Three time scale singular perturbation problems and nonsmooth dynamical systems

Cardin, Pedro; da Silva, Paulo; Teixeira, Marco

doi:10.1090/S0033-569X-2014-01360-X

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
MathSciNet review: 3291821
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Abstract: In this paper we study three time scale singular perturbation problems \[ \varepsilon x’ = f(\mathbf {x},\varepsilon ,\delta ), \qquad y’ = g(\mathbf {x},\varepsilon ,\delta ), \qquad z’ = \delta h(\mathbf {x},\varepsilon ,\delta ), \] 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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