Three time scale singular perturbation problems and nonsmooth dynamical systems

Abstract

In this paper we study three time scale singular perturbation problems \[ ε x ′ = f ( x , ε , δ ) , y ′ = g ( x , ε , δ ) , z ′ = δ h ( x , ε , δ ) , \varepsilon x’ = f(\mathbf {x},\varepsilon ,\delta ), \qquad y’ = g(\mathbf {x},\varepsilon ,\delta ), \qquad z’ = \delta h(\mathbf {x},\varepsilon ,\delta ), \] where x = ( x , y , z ) ∈ R n × R m × R p \mathbf {x} = (x,y,z) \in \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^p , ε \varepsilon and δ \delta are two independent small parameters ( 0 > ε (0>\varepsilon , δ ≪ 1 \delta \ll 1 ), and f f , g g , h h are C r C^r functions, where r r is big enough for our purposes. We establish conditions for the existence of compact invariant sets (singular points, periodic and homoclinic orbits) when ε , δ > 0 \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.