On the susceptibility of numerical methods to computational chaos and superstability

• Computational chaos and superstability are studied theoretically and numerically.
• The Backward Euler method is shown to suppress chaos.
• The Forward Euler method is shown to destabilize stable periodic orbits.
• Numerical results hint that this destabilization can engender computational chaos.

In the present study, the susceptibility of the forward and the backward Euler methods to computational chaos and superstability is investigated via the means of both a theoretical analysis and numerical experiments. A linear stability analysis of the fixed points and the periodic orbits of the maps induced by these methods asserts that, for large enough time-steps Δt, these maps undergo bifurcations and as result the acquired solutions are spurious. More specifically, it is shown that the backward Euler method suppresses chaotic behavior, whereas the forward Euler renders all linearly stable fixed points and periodic orbits of its induced map linearly unstable. Numerical experiments that illustrate the validity of the theoretical analysis are also presented and discussed. For the forward Euler method, in particular, the computation of bifurcation diagrams, the Maximum Lyapunov exponent and the Kolmogorov–Sinai entropy suggest that it can engender computational chaos.