Post-Stabilization of Invariants and Application to Numerical Analysis of Chaos for Some 3-Dimensional Systems

This research relates to a numerical integrator with post-stabilization of several constraints for an autonomous dynamical system. A generally analytical approach shows that the total energy correction is not valid in most cases, while post-stabilization of each independent energy is. As a typical test example, we consider a non-integrable Hamiltonian system of three degrees of freedom, which can be split into two independent pieces, one 1D harmonic oscillator and another 2D non-integrable system, by using a transformation of variables. Phase portraits on Poincaré sections about the 2D system manifest that our analysis is reasonable. In addition, a problem how to compute Lyapunov exponents in constrained systems is proposed. As a suggestion, it is best to stabilize all constraints involving each energy integral and its corresponding variation in order to avoid spurious Lyapunov exponents. Because an appropriately larger time step is acceptable in this sense, it is not expensive to use the fast Lyapunov indicators to describe the global dynamics of phase space for the 3D system, where regions of chaos and order are clearly identified.