Strong Stochastic Stability Research Articles

On stochastic stability of non-uniformly expanding interval maps

We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic stability.

Strong stochastic stability for non-uniformly expanding maps

AbstractWe consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps.Astérisque 286(2003), 25–62], where the stochastic stability in the$\mathrm {weak}^*$topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the$L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding.Invent. Math. 140(2000), 351–398] and the second one is the class ofViana maps.

The method of parametric Lyapunov functions for Markov dynamic systems

The notion of parametric Lyapunov function is introduced for Markov dynamic systems. The existence of a function of this kind is shown to be a necessary and sufficient condition for the strong stochastic stability of an equilibrium. In terms of parametric Lyapunov functions, a sufficient criterion is proved for asymptotic strong stochastic stability in the case of Feller Markov chains. Some examples are given showing the efficiency of the method proposed.

Uncertain jumping systems with strong and weak functional delays

This paper provides complete results on the stability behavior of a class of uncertain dynamical systems with jumping parameters and functional time-delays. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. The parametric uncertainties are norm-bounded appearing in all system matrices and the delay factor depends on the mode of operation. Notions of weak and strong stochastic stability for the jumping system are developed depending on the available information using a prescribed H ∞ -performance. Memoryless and delayed-state feedback are considered to guarantee the closed-loop stability. All the results are cast into linear matrix inequalities format. A numerical example is given to illustrate the developed results.

Stochastic Stability¶for Contracting Lorenz Maps and Flows

In a previous work [M], we proved the existence of absolutely continuous invariant measures for contracting Lorenz-like maps, and constructed Sinai–Ruelle–Bowen measures f or the flows that generate them. Here, we prove stochastic stability for such one-dimensional maps and use this result to prove that the corresponding flows generating these maps are stochastically stable under small diffusion-type perturbations, even though, as shown by Rovella [Ro], they are persistent only in a measure theoretical sense in a parameter space. For the one-dimensional maps we also prove strong stochastic stability in the sense of Baladi and Viana[BV].

Strong stochastic stability and rate of mixing for unimodal maps

We consider small random perturbations of a large class of nonuniformly hyperbolic unimodal maps and prove stochastic stability in the strong sense (L -convergence of invariant densities) and uniform bounds for the exponential rate of decay of correlations. Our method is based on an analysis of the spectrum of a modified Perron-Frobenius operator for a tower extension of the Markov chain.