NONSTANDARD DISCRETIZATION SCHEMES APPLIED TO THE CONSERVATIVE HÉNON–HEILES SYSTEM

Abstract

The discretization of ordinary differential equations is investigated for the case of the conservative Hénon–Heiles system. Starting from a discrete Hamiltonian function, which is invariant under time reversal, discrete equations of motion are analytically obtained using three different discretization schemes recently proposed and investigated in the literature. In the case where the discretization scheme successfully provide discrete systems in which the trace of the Jacobian matrix corresponding to the property required by a conservative system is preserved, it is shown that they are not necessarily invariant to time reversal. Such models are however quite robust when the time step is increased. For the schemes where the trace of Jacobian matrix does not match the condition required by conservative systems, it is shown that energy conservation is not achieved and the original dynamics is lost. Steps toward the solution to this problem are given.