The Gross Pitaevski map as a chaotic dynamical system

TL;DR

This paper analyzes the Gross Pitaevski map as a classical dynamical system, revealing strong chaos through Lyapunov exponents, exponential energy growth, and an integrable-chaotic transition in resonant cases.

Contribution

It provides a systematic analysis of the Gross Pitaevski map's chaotic behavior, including analytical computation of Lyapunov spectra and exploration of resonant and non-resonant cases.

Findings

Strong chaotic behavior evidenced by Lyapunov exponents

Exponential energy growth linked to rotational invariance

Resonant cases show an integrable-chaotic transition

Abstract

The Gross Pitaevski map is a discrete time, split operator version of the Gross Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own right. A systematic analysis of Lyapunov exponents exposes strongly chaotic behavior. Exponential growth of energy is then shown to be a direct consequence of rotational invariance and for stationary solutions the full spectrum of Lyapunov exponents is analytically computed. The present analysis includes the resonant case, when the free rotation period is commensurate to $2\pi$, and the map has countably many constants of the motion. Except for lowest order resonances, this case exhibits an integrable-chaotic transition.