Central difference scheme and chaos

Ushiki, Shigehiro

doi:10.1016/0167-2789(82)90044-6

Cited by 57 publications

(22 citation statements)

References 6 publications

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“…[5,6,11,12] that the central difference method (3) produces a discrete-time system with local dynamics that are inconsistent with the continuous-time system, and even produces chaos when used as a numerical method. Mickens [6] applied the above method for the decay equation dx/dt ¼ 2 x and showed that the numerical scheme for the decay equation has numerical instabilities regardless of the chosen step-size.…”

Section: Introductionmentioning

confidence: 98%

Preservation of local dynamics when applying central difference methods: application to SIR model

Roeger

Barnard

2007

Journal of Difference Equations and Applications

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We apply the central difference method (u tþ1 2 u t21 )/(2Dt) ¼ f (u t ) to an epidemic SIR model and show how the local stability of the equilibria is changed after applying the numerical method. The above central difference scheme can be used as a numerical method to produce a discrete-time model that possesses interesting local dynamics which appears inconsistent with the continuous model. Any fixed point of a differential equation will become an unstable saddle node after applying this method. Two other implicitly defined central difference methods are also discussed here. These two methods are more efficient for preserving the local stability of the fixed points for the continuous models. We apply conformal mapping theory in complex analysis to verify the local stability results.

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Section: Introductionmentioning

confidence: 98%

Preservation of local dynamics when applying central difference methods: application to SIR model

Roeger

Barnard

2007

Journal of Difference Equations and Applications

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“…Since the standard approach to constructing finite difference methods for solving differential equations can lead to incorrect behavior in the solutions (e.g. "ghost solutions", numerical instabilities and chaotic behavior [37]), Mickens, using the exact difference equations as a guide, proposed the modeling rules for constructing Nonstandard finite-difference (NSFD) methods as follows [23,22]:…”

Section: Introductionmentioning

confidence: 99%

Universal approaches to approximate biological systems with nonstandard finite difference methods

Wood

Kojouharov

Dimitrov

2017

Mathematics and Computers in Simulation

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“…Whitehead and MacDonald 6 have produced chaos by applying the Euler dierencing scheme. Ushiki 7 has shown an application of a centred dierencing scheme to produce chaos. Lorenz 4 describes how computational chaos can occur in models that are used to represent physical systems.…”

Section: Introductionmentioning

confidence: 99%

Efficacy of some numerical integration schemes to simulate chaotic behaviour

Chandra¹

1997

Commun. Numer. Meth. Engng.

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SUMMARYChaotic behaviour of physical systems has been observed experimentally and numerically. It has also been observed that chaotic systems are extremely sensitive to initial conditions. In this paper, the ecacy of using dierent numerical integration schemes, some of which are preferred in ®nite element software for transient dynamics, is discussed using techniques to detect chaotic behaviour such as PoincareÂ maps, FFTs, the Lyapunov exponent and fractal dimensions. #

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Central difference scheme and chaos

Cited by 57 publications

References 6 publications

Preservation of local dynamics when applying central difference methods: application to SIR model

Preservation of local dynamics when applying central difference methods: application to SIR model

Universal approaches to approximate biological systems with nonstandard finite difference methods

Efficacy of some numerical integration schemes to simulate chaotic behaviour

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