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

TL;DR

This paper provides examples demonstrating that normal hyperbolicity can fail in families of invariant manifolds even when Lyapunov-type numbers stay bounded away from critical thresholds, showing breakdowns at critical parameters.

Contribution

It introduces three explicit examples illustrating breakdown of normal hyperbolicity despite bounded Lyapunov-type numbers, highlighting subtle failure mechanisms.

Findings

Normal hyperbolicity can break down at critical parameters.

Invariant manifolds may become non-smooth or cease to be hyperbolic.

Lyapunov-type numbers may remain bounded even as hyperbolicity fails.

Abstract

We present three examples to illustrate that in the continuation of a family of normally hyperbolic $C^1$ 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 $C^1$ 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 $C^1$ manifolds converges to a nonsmooth invariant set, for which generalized Lyapunov-type numbers are undefined.