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Technical Brief

On the Approximation of Delayed Systems by Taylor Series Expansion

[+] Author and Article Information
Tamas Insperger

Department of Applied Mechanics,
Budapest University of Technology and Economics,
Budapest, Hungary
e-mail: insperger@mm.bme.hu

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received December 31, 2013; final manuscript received March 12, 2014; published online January 12, 2015. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 10(2), 024503 (Mar 01, 2015) (4 pages) Paper No: CND-13-1331; doi: 10.1115/1.4027180 History: Received December 31, 2013; Revised March 12, 2014; Online January 12, 2015

It is known that stability properties of delay-differential equations are not preserved by Taylor series expansion of the delayed term. Still, this technique is often used to approximate delayed systems by ordinary differential equations in different engineering and biological applications. In this brief, it is demonstrated through some simple second-order scalar systems that low-order Taylor series expansion of the delayed term approximates the asymptotic behavior of the original delayed system only for certain parameter regions, while for high-order expansions, the approximate system is unstable independently of the system parameters.

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Figures

Grahic Jump Location
Fig. 1

Stability diagram and the number of unstable characteristic exponents for the delayed oscillator (4) with a = 1 and its Taylor series approximations of different order. Stable domains are indicated by gray shading.

Grahic Jump Location
Fig. 2

Stability diagram and the number of unstable characteristic exponents for Eq. (6) with a = 1 and τ=0.5 and its Taylor series approximations of different order. Stable domains are indicated by gray shading. The critical delay for the different approximations is also shown.

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