Invariance Principle For Dynamical Systems Research Articles

Rates in Almost Sure Invariance Principle for Dynamical Systems with Some Hyperbolicity

We prove the almost sure invariance principle with rate $${o(n^{\varepsilon})}$$ for every $${\varepsilon > 0}$$ for Hölder continuous observables on nonuniformly expanding and nonuniformly hyperbolic transformations with exponential tails. Examples include Gibbs–Markov maps with big images, Axiom A diffeomorphisms, dispersing billiards and a class of logistic and Hénon maps. The best previously proved rate is $${O(n^{1/4} (\log n)^{1/2} (\log \log n)^{1/4})}$$ . As a part of our method, we show that nonuniformly expanding transformations are factors of Markov shifts with simple structure and natural metric (similar to the classical Young towers). The factor map is Lipschitz continuous and probability measure preserving. For this we do not require the exponential tails.

Almost sure invariance principle for dynamical systems by spectral methods

We prove the almost sure invariance principle for stationary R^d--valued processes (with dimension-independent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments.

Almost sure invariance principle for dynamical systems with stretched exponential mixing rates

We prove the almost sure invariance principle for a class of abstract dynamical systems including dynamical systems with stretched exponential mixing rates. The result can be applied to chaotic billiards and hyperbolic attractors with Markov sieves as well as expanding maps of the interval and Axiom A diffeomorphisms.

An invariance principle for nonlinear hybrid and impulsive dynamical systems

In this paper we develop an invariance principle for dynamical systems possessing left-continuous flows. Specifically, we show that left-continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi-infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems. As a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems; namely, state-dependent impulsive dynamical systems. These results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems.

Limit theorems and Markov approximations for chaotic dynamical systems

We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.

An application of the invariance principle to a non-linear, first order evolution problem: Asymptotic stability in non-isothermal magnetohydrodynamics

The purpose of this paper is to study, with the aid of the invariance principle for dynamical systems stated by Dafermos, the asymptotic stability of the null solution of a mixed, non-linear, first order evolution problem. As a particular case, one obtains a theorem on unconditional, asymptotic stability of stationary flows of non-isothermal magnetohydrodynamics in the Boussinesq approximation.

Asymptotic behavior of a class of abstract dynamical systems

In recent years the extension of the second or direct method of Liapunov to distributed parameter systems has gained much attention; specifically Movchan [I] and Zubov [2] have given results paralleling Liapunov’s classical theorems. However, in dealing with the question of asymptotic stability of equilibria, these theorems are not always applicable because of their rather stringent hypotheses on the negative definiteness of the time derivative of Liapunov type functionals. In [3] Barbashin and Krasovskii did weaken somewhat the hypotheses of Liapunov’s theorems for autonomous and periodic systems but it was LaSalle [46] who exploited the invariance of limit sets to obtain a much more general theory, his principal theorem being stated in the form of an invariance principle. This was then extended by Hale [7] for the infinite dimensional space of autonomous functional differential equations and by Miller in [8,9] f or almost periodic ordinary and functional differential equations. These developments suggest the desirability of formulating invariance principles for infinite dimensional spaces suitable for partial differential equations. This paper is in that direction. The direct generalization of autonomous systems of ordinary differential equations to infinite dimensions is the abstract dynamical system. Thus, one would like to state an invariance principle similar to the theorem of LaSalle for autonomous ordinary differential equations which is valid for dynamical systems in general. Brayton and Miranker [lo] stated a type of invariance principle for abstract dynamical systems but their work was limited to a particular class of systems