Fibrations of the Persistent Invariant Manifolds

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


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:


Base Point Unique Fixed Point Lipschitz Norm Invariant Subbundle Fibration Theorem 
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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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