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

A discrete dynamical system is derived, via a Fourier-Galerkin procedure, from three-dimensional equations describing incompressible plasmas in the magnetohydrodynamic (MHD) framework. The obtained six-dimensional (6D) map, consisting of logistic and nonlinear terms, can provide useful insights into incompressible plasmas dynamics when bifurcation parameters, controlling dissipative and coupling terms, are changed. The map preserves the total energy in the ideal MHD approximation (i.e., by neglecting dissipative terms), manifests a sensitive dependence to the initial conditions as well as at least one Lyapunov exponent is positive (as for chaotic systems), and is characterized by a dissipative nature of its phase space. Moreover, all fixed points of the usual MHD equations are recovered, including the fluid fixed point, the Alfvénic point, and the Taylor force-free solution. Finally, also some interesting properties, as the existence of a kinematic dynamo action, are evidenced, suggesting that discrete dynamical systems deserve consideration for the description of incompressible plasmas.