Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system

Abstract

Chaotic trajectories in Hamiltonian systems may have a peculiar evolution, owing to stickiness effects or migration to adjacent stochastic regions. As a result, the function χ( t), which measures the exponential divergence of nearby trajectories, changes its behaviour within different time intervals. We obtain such trajectories, through numerical integration, for a model 3D Hamiltonian system. Having the plots of χ( t) as a guide, we divide trajectories into segments, each one being assigned an Effective Lyapunov Number (ELN), λ i . We monitor the evolution of the trajectories through a “quasi-integral” time series, which can follow trapping or escape events. Using the time-delay reconstruction scheme, we calculate the correlation dimension, D (2), of each trajectory segment. Our numerical results show that, as the ELN of different segments increases, the correlation dimension of the set on which the trajectory segment is embedded, also tends to increase by a statistically significant amount. This result holds only if the differences of the ELN are relatively large, reflecting motion within different phase-space domains, indicating that the transport process does not have the same statistical properties throughout the phase space. As a consequence, D (2) can serve as a scalar index to descriminate between regions of stickiness and regions of unimpeded transport.