDSs
7.3 Uniformly Stable Monotone Nonautonomous Dynamical Systems
7.4 Sub-Linear Nonautonomous Dynamical Systems
7.5 Comparability by the Character of Recurrence of Motions for Sub-Linear Cocycles
7.6 Application
7.6.1 Ordinary Differential Equations
7.6.2 Functional Differential Equations with Finite Delay
7.6.3 Difference Equations
7.7 Translation-Invariant Monotone Systems
7.8 Time-Dependent Chemical Reaction Networks
7.9 Translation-Invariant Discrete Monotone Systems
Bibliography
Index