Footprints of sticky motion in the phase space of higher dimensional nonintegrable conservative systems

TL;DR

This paper investigates sticky motion in high-dimensional conservative systems, revealing how it affects phase space dynamics and can be detected using statistical measures related to Lyapunov exponents.

Contribution

It introduces four measures to quantify sticky motion in high-dimensional phase spaces and systematically studies their effectiveness across various coupled maps.

Findings

Sticky motion occurs above a certain nonlinearity threshold in high dimensions.

Transition from quasiregular to chaotic motion is simultaneous across all unstable directions.

Statistical measures effectively detect sticky motion in high-dimensional systems.

Abstract

"Sticky" motion in mixed phase space of conservative systems is difficult to detect and to characterize, in particular for high dimensional phase spaces. Its effect on quasi-regular motion is quantified here with four different measures, related to the distribution of the finite time Lyapunov exponents. We study systematically standard maps from the uncoupled two-dimensional case up to coupled maps of dimension $20$. We find that sticky motion in all unstable directions above a threshold $K_{d}$ of the nonlinearity parameter $K$ for the high dimensional cases $d=10,20$. Moreover, as $K$ increases we can clearly identify the transition from quasiregular to totally chaotic motion which occurs simultaneously in all unstable directions. The results show that all four statistical measures sensitively probe sticky motion in high dimensional systems.