Dynamical systems of Langevin type

Dynamical systems of Langevin type are deterministic mappings that arise if the Gaussian white noise of a Langevin equation is replaced by a deterministic chaotic dynamics. We describe the various types of mappings that can be obtained in this way, and investigate typical transition scenarios from complicated non-Gaussian to Langevin-like behavior that occur if a time scale parameter is changed. We define a new characteristic quantify fir ergodic mappings called ‘effective Langevin radius’, which is related to the critical time scale where the marginal invariant density of the dynamical system of Langevin type loses its differentiability. The invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force. Some general symmetry properties of the invariant densities are discussed.