Low-dimensional chaos in the single wave model for self-consistent wave-particle Hamiltonian

TL;DR

This paper investigates the nonlinear dynamics and chaos in the self-consistent wave-particle Hamiltonian model, revealing how chaos emerges with increasing particles and under specific energetic conditions.

Contribution

It provides a detailed analysis of chaos development in the single wave model, highlighting mechanisms like homoclinic tangles and resonance overlap.

Findings

Chaos arises from homoclinic tangles and resonance overlap.

High energy levels can cause the wave amplitude to vanish, inducing strong chaos.

The model transitions from integrable to chaotic as particle number increases.

Abstract

We analyze nonlinear aspects of the self-consistent wave-particle interaction using Hamiltonian dynamics in the single wave model, where the wave is modified due to the particle dynamics. This interaction plays an important role in the emergence of plasma instabilities and turbulence. The simplest case, where one particle (N = 1) is coupled with one wave (M = 1), is completely integrable, and the nonlinear effects reduce to the wave potential pulsating while the particle either remains trapped or circulates forever. On increasing the number of particles (N = 2, M = 1), integrability is lost and chaos develops. Our analyses identify the two standard ways for chaos to appear and grow (the homoclinic tangle born from a separatrix, and the resonance overlap near an elliptic fixed point). Moreover, a strong form of chaos occurs when the energy is high enough for the wave amplitude to vanish occasionally.