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Über dieses Buch

The aim of this book is to present a recently developed approach suitable for investigating a variety of qualitative aspects of order-preserving random dynamical systems and to give the background for further development of the theory. The main objects considered are equilibria and attractors. The effectiveness of this approach is demonstrated by analysing the long-time behaviour of some classes of random and stochastic ordinary differential equations which arise in many applications.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Abstract not available
Igor Čhuešhov

1. General Facts about Random Dynamical Systems

Abstract
  • 1.1 Metric Dynamical Systems
  • 1.2 Concept of RDS
  • 1.3 Random Sets
  • 1.4 Dissipative, Compact and Asymptotically Compact RDS
  • 1.5 Trajectories
  • 1.6 Omega-limit Sets
  • 1.7 Equilibria
  • 1.8 Random Attractors
  • 1.9 Dissipative Linear and Affine RDS
  • 1.10 Connection Between Attractors and Invariant Measures
Igor Čhuešhov

2. Generation of Random Dynamical Systems

Abstract
  • 2.1 RDS Generated by Random Differential Equations
  • 2.2 Deterministic Invariant Sets
  • 2.3 The Itô and Stratonovich Stochastic Integrals
  • 2.4 RDS Generated by Stochastic Differential Equations
  • 2.5 Relations Between RDE and SDE
Igor Čhuešhov

3. Order-Preserving Random Dynamical Systems

Abstract
  • 3.1 Partially Ordered Banach Spaces
  • 3.2 Random Sets in Partially Ordered Spaces
  • 3.3 Definition of Order-Preserving RDS
  • 3.4 Sub-Equilibria and Super-Equilibria
  • 3.5 Equilibria
  • 3.6 Properties of Invariant Sets of Order-Preserving RDS
  • 3.7 Comparison Principle
Igor Čhuešhov

4. Sublinear Random Dynamical Systems

Abstract
  • 4.1 Sublinear and Concave RDS
  • 4.2 Equilibria and Semi-Equilibria for Sublinear RDS
  • 4.3 Almost Equilibria
  • 4.4 Limit Set Trichotomy for Sublinear RDS
  • 4.5 Random Mappings
  • 4.6 Positive Affine RDS
Igor Čhuešhov

5. Cooperative Random Differential Equations

Abstract
  • 5.1 Basic Assumptions and the Existence Theorem
  • 5.2 Generation of RDS
  • 5.3 Random Comparison Principle
  • 5.4 Equilibria, Semi-Equilibria and Attractors
  • 5.5 Random Equations with Concavity Properties
  • 5.6 One-Dimensional Explicitly Solvable Random Equations
  • 5.7 Applications
    • 5.7.1 Random Biochemical Control Circuit
    • 5.7.2 Random Gonorrhea Model
    • 5.7.3 Random Model of Symbiotic Interaction
    • 5.7.4 Random Gross-Substitute System
  • 5.8 Order-Preserving RDE with Non-Standard Cone
Igor Čhuešhov

6. Cooperative Stochastic Differential Equations

Abstract
  • 6.1 Main Assumptions
  • 6.2 Generation of Order-Preserving RDS
  • 6.3 Conjugacy with Random Differential Equations
  • 6.4 Stochastic Comparison Principle
  • 6.5 Equilibria and Attractors
  • 6.6 One-Dimensional Stochastic Equations
    • 6.6.1 Stochastic Equations on \(\mathbb{R}_+\)
    • 6.6.2 Stochastic Equations on a Bounded Interval
  • 6.7 Stochastic Equations with Concavity Properties
  • 6.8 Applications
    • 6.8.1 Stochastic Biochemical Control Circuit
    • 6.8.2 Stochastic Gonorrhea Model
    • 6.8.3 Stochastic Model of Symbiotic Interaction
    • 6.8.4 Lattice Models of Statistical Mechanics
Igor Čhuešhov

References

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Abstract not available
Igor Čhuešhov

Index

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Abstract not available
Igor Čhuešhov

Backmatter

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