Mathematics – Dynamical Systems
Scientific paper
2009-12-30
Mathematics
Dynamical Systems
13 pages, 6 figures, pdfLaTeX
Scientific paper
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.
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