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Discrete Dynamics in Nature and Society
Volume 4 (2000), Issue 1, Pages 21-28
http://dx.doi.org/10.1155/S1026022600000029

Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations

Centro de Estudios Avanzados en Optica and Centro de Astrofísica Teórica, Fac. de Ciencias, La Hechicera, Universidad de los Andes, AP 26, Mérida 5101, Venezuela

Received 22 October 1998

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For computational purposes, a numerical algorithm maps a differential equation into an often complex difference equation whose structure and stability depends on the scheme used. When considering nonlinear models, standard and nonstandard integration routines can act invasively and numerical chaotic instabilities may arise. However, because nonstandard schemes offer a direct and generally simpler finite-difference representations, in this work nonstandard constructions were tested over three different systems: a photoconductor model, the Lorenz equations and the Van der Pol equations. Results showed that although some nonstandard constructions created a chaotic dynamics of their own, there was found a construction in every case that greatly reduced or successfully removed numerical chaotic instabilities. These improvements represent a valuable development to incorporate into more sophisticated algorithms.