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On singularly perturbed Filippov systems

Published online by Cambridge University Press:  08 July 2013

PEDRO T. CARDIN ,
PAULO R. DA SILVA  and
MARCO A. TEIXEIRA
PEDRO T. CARDIN
Affiliation:
Departamento de Matemática, Faculdade de Engenharia de Ilha Solteira, UNESP – Univ Estadual Paulista, Rua Rio de Janeiro, 266, CEP 15385-000 Ilha Solteira, São Paulo, Brazil email: pedrocardin@mat.feis.unesp.br
PAULO R. DA SILVA
Affiliation:
Departamento de Matemática, Instituto de Biociências, Letras e Ciências Exatas, UNESP – Univ Estadual Paulista, Rua C. Colombo, 2265, CEP 15054-000 S. J. Rio Preto, São Paulo, Brazil email: prs@ibilce.unesp.br
MARCO A. TEIXEIRA
Affiliation:
IMECC–UNICAMP, CEP 13081-970, Campinas, São Paulo, Brazil email: teixeira@ime.unicamp.br
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Abstract

In this paper, we study singularly perturbed Filippov systems. More specifically, our main question is to know how the dynamics of Filippov systems is affected by singular perturbations. We extend the Fenichel theory developed in Fenichel (J. Differ. Equ., 1979, Vol. 31, pp. 53–98) to these systems. In addition, the study of non-smooth constrained systems is considered.

Keywords

Filippov systemsSliding modeSingular perturbationConstrained systemsImpasse set

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Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 6 , December 2013 , pp. 835 - 856
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Cardin, P. T., da Silva, P. R. & Teixeira, M. A. (2012) Implicit differential equations with impasse singularities and singular perturbation problems. Isr. J. Math. 189, 307–322.CrossRefGoogle Scholar
[2]di Bernardo, M., Budd, C. J., Champneys, A. R., Kowalczyk, P., Nordmark, A. B., OlivarTost, G. Tost, G. & Piiroinen, P. T. (2005) Bifurcations in non-smooth dynamical systems. SIAM Rev. 50, 629–701.CrossRefGoogle Scholar
[3]Dumortier, F. & Roussarie, R. (1996) Canard cycles and center manifolds. With an appendix by Cheng Zhi Li. Mem. Amer. Math. Soc. 121, no. 577, x+100 pp.CrossRefGoogle Scholar
[4]Eckhaus, W. (1983) Relaxation oscillations including a standard chase on French ducks. In: Asymptotic Analysis II, 449–494, Lecture Notes in Math., 985, Springer, Berlin.CrossRefGoogle Scholar
[5]Fenichel, N. (1979) Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98.CrossRefGoogle Scholar
[6]Filippov, A. F. (1988) Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications (Soviet Series), Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
[7]Fridman, L. M. (2003) Application of the method of boundary functions for finding slow periodic solutions of singularly perturbed bang-bang systems. Differ. Equ. 39 (2), 256–266.CrossRefGoogle Scholar
[8]Rabier, P. J. & Rheinboldt, W. C. (1994) On impasse points of quasilinear differential-algebraic equations. J. Math. Anal. Appl. 181, 429–454.CrossRefGoogle Scholar
[9]Sieber, J. & Kowalczyk, P. (2009) Small-scale instabilities in dynamical systems with sliding. Physica D. Nonlinear Phenom. 239, 44–57.CrossRefGoogle Scholar
[10]Szmolyan, P. (1990) Transversal heteroclinic and homoclinic orbits in singular perturbation problems. J. Differ. Equ. 92, 252–281.CrossRefGoogle Scholar
[11]Zhitomirskii, M. (1993) Local normal forms for constrained systems on 2-manifolds. Bol. Soc. Bras. Mat. 24, 211–232.CrossRefGoogle Scholar