Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems

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2009, 2009(Special): 34-43. doi: 10.3934/proc.2009.2009.34

Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems

Roumen Anguelov ^1, , Jean M.-S. Lubuma ^1, and Meir Shillor ^2,

1.	Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa, South Africa
2.	Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309

Received July 2008 Revised March 2009 Published September 2009

Abstract
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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: nonstandard finite difference method, dynamical systems, dynamic consistency.

Mathematics Subject Classification: Primary: 65L12; Secondary: 37B99.

Citation: Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34

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