Breakdown of Normal Hyperbolicity for a Family of Invariant Manifolds with Generalized Lyapunov-Type Numbers Uniformly Bounded below Their Critical Values

摘要

We present three examples to illustrate that in the continuation of a familyof normally hyperbolic $C^1$ manifolds, the normal hyperbolicity may break downas the continuation parameter approaches a critical value even though thecorresponding generalized Lyapunov-type numbers remain uniformly bounded belowtheir critical values throughout the process. In the first example, a $C^1$manifold still exists at the critical parameter value, but it is no longernormally hyperbolic. In the other two examples, at the critical parameter valuethe family of $C^1$ manifolds converges to a nonsmooth invariant set, for whichgeneralized Lyapunov-type numbers are undefined.