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Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems

Pages: 34 - 43, Issue Special, September 2009

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Roumen Anguelov - Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa (email)
Jean M.-S. Lubuma - Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa (email)
Meir Shillor - Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States (email)

Abstract: This work deals with the relationship between a continuous dynamical system and numerical methods for its computer simulations, viewed as discrete dynamical systems. The term 'dynamic consistency' of a numerical scheme with the associated continuous system is usually loosely defined, meaning that the numerical solutions replicate some of the properties of the solutions of the continuous system. Here, this concept is replaced with topological dynamic consistency, which is defined in precise terms through the topological equivalence of maps. This ensures that all the topological properties (e.g., fixed points and their stability, periodic solutions, invariant sets, etc.) are preserved. Two examples are provided which demonstrate that numerical schemes satisfying this strong notion of dynamic consistency can be constructed using the nonstandard finite difference method.

Keywords:  dynamical systems, dynamic consistency, nonstandard finite difference method
Mathematics Subject Classification:  Primary: 65L12; Secondary: 37B99

Received: July 2008;      Revised: March 2009;      Published: September 2009.