Discrete Dynamics in Nature and Society
Volume 6 (2001), Issue 2, Pages 121-128
Detection of the onset of numerical chaotic instabilities by lyapunov exponents
Centro de Estudios Avanzados en Optica and Centro de Astrofisica Teórica, Facultad de Ciencias, La Hechicera, Universidad de Los Andes, Mérida 5051, Venezuela
Received 2 July 2000
Copyright © 2001 Alicia Serfaty De Markus. 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.
It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that
establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.