Chaos and complexity in the dynamics of nonlinear Alfv\'en waves in a magnetoplasma

TL;DR

This paper investigates the nonlinear dynamics of Alfvén wave envelopes in magnetoplasma, revealing conditions for periodic, quasi-periodic, and chaotic states, and linking chaos to potential wave turbulence in space plasmas.

Contribution

It introduces a low-dimensional dynamical model for Alfvén wave interactions, identifying chaos emergence and characterizing complex behaviors in plasma wave dynamics.

Findings

Chaotic states occur at certain wave numbers with high modulational instability.

Lyapunov exponents and bifurcation analysis confirm the transition to chaos.

Chaotic phase space complexity aligns with known dynamical systems like Hénon and Lorenz.

Abstract

The nonlinear dynamics of circularly polarized dispersive Alfv{\'e}n wave (AW) envelopes coupled to the driven ion sound waves of plasma slow response is studied in a uniform magnetoplasma. By restricting the wave dynamics to a few number of harmonic modes, a low-dimensional dynamical model is proposed to describe the nonlinear wave-wave interactions. It is found that two subintervals of the wave number of modulation $k$ of AW envelope exists, namely $(3/4)k_c<k<k_c$ and $0<k<(3/4)k_c$, where $k_c$ is the critical value of $k$ below which the modulational instability (MI) occurs. In the former, where the MI growth rate is low, the periodic and/or quasi-periodic states are shown to occur, whereas the latter, where the MI growth is high, brings about the chaotic states. The existence of these states is established by the analyses of Lyapunov exponent spectra together with the bifurcation diagram and phase-space portraits of dynamical variables. Furthermore, the complexities of chaotic phase spaces in the nonlinear motion are measured by the estimations of the correlation dimension (CD) as well as the approximate entropy (ApEn), and compared with those for the known H{\'e}non map and the Lorenz system in which a good qualitative agreement is noted. The chaotic motion thus predicted in a low-dimensional model can be a prerequisite for the onset of Alfv{\'e}nic wave turbulence to be observed in a higher dimensional model that and is relevant in the Earth's ionosphere and magnetosphere.