Chaos and complexity in the dynamics of nonlinear Alfvén waves in a magnetoplasma

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.