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.
Author(s): Igor Chueshov
Series: Lecture Notes in Mathematics
Edition: 1
Publisher: Springer
Year: 2002
Language: English
Pages: 234
front-matter......Page 1
0. Introduction......Page 9
1.1 Metric Dynamical Systems......Page 16
1.2 Concept of RDS......Page 20
1.3 Random Sets......Page 25
1.4 Dissipative, Compact and Asymptotically Compact RDS......Page 31
1.5 Trajectories......Page 39
1.6 Omega-limit Sets......Page 41
1.7 Equilibria......Page 45
1.8 Random Attractors......Page 48
1.9 Dissipative Linear and Affine RDS......Page 52
1.10 Connection Between Attractors and Invariant Measures......Page 56
2.1 RDS Generated by Random Differential Equations......Page 61
2.2 Deterministic Invariant Sets......Page 67
2.3 The Itô and Stratonovich Stochastic Integrals......Page 71
2.4 RDS Generated by Stochastic Differential Equations......Page 76
2.5 Relations Between Random and Stochastic Differential Equations......Page 82
3.1 Partially Ordered Banach Spaces......Page 88
3.2 Random Sets in Partially Ordered Spaces......Page 93
3.3 Definition of Order-Preserving RDS......Page 98
3.4 Sub-Equilibria and Super-Equilibria......Page 100
3.5 Equilibria......Page 105
3.6 Properties of Invariant Sets of Order-Preserving RDS......Page 110
3.7 Comparison Principle......Page 114
4.1 Sublinear and Concave RDS......Page 117
4.2 Equilibria and Semi-Equilibria for Sublinear RDS......Page 120
4.3 Almost Equilibria......Page 126
4.4 Limit Set Trichotomy for Sublinear RDS......Page 129
4.5 Random Mappings......Page 136
4.6 Positive Affine RDS......Page 142
5.1 Basic Assumptions and the Existence Theorem......Page 146
5.2 Generation of RDS......Page 148
5.3 Random Comparison Principle......Page 153
5.4 Equilibria, Semi-Equilibria and Attractors......Page 159
5.5 Random Equations with Concavity Properties......Page 163
5.6 One-Dimensional Explicitly Solvable Random Equations......Page 169
5.7.1 Random Biochemical Control Circuit......Page 174
5.7.2 Random Gonorrhea Model......Page 178
5.7.3 Random Model of Symbiotic Interaction......Page 179
5.7.4 Random Gross-Substitute System......Page 181
5.8 Order-Preserving RDE with Non-Standard Cone......Page 183
6.1 Main Assumptions......Page 187
6.2 Generation of Order-Preserving RDS......Page 188
6.3 Conjugacy with Random Differential Equations......Page 190
6.4 Stochastic Comparison Principle......Page 194
6.5 Equilibria and Attractors......Page 196
6.6 One-Dimensional Stochastic Equations......Page 201
6.7 Stochastic Equations with Concavity Properties......Page 216
6.8.1 Stochastic Biochemical Control Circuit......Page 221
6.8.2 Stochastic Gonorrhea Model......Page 223
6.8.3 Stochastic Model of Symbiotic Interaction......Page 224
6.8.4 Lattice Models of Statistical Mechanics......Page 225
7. References......Page 228
8. Index......Page 233
