Inventiones mathematicae

, Volume 174, Issue 2, pp 355–433

Approximately invariant manifolds and global dynamics of spike states


DOI: 10.1007/s00222-008-0141-y

Cite this article as:
Bates, P., Lu, K. & Zeng, C. Invent. math. (2008) 174: 355. doi:10.1007/s00222-008-0141-y


We investigate the existence of a true invariant manifold given an approximately invariant manifold for an infinite-dimensional dynamical system. We prove that if the given manifold is approximately invariant and approximately normally hyperbolic, then the dynamical system has a true invariant manifold nearby. We apply this result to reveal the global dynamics of boundary spike states for the generalized Allen–Cahn equation.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsBrigham Young UniversityProvoUSA
  3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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