BREAKDOWN OF NORMAL HYPERBOLICITY FOR A FAMILY OF INVARIANT MANIFOLDS WITH GENERALIZED LYAPUNOV-TYPE NUMBERS UNIFORMLY BOUNDED BELOW THEIR CRITICAL VALUES

Abstract. We present three examples to illustrate that in the continuation of a family of normally hyperbolic C1 manifolds, the normal hyperbolicity may break down as the continuation parameter approaches a critical value even though the corresponding generalized Lyapunov-type numbers remain uniformly bounded below their critical values throughout the process. In the first example, a C1 manifold still exists at the critical parameter value, but it is no longer normally hyperbolic. In the other two examples, at the critical parameter value the family of C1 manifolds converges to a nonsmooth invariant set, for which generalized Lyapunov-type numbers are undefined. 1