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.
Highlights
Realistic modeling of a physical or social dynamical process involves the formulation of a nonlinear system of differential equations, which in general has to be solved by numerical techniques
In this case the choice of the numerical algorithm is crucial as the scheme itself could importantly interact with the model, producing a dynamics of its own
Such interactions may arise mainly in the transition from a continuous-time dynamic model to a discrete one, as the numerical algorithm maps a differential equation into a difference equation
Summary
Realistic modeling of a physical or social dynamical process involves the formulation of a nonlinear system of differential equations, which in general has to be solved by numerical techniques. The purpose of this work is to handle these instabilities through nonstandard simple finite representations [1], with the goal that if significant improvement is achieved, nonstandard constructions could replace standard building blocks of far more advanced algorithms. To this end, nonstandard finite-difference schemes, to solve a system of ordinary differential equations (ODEs) of well-known examples for which chaotic and nonchaotic dynamics are well established, are constructed. As there is no unique representation, the "right" finite scheme could be determined (or discarded) by requesting the same stability properties of the continuos-time system
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