Influence of numerical discretization on chaos: a quantitative study of Lyapunov dynamics and bifurcation behavior

Abstract

The sensitivity of chaotic dynamical systems to the initial conditions makes the long-term numerical simulations of such systems challenging since the discretization error can either dampen true chaos or produce artificial chaos. In this paper, we examine the capability of the conventional, advanced, and trigonometric polynomial-based methods in preserving chaos within oscillatory nonlinear systems during lengthy simulation periods. The research procedure is divided into three parts: selecting chaotic systems and numerical techniques, conducting lengthy numerical simulations, and analyzing results based on phase spaces, Lyapunov exponents, Kolmogorov–Sinai entropy, and computation cost. Numerical experiments have been done on the Lorenz, Rossler, Chen, and Chua systems over 50,000s time intervals. It is shown that the backward Euler method fails to preserve the real dynamics in chaotic systems if the appropriate integration steps are not chosen, whereas the forward Euler method creates artificial chaos under instability conditions of the discretization process. Among all numerical methods, the Gautschi scheme offers the best computational efficiency, requiring only 2.7% of the number of computations needed by AB–AM in the case of the Lorenz system but still keeping positive Lyapunov exponents and K-S entropy.