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Towards Calculating the Basin of Attraction of Non-Smooth Dynamical Systems Using Radial Basis Functions

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Approximation Algorithms for Complex Systems

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 3))

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Summary

We consider a special type of non-smooth dynamical systems, namely x? = f(t, x), where x ? R, f is t-periodic with period T and non-smooth at x = 0. In [6] a sufficient Borg-like condition to determine a subset of its basin of attraction was given. The condition involves a function W and its partial derivatives; the function W is t-periodic and non-smooth at x = 0.

In this article, we describe a method to approximate this function W using radial basis functions. The challenges that W is non-smooth at x = 0 and a time-periodic function are overcome by introducing an artificial gap in x-direction and using a time-periodic kernel. The method is applied to an example which models a motor with dry friction.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Sussex, Falmer, BN1 9QH, UK

    Peter Giesl

Authors
  1. Peter Giesl
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Editor information

Editors and Affiliations

  1. , Dept. of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Emmanuil H Georgoulis

  2. Abt. Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Armin Iske

  3. Dept. Mathematics and, Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

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Giesl, P. (2011). Towards Calculating the Basin of Attraction of Non-Smooth Dynamical Systems Using Radial Basis Functions. In: Georgoulis, E., Iske, A., Levesley, J. (eds) Approximation Algorithms for Complex Systems. Springer Proceedings in Mathematics, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16876-5_9

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