Noise-induced phenomena in one-dimensional maps
Interaction between deterministic chaos and stochastic randomness has been an important problem in nonlinear dynamical systems studies. Noise-induced phenomena are understood as drastic change of natural invariant densities by adding external noise to a deterministic dynamical systems, resulting qualitative transition of observed nonlinear phenomena. Stochastic resonance, noise-induced synchronization, and noise-induced chaos are typical examples in this scheme. The simplest mathematical model for problem is one-dimensional map stochastically perturbed by noise. In this presentation, we discuss typical behavior of noised dynamical systems based on numerically observed noise-induced phenomena in logistic map, Belousov-Zhabotinsky map and modified Lasota-Mackey map. Our observation indicates that (i) both noise-induced chaos and noise-induced order may coexist, and that (ii) asymptotical periodicity of densities varies according to noise amplitude. An application to time-series analysis of rotating fluid is also exhibited.