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Discrete Dynamical Systems with Random Impulses

J. Kováč, J. Veselý, K. Janková

Abstract

We study the behaviour of discrete dynamical systems generated by a continuous map $f$ of a compact real interval into itself where at randomly chosen times a function different from $f$ - so called impulse function is applied. We show that both the splittting property and the average contraction property guarantee the stability of the system. We give a number of examples where the verification of these properties is simple.

Discrete Dynamical Systems with Random Impulses

Abstract

We study the behaviour of discrete dynamical systems generated by a continuous map of a compact real interval into itself where at randomly chosen times a function different from - so called impulse function is applied. We show that both the splittting property and the average contraction property guarantee the stability of the system. We give a number of examples where the verification of these properties is simple.

Paper Structure

This paper contains 10 sections, 12 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Impulse system as a dynamical system controlled by a Markov chain
  3. Invariant distribution and reversed chain
  4. Main results
  5. Proof of Theorem 1
  6. Proofs of the auxiliary lemmas
  7. Proof of Theorem 2
  8. Proof of Theorem 3
  9. Final remarks and examples
  10. Appendix

Key Result

Theorem 1.1

If $\mathbb{P}^{-}(S)=1$, then for any probability measure $\hat{\mu}$ defined on $(\hat{M},\hat{\mathcal{S}})$ and for any continuous and bounded function $\hat{h}:\hat{M}\to \mathbb{R}$ where $\hat{\pi}:S\to \hat{M}$ is a function given by $\hat{\pi}(\xi)=(\xi_0,\pi(\xi))$ and $\hat{\pi}\odot \mathbb{P}^{-}$ denotes a pushforward measure on $(\hat{M},\hat{\mathcal{S}})$ defined as for $\hat{B}\

Figures (2)

  • Figure 1: The functions $f(x)=\left(1-\frac{\sqrt{2}}{2}\right)x+\sqrt{2}$ (blue) and $g(x)=\sqrt{x}$ (red) on the interval $\left<0,2\right>$.
  • Figure 2: Comparison of the empirical distribution functions with the distribution function $F$ of of the limit distribution $\nu$.

Theorems & Definitions (26)

  • Remark 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 2.6
  • ...and 16 more