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

Denis Butusov,Aleksandra Tutueva,Artur Karimov,Dmitry Kaplun,Erivelton G Nepomuceno

doi:10.3390/e21040362

Denis Butusov, Aleksandra Tutueva + Show 3 more

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https://doi.org/10.3390/e21040362

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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.

Highlights

The influence of numerical methods on discrete models of chaotic systems is widely studied
ODE solvers are represented by the explicit Runge–Kutta 4 (RK4) method [22] and Padé 4 method given by the formula (5)
The nonlinear integration techniques based on a direct Padé approximation of the solution have been applied to the conservative nonlinear chaotic system

Summary

Introduction

The influence of numerical methods on discrete models of chaotic systems is widely studied. While highly accurate numerical methods for chaotic problems integration have been recently developed [1,2], some studies reveal the negative aspects of popular discretization techniques [3,4] and discover the additional properties introduced by numerical errors [5]. When new class of integration methods appears, the collateral numerical effects are of certain interest. A relationship between Padé approximation and numerical solution of an ordinary differential equation is well known [6,7]. From this point of view, various integration methods implement various types of approximation leading to different properties, such as the order of accuracy and A-stability [8].

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