Minimal sets in singularly perturbed systems with three time-scales

Abstract

In this work we study three time scale singular perturbation problems

"x
′
= f(x; "; ); y
′
= g(x; "; ); z
′
= h(x; "; );
where x = (x; y; z) ∈ Rn × Rm × Rp, " and are two independent small parameter (0 < ", ≪ 1), and f, g, h are Cr functions, with r ≥ 1. We establish conditions for the existence of compact invariant sets (singular points, periodic and homoclinic orbits) when "; > 0. Our main strategy is to consider three time scales which generate three different limit problems.