Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intermittent two-point dynamics at the transition to chaos for random circle endomorphisms

Vincent P. H. Goverse, Ale Jan Homburg, Jeroen S. W. Lamb

TL;DR

This work analyzes the transition to chaos in random circle endomorphisms via intermittent two-point dynamics. By formulating annealed Koopman operators and their moment Lyapunov function, it connects the sign of the Lyapunov exponent $\lambda$ to distinct dynamical regimes: synchronisation ($\lambda<0$), intermittency at the transition ($\lambda=0$), and chaos ($\lambda>0$). The authors prove the existence of infinite stationary measures off the diagonal at $\lambda=0$ and provide precise growth rates near the diagonal, with refinements in the chaotic regime governed by the negative zero $\gamma$ of $\Lambda$. Their inducing constructions yield stationary measures for the two-point motion with full support off the diagonal and reveal how near-diagonal mass scales with $\varepsilon$ in different $\lambda$-regimes. Overall, the paper develops a spectral-analytic framework for two-point random dynamics that captures intermittency and infinite ergodic behavior at the onset of chaos, with potential extensions to higher dimensions and broader noise models.

Abstract

We establish the existence of intermittent two-point dynamics and infinite stationary measures for a class of random circle endomorphisms with zero Lyapunov exponent, as a dynamical characterisation of the transition from synchronisation (negative Lyapunov exponent) to chaos (positive Lyapunov exponent).

Intermittent two-point dynamics at the transition to chaos for random circle endomorphisms

TL;DR

This work analyzes the transition to chaos in random circle endomorphisms via intermittent two-point dynamics. By formulating annealed Koopman operators and their moment Lyapunov function, it connects the sign of the Lyapunov exponent to distinct dynamical regimes: synchronisation (), intermittency at the transition (), and chaos (). The authors prove the existence of infinite stationary measures off the diagonal at and provide precise growth rates near the diagonal, with refinements in the chaotic regime governed by the negative zero of . Their inducing constructions yield stationary measures for the two-point motion with full support off the diagonal and reveal how near-diagonal mass scales with in different -regimes. Overall, the paper develops a spectral-analytic framework for two-point random dynamics that captures intermittency and infinite ergodic behavior at the onset of chaos, with potential extensions to higher dimensions and broader noise models.

Abstract

We establish the existence of intermittent two-point dynamics and infinite stationary measures for a class of random circle endomorphisms with zero Lyapunov exponent, as a dynamical characterisation of the transition from synchronisation (negative Lyapunov exponent) to chaos (positive Lyapunov exponent).

Paper Structure

This paper contains 20 sections, 31 theorems, 254 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Context
  4. Organization of the paper
  5. Setting and strategy of the proof
  6. Hypotheses
  7. Strategy of the proof
  8. Koopman operators
  9. Twisted Koopman operator and moment Lyapunov function
  10. Linearized Koopman operator
  11. Two-point Koopman operator
  12. Topological random dynamics
  13. Zero Lyapunov exponent
  14. Positive Lyapunov exponent
  15. Negative Lyapunov exponent
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $T_\omega$ be a random dynamical system satisfying hypotheses h:endo to h:openaftertwo, $\lambda$ be its Lyapunov exponent, and let $T^{(2)}_\omega$ be the corresponding two-point random dynamical system. Then,

Figures (3)

  • Figure 1: The graph of $T_{\nu}$\ref{['eq:chatgpt']} at $\nu=0.6$.
  • Figure 2: Illustration of aspects of the dynamics in Example \ref{['ex:T_nu']} regarding the family of random maps $T_{\nu,\omega}$ with $\nu = 0.6$.
  • Figure 3: Illustration of aspects of the dynamics of Example \ref{['ex:T_nu']} regarding the family of random maps $T_{\nu,\omega}$, with $\vartheta = 0.5$ and varying $\nu$.

Theorems & Definitions (68)

  • Theorem 1.1: Topological random dynamics
  • Theorem 1.2: Stationary measures
  • Example 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 58 more