Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Impulsive Lorenz semiflows: Physical measures, statistical stability and entropy stability

José F. Alves, Wael Bahsoun

Abstract

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures of the semiflows are close, in the weak* topology, to the unique physical measure of the Lorenz flow. A similar conclusion holds for the entropies associated with the physical measures.

Impulsive Lorenz semiflows: Physical measures, statistical stability and entropy stability

Abstract

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures of the semiflows are close, in the weak* topology, to the unique physical measure of the Lorenz flow. A similar conclusion holds for the entropies associated with the physical measures.
Paper Structure (14 sections, 18 theorems, 75 equations, 2 figures)

This paper contains 14 sections, 18 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Physical measures and $u$-Gibbs measures
  3. Impulsive semiflows
  4. Lorenz flows
  5. Piecewise hyperbolic diffeomorphisms
  6. Physical and $u$-Gibbs measures
  7. Lorenz maps
  8. Statistical stability
  9. Impulsive Lorenz semiflows
  10. Poincaré return map
  11. Physical measures
  12. Impulsive stability of Lorenz flows
  13. Statistical stability
  14. Entropy stability

Key Result

Theorem A

Let $X$ be a Lorenz flow and $(M,X,\Sigma,\varphi)$ be an impulsive dynamical system such that $\varphi(\Sigma)\subset B^+(\Sigma).$ If $\varphi$ is a $C^2$ map sufficiently close to $\operatorname{inc}_\Sigma$ in $\operatorname{Emb}^1(\Sigma,M)$, then the semiflow associated with $(M,X,\Sigma,\varp

Figures (2)

  • Figure 2: The maps $\psi$ and $\psi^{-1}$
  • Figure 3: The maps $F_Y$ and $\tilde{F}_Y$

Theorems & Definitions (31)

  • Theorem A
  • Theorem B
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 21 more