Qualitative and Asymptotic Analysis of Differential Equations With Random Perturbations

by Samoilenko, Anatoliy M.; Stanzhytskyi, Oleksandr

• ISBN13: 9789814329064
• ISBN10: 9814329061
• Format: Hardcover
• Copyright: 2011-06-30
• Publisher: World Scientific Pub Co Inc

Summary

Differential equations with random perturbations are the mathematical models of real-world processes that cannot be described via deterministic laws, and their evolution depends on the random factors. The modern theory of differential equations with random perturbations is on the edge of two mathematical disciplines: random processes and ordinary differential equations. Consequently, the sources of these methods come both from the theory of random processes and from the classic theory of differential equations.This work focuses on the approach to stochastic equations from the perspective of ordinary differential equations. For this purpose, both asymptotic and qualitative methods which appeared in the classical theory of differential equations and nonlinear mechanics are developed.

Table of Contents

Introduction	p. vii
Differential equations with random right-hand sides and impulsive effects	p. 1
An impulsive process as a solution of an impulsive system	p. 2
Dissipativity	p. 3
Stability and Lyapunov functions	p. 10
Stability of systems with permanently acting random perturbations	p. 20
Solutions periodic in the restricted sense	p. 23
Periodic solutions of systems with small perturbations	p. 28
Periodic solutions of linear impulsive systems	p. 34
Weakly nonlinear systems	p. 40
Comments and References	p. 49
Invariant sets for systems with random perturbations	p. 53
Invariant sets for systems with random right-hand sides	p. 54
Invariant sets for stochastic Ito systems	p. 60
The behaviour of invariant sets under small perturbations	p. 64
A study of stability of an equilibrium via the reduction principle for systems with regular random perturbations	p. 72
Stability of an equilibrium and the reduction principle for Ito type systems	p. 76
A study of stability of the invariant set via the reduction principle. Regular perturbations	p. 84
Stability of invariant sets and the reduction principle for Ito type systems	p. 92
Comments and References	p. 101
Linear and quasilinear stochastic Ito systems	p. 105
Mean square exponential dichotomy	p. 106
A study of dichotomy in terms of quadratic forms	p. 115
Linear system solutions that are mean square bounded on the semiaxis	p. 127
Quasilinear systems	p. 135
Linear system solutions that are probability bounded on the axis. A generalized notion of a solution	p. 138
Asymptotic equivalence of linear systems	p. 148
Conditions for asymptotic equivalence of nonlinear systems	p. 178
Comments and References	p. 185
Extensions of Ito systems on a torus	p. 189
Stability of invariant tori	p. 190
Random invariant tori for linear extensions	p. 196
Smoothness of invariant tori	p. 205
Random invariant tori for nonlinear extensions	p. 209
An ergodic theorem for a class of stochastic systems having a toroidal manifold	p. 213
Comments and References	p. 222
The averaging method for equations with random perturbations	p. 225
A substantiation of the averaging method for systems with impulsive effect	p. 226
Asymptotics of normalized deviations of averaged solutions	p. 232
Applications to the theory of nonlinear oscillations	p. 247
Averaging for systems with impulsive effects at random times	p. 259
The second theorem of M. M. Bogolyubov for systems with regular random perturbations	p. 267
Averaging for stochastic Ito systems. An asymptotically finite interval	p. 276
Averaging on the semiaxis	p. 282
The averaging method and two-sided bounded solutions of Ito systems	p. 285
Comments and References	p. 291
Bibliography	p. 295
Index	p. 311
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