Monotone Random Systems Theory and Applications

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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Igor Čhuešhov

Backmatter