Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Three time scale singular perturbation problems and nonsmooth dynamical systems

Authors: Pedro T. Cardin, Paulo R. da Silva and Marco A. Teixeira
Journal: Quart. Appl. Math. 72 (2014), 673-687
MSC (2010): Primary 34D15, 34N05, 34C20
DOI: https://doi.org/10.1090/S0033-569X-2014-01360-X
Published electronically: September 17, 2014
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Abstract: In this paper we study three time scale singular perturbation problems

$\displaystyle \varepsilon x' = f(\mathbf {x},\varepsilon ,\delta ), \qquad y' =... ...\varepsilon ,\delta ), \qquad z' = \delta h(\mathbf {x},\varepsilon ,\delta ), $

where $ \mathbf {x} = (x,y,z) \in \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p$, $ \varepsilon $ and $ \delta $ are two independent small parameters $ (0<\varepsilon $, $ \delta \ll 1$), and $ f$, $ g$, $ h$ are $ C^r$ functions, where $ r$ is big enough for our purposes. We establish conditions for the existence of compact invariant sets (singular points, periodic and homoclinic orbits) when $ \varepsilon , \delta > 0$. Our main strategy is to consider three time scales which generate three different limit problems. In addition, we prove that double regularization of nonsmooth dynamical systems with self-intersecting switching variety provides a class of three time scale singular perturbation problems.

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Additional Information

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 Cristóvão 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

DOI: https://doi.org/10.1090/S0033-569X-2014-01360-X
Keywords: Geometric theory, singular perturbations, three time scales, nonsmooth dynamical systems
Received by editor(s): October 12, 2012
Published electronically: September 17, 2014
Article copyright: © Copyright 2014 Brown University

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