Poincaré recurrence theory and its applications to nonlinear physics

Theoretical results concerning the Poincaré recurrence problem and their application to problems in nonlinear physics are reviewed. The effects of noise, nonhyperbolicity, and the size of the recurrence region on the characteristics of the recurrence time sequence are examined. Relations of the recurrence time sequence dimension to the Lyapunov exponents and the Kolmogorov entropy are demonstrated. Methods for calculating the local and global attractor dimensions and the Afraimovich – Pesin dimension are presented. Methods using the Poincaré recurrence times to diagnose the stochastic resonance and the synchronization of chaos are described.