Small perturbations of chaotic dynamical systems. (English. Russian original) Zbl 0702.58063

Blank, M. L.

Russ. Math. Surv. 44, No. 6, 1-33 (1989); translation from Usp. Mat. Nauk 44, No. 6(270), 3-28 (1989).

The author mainly investigates statistical properties of chaotic dynamical systems such as the existence and properties of invariant measures on a stochastic attractor, the existence and estimation of the rate of correlation decay on these measures, the applicability of the central limit theorem, and so on. It is possible to investigate all these questions thoroughly by the method of symbolic dynamics.
The paper has the following structure. In the first section the author presents a description of the main constructions, methods, and results connected with stochastic attractors. In the second section he investigates the stability of stochastic attractors relative to small random perturbations. In the third section various effects in space discretization of dynamical systems are investigated. In the fourth section time discretizations of dynamical systems are studied.

MSC:

37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37E99	Low-dimensional dynamical systems
28D05	Measure-preserving transformations

Keywords:

hyperbolic systems; chaotic dynamical systems; stochastic attractor; symbolic dynamics; stability; small random perturbations; space discretization; time discretizations