Narrow Results

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincaré maps in models of nonholonomic mechanics.

The influence of white and colored noise on the dynamics of self-sustained oscillator is considered in the regime of spiral (phase-coherent) chaos. The effective phase diffusion coefficient and power spectra are analyzed for noisy chaotic self-sustained oscillations. We show that chaotic oscillations can be synchronized by external narrow-band noise. Effects of chaos synchronization are compared for narrow-band noise signals with similar spectra but distinct probability distributions.