Fast chaos versus white noise: entropy analysis and a Fokker–Planck model for the slow dynamics

We show in detail how methods of time series analysis such as dimension and entropy estimates support the idea that fast low-dimensional chaos can be modelled properly by noise. Motivated by this observation, we derive a formalism by which fast chaotic degrees of freedom in systems with time scale separation can be replaced by a suitable stochastic process. A Fokker–Planck equation for the phase space density of the slow degrees of freedom is derived from the full deterministic system by a projection operator technique together with a perturbation expansion. We compare two different projection strategies and illustrate the resulting equations by specifying them for a simple model system.