Dynamical aspects of chaotic dynamical systems just after band crises

Abstract

Dynamical aspects of chaotic dynamical systems just after the three band crises are studied by extracting various correlations of dynamical systems. Chaotic systems have both periodic and non-periodic motions of phase points just after the band crises. We decompose the behavior of phase points for dissipative systems into the motions in the three-band attractor and in the repeller. Logistic map and Hénon map are studied as examples of low-dimensional chaotic maps. Driven damped pendulum is taken up as an example of nonlinear ordinary differential equations. Every system exhibits the three-period motion when the local Lyapunov exponent is small and does the non-periodic behavior when the local Lyapunov exponent is large. The generalized power spectrum is used for analyzing the behavior of trajectories. We will analyze the intermittent switching between periodic and non-periodic behaviors by the generalized power spectrum in these dissipative systems. This behavior cannot be found directly only by looking at the thermodynamic functions defined in the frame of the thermodynamic formalism. It is shown that the peculiar behaviors of trajectories in the repeller is clearly extracted.