A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities

Abstract

7 pages; 3 figures.International audienceIn this article the authors describe the method of construction of approximate chaotic solutions of dynamical model equations with quadratic nonlinearities in a general form using a new accurate numerical method. Numerous systems of chaotic dynamical systems of this type are studied in modern literature.The authors find the region of convergence of the method and offer an algorithm of construction and several criteria to check the accuracy of the approximate chaotic solutions

An Accurate Numerical Method and Algorithm for Constructing Solutions of Chaotic Systems

“…In the article [39], the authors proved the theorem about estimation a region of convergence for the ODEs with any quadratic nonlinearities. In particular, in this case (for I ≥ 0, H > 1, N > 0)…”

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

“…where Θ h denotes a special numerator increment function providing a required order of accuracy. Up to date, several methods of a class (2) have been derived. The most popular one is a simple nonlinear method proposed by L. Wuytack, C. Brezinski and some other researchers [8,9]:…”

“…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]. Thus, when new class of integration methods appears, the collateral numerical effects are of certain interest.…”

The lower bound error as an auxiliary technique to select the integration step-size in the simulation of chaotic systems

“…One of the first papers to deal with the problem of rounding off errors for the linear case in a digital computer is [7], where the author studies the effects of round-off in the floating point realization of a general linear discrete filter governed by a stable difference equation. Besides this, there are also recent works aiming to understand and minimize the round-off error [8][9][10][11][12]. To investigate the round-off errors in recursive functions, [13] introduced a technique to evaluate the lower bound error based on the on the fact that the interval extensions [14] are mathematically equivalent but may exhibit different computer simulation results.…”