The Effects of Padé Numerical Integration in Simulation of Conservative Chaotic Systems

Abstract

In this paper, we consider nonlinear integration techniques, based on direct Padé approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtained by nonlinear integration are studied, including phase space volume dynamics, bifurcation diagrams, spectral entropy, and the Lyapunov spectrum. We also plot 2D dynamical maps to enlighten the features introduced by nonlinear integration techniques. The comparative study of classical integration methods and Padé approximation methods is given. It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy. This property reduces the applicability of numerical methods based on Padé approximation to the chaotic system simulation but it is still useful for construction of pseudo-random number generators that are resistive to chaos degradation or discrete maps with highly nonlinear properties.