Abstract
In this chapter we discuss some ways to extend the main theorem to slightly more general situations, such as non-autonomous systems, overflowing invariant manifolds, and smooth parameter dependence. These extensions are known from the compact and Euclidean settings, but a bit scattered over the literature. We try to collect a number of these results here, while extending them to our noncompact setting.
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Notes
- 1.
Note that since \(t \le \tau \), we have a reverse flow \(\Psi ^+(t,\tau )\) for the unstable directions, which indeed satisfies the growth estimates (1.10).
- 2.
We only can (and need to) perform the pullback in an \(\eta \)-sized tubular neighborhood of \(\gamma \). Outside this neighborhood we smoothly cut off the vector field to a suitable linearization. Therefore we only recover the local stable manifold.
- 3.
We immediately recover higher smoothness instead of having to go through an elaborate scheme involving the fiber contraction theorem as in Sect. 3.7. The reason is that \(B^{k\rho }({\mathbb{R }}_{\ge 0};\mathcal E _m) \hookrightarrow B^\rho ({\mathbb{R }}_{\ge 0};\mathcal E _m)\) is a continuous embedding when \(\rho < 0\) and we look at \({\mathbb{R }}_{\ge 0}\), that is, higher powers of negative exponential growth decay even stronger.
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Eldering, J. (2013). Extension of Results. In: Normally Hyperbolic Invariant Manifolds. Atlantis Series in Dynamical Systems, vol 2. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-003-4_4
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DOI: https://doi.org/10.2991/978-94-6239-003-4_4
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Publisher Name: Atlantis Press, Paris
Print ISBN: 978-94-6239-002-7
Online ISBN: 978-94-6239-003-4
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