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Fibrations of the Persistent Invariant Manifolds

  • Charles Li
  • Stephen Wiggins
Part of the Applied Mathematical Sciences book series (AMS, volume 128)

Abstract

We continue to work in \({{\tilde{D}}_{k}}\), where the bumped perturbed flow (2.6.27) is defined. From Proposition 3.1.1, we know the existence of the C n codimension 1 center-unstable manifold \(W_{{{{\delta }_{1}},\delta }}^{{cu}}\), the C n codimension 1 center-stable manifold \(W_{{{{\delta }_{1}},\delta }}^{{cs}}\), and the C n codimension 2 center manifold \({{W}_{{{{\delta }_{1}},\delta }}}\), under the bumped perturbed flow (2.6.27). More specifically, \(W_{{{{\delta }_{1}},\delta }}^{{cu}}\) exists in \(\tilde{D}_{k}^{{(1)}}\); moreover, it is overflowing invariant. \(W_{{{{\delta }_{1}},\delta }}^{{cs}}\) exists in \(\tilde{D}_{k}^{{(2)}}\); moreover, it is inflowing invariant. Then \({{W}_{{{{\delta }_{1}},\delta }}} \equiv W_{{{{\delta }_{1}},\delta }}^{{cu}} \cap W_{{{{\delta }_{1}},\delta }}^{{cs}}\) exists in \(\tilde{D}_{k}^{{(2)}}\), and it is inflowing invariant. Since the fibration theorem is concerned with the fiber representations of \(W_{{{{\delta }_{1}},\delta }}^{{cu}}\) and \(W_{{{{\delta }_{1}},\delta }}^{{cs}}\) with respect to \({{W}_{{{{\delta }_{1}},\delta }}}\) as the base, we have to work in a region where \({{W}_{{{{\delta }_{1}},\delta }}}\) exists. Therefore, we can work only inside \(\tilde{D}_{k}^{{(2)}}\). We know that \({{W}_{{{{\delta }_{1}},\delta }}}\) is inflowing invariant in \(\tilde{D}_{k}^{{(2)}}\). Next, we will prove the following lemma:

Keywords

Base Point Unique Fixed Point Lipschitz Norm Invariant Subbundle Fibration Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Charles Li
    • 1
  • Stephen Wiggins
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Applied MechanicsCalifornia Institute of TechnologyPasadenaUSA

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