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.

A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

“…The effects of Padé integration in the study of the chaotic behavior of conservative nonlinear chaotic systems have been reported by Butusov et al [18]. The comparative study of the Runge-Kutta methods versus Padé methods shows that chaotic behavior appears in models obtained by nonlinear integration techniques where chaos does not appear in conventional methods.…”

“…where ω > 0 is a relaxation parameter and Φ ω is an increment function [17]. Another discretization technique is based on Padé approximation in the following form [18]:…”

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“…Ó ïðàöÿõ [17,19,20], âèêîðèñòîâóþ÷è ðiçíi àïðîêñèìàíòè Ïàäå, äîñëiäaeóþòüñÿ êîíêðåòíi ïðèêëàäíi çàäà÷i.  [22] ïîáóäîâàíî íåÿâíi ÷èñåëüíi ìåòîäè äðóãîãî ïîðÿäêó òî÷íîñòi ç ìiíiìàëüíîþ ïîõèáêîþ äèñêðåòèçàöi¨i¨õ çàñòîñîâàíî äëÿ àíàëiçó äèíàìi÷íèõ ñèñòåì, à â [21] ïîáóäîâàíî äâîñòîðîííié àëãîðèòì äðóãîãî ïîðÿäêó òî÷íîñòi, ÿêèé ðóíòó¹òüñÿ íà íåïåðåðâíèõ äðîáàõ.…”

Nonlinear Dynamics of Exclusive Excited-State Emission Quantum Dot Lasers Under Optical Injection

“…where aES denotes the differential gain of ES, ξ represents the gain limiting factor, VB is the total volume of QDs, and VS denotes the intra-cavity laser field volume. Numerical methods for the solution of ordinary differential equations are the main tools to investigate the nonlinear dynamical systems [40,41]. In this work, a desktop PC with a six-core processor (AMD Ryzen 5 1600X, Advanced Micro Devices Inc., Santa Clara, CA, USA) and 16GB installed memory is used to perform the simulation, and we adopt the ode45 solver (Fourth-Fifth order Runge-Kutta algorithm, where the fourth-order provides the candidate solutions and the fifth-order controls the errors) in MATLAB software to solve the above differential equations, after taking into account the accuracy and speed of the calculations.…”