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

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 Nth order Chebychev maps. The leading order corrections are universal for 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.