Finding the effective dynamics to make rare events typical in chaotic maps

TL;DR

This paper introduces a framework to transform chaotic maps into effective models where rare, atypical trajectories become typical, enabling better understanding and control of rare events in chaotic systems.

Contribution

It proposes a method to find topologically-conjugate maps that make rare events typical, extending large-deviation techniques to chaotic dynamics.

Findings

Effective maps can replicate atypical trajectories as typical ones.

The method parallels the Doob transform used in stochastic processes.

Application to phase transitions in finite-time Lyapunov exponents.

Abstract

Dynamical fluctuations or rare events associated with atypical trajectories in chaotic maps due to specific initial conditions can crucially determine their fate, as the may lead to stability islands or regions in phase space otherwise displaying unusual behavior. Yet, finding such initial conditions is a daunting task precisely because of the chaotic nature of the system. In this work, we circumvent this problem by proposing a framework for finding an effective topologically-conjugate map whose typical trajectories correspond to atypical ones of the original map. This is illustrated by means of examples which focus on counterbalancing the instability of fixed points and periodic orbits, as well as on the characterization of a dynamical phase transition involving the finite-time Lyapunov exponent. The procedure parallels that of the application of the generalized Doob transform in the stochastic dynamics of Markov chains, diffusive processes and open quantum systems, which in each case results in a new process having the prescribed statistics in its stationary state. This work thus brings chaotic maps into the growing family of systems whose rare fluctuations -- sustaining prescribed statistics of dynamical observables -- can be characterized and controlled by means of a large-deviation formalism.