Stochastic differential equations driven by deterministic chaotic maps: analytic solutions of the Perron-Frobenius equation

TL;DR

This paper derives analytic solutions for the Perron-Frobenius equation in discrete-time dynamical systems driven by chaotic maps, revealing universal corrections and the impact of correlations on Gaussian convergence.

Contribution

It provides the first analytic solutions to the Perron-Frobenius equation for systems driven by Chebychev maps, including corrections to Fokker-Planck equations and analysis of correlated driving forces.

Findings

Universal leading order corrections for N≥4 Chebychev maps.

Different corrections for N=2 and N=3 Chebychev maps.

Strong correlations can prevent Gaussian limit convergence.

Abstract

We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to ordinary Fokker-Planck equations in leading order of the time scale separation parameter. We present analytic solutions to the equations for the example of driving forces generated by N-th order Chebychev maps. The leading order corrections are universal for N larger or equal to 4 but different for N=2 and N=3. We also study diffusively coupled Chebychev maps as driving forces, where strong correlations may prevent convergence to Gaussian limit behavior.