Qualitative and asymptotic analysis of differential equations with random perturbations / Anatoliy M. Samoilenko, Oleksandr Stanzhytskyi.

Format:

Book

Author/Creator:

Samoĭlenko, A. M. (Anatoliĭ Mikhaĭlovich)

Contributor:

Stanzhytskyi, Oleksandr.

Series:

World Scientific series on nonlinear science. Monographs and treatises ; Series A, v. 78.

World Scientific series on nonlinear science. Series A, Monographs and treatises, 1793-1010 ; v. 78

Language:

English

Subjects (All):

Differential equations, Nonlinear.

Perturbation (Mathematics).

Physical Description:

1 online resource (323 p.)

Edition:

1st ed.

Place of Publication:

Singapore : World Scientific, c2011.

Language Note:

English

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 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.

Contents:

Contents; Introduction; Chapter 1 Differential equations with random right-hand sides and impulsive effects; 1.1 An impulsive process as a solution of an impulsive system; 1.2 Dissipativity; 1.3 Stability and Lyapunov functions; 1.4 Stability of systems with permanently acting random perturbations; 1.5 Solutions periodic in the restricted sense; 1.6 Periodic solutions of systems with small perturbations; 1.7 Periodic solutions of linear impulsive systems; 1.8 Weakly nonlinear systems; 1.9 Comments and References; Chapter 2 Invariant sets for systems with random perturbations

2.1 Invariant sets for systems with random right-hand sides2.2 Invariant sets for stochastic Ito systems; 2.3 The behaviour of invariant sets under small perturbations; 2.4 A study of stability of an equilibrium via the reduction principle for systems with regular random perturbations; 2.5 Stability of an equilibrium and the reduction principle for Ito type systems; 2.6 A study of stability of the invariant set via the reduction principle. Regular perturbations; 2.7 Stability of invariant sets and the reduction principle for Ito type systems; 2.8 Comments and References

Chapter 3 Linear and quasilinear stochastic Ito systems3.1 Mean square exponential dichotomy; 3.2 A study of dichotomy in terms of quadratic forms; 3.3 Linear system solutions that are mean square bounded on the semiaxis; 3.4 Quasilinear systems; 3.5 Linear system solutions that are probability bounded on the axis. A generalized notion of a solution; 3.6 Asymptotic equivalence of linear systems; 3.7 Conditions for asymptotic equivalence of nonlinear systems; 3.8 Comments and References; Chapter 4 Extensions of Ito systems on a torus; 4.1 Stability of invariant tori

4.2 Random invariant tori for linear extensions4.3 Smoothness of invariant tori; 4.4 Random invariant tori for nonlinear extensions; 4.5 An ergodic theorem for a class of stochastic systems having a toroidal manifold; 4.6 Comments and References; Chapter 5 The averaging method for equations with random perturbations; 5.1 A substantiation of the averaging method for systems with impulsive effect; 5.2 Asymptotics of normalized deviations of averaged solutions; 5.3 Applications to the theory of nonlinear oscillations; 5.4 Averaging for systems with impulsive effects at random times

5.5 The second theorem of M. M. Bogolyubov for systems with regular random perturbations5.6 Averaging for stochastic Ito systems. An asymptotically finite interval; 5.7 Averaging on the semiaxis; 5.8 The averaging method and two-sided bounded solutions of Ito systems; 5.9 Comments and References; Bibliography; Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786613433541

9781283433549

1283433540

9789814329071

981432907X

OCLC:

774956316