A discrete dynamical system: The poor man's magnetohydrodynamic (PMMHD) equations

Area of Science:

• Plasma Physics
• Dynamical Systems Theory
• Computational Physics

Background:

• Incompressible plasma dynamics are complex and often studied using magnetohydrodynamics (MHD).
• Traditional analytical and numerical methods face challenges in capturing the full spectrum of plasma behaviors.
• Discrete dynamical systems offer a simplified yet insightful framework for complex phenomena.

Purpose of the Study:

• To derive a discrete dynamical system from 3D incompressible MHD equations.
• To analyze the properties of the resulting six-dimensional (6D) map.
• To explore the system's potential for understanding plasma dynamics and related phenomena.

Main Methods:

• Application of a Fourier-Galerkin procedure to 3D incompressible MHD equations.
• Derivation of a 6D discrete map with logistic and nonlinear terms.
• Analysis of the map's energy conservation, Lyapunov exponents, and phase space behavior.

Main Results:

• The 6D map preserves total energy in the ideal MHD approximation.
• The system exhibits sensitive dependence on initial conditions and positive Lyapunov exponents, indicating chaotic behavior.
• All fixed points of the standard MHD equations, including fluid, Alfvénic, and Taylor force-free solutions, are recovered.
• Evidence of kinematic dynamo action is observed.

Conclusions:

• Discrete dynamical systems, like the derived 6D map, are valuable tools for studying incompressible plasmas.
• The model provides insights into plasma dynamics, energy conservation, and chaotic properties.
• The framework supports further investigation into phenomena such as kinematic dynamo action in plasmas.