Tori in Resonance

Abstract

This paper gives three examples of ordinary differential equations which depend on one or more parameters and which admit invariant tori for some values of the parameters. These examples illustrate how invariant tori evolve as the parameters are changed; in particular how they disappear, bifurcate and lose smoothness. The equations presented are choosen to be as simple as possible in order to clearly show the interesting phenomenon without unnecessary details. However, the theory of normal forms and unfoldings was used to select typical examples, but no attempt will be made to define precisely the universe of discourse where these examples are generic. The unfolding of invariant tori would consist of a mutitude of cases not all of which are that interesting.