Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

Jan Bouwe van den Berg, Jonathan Jaquette, J. D.Mireles James

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional “tail”. We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation.

Original languageEnglish (US)
Pages (from-to)3589-3649
Number of pages61
JournalJournal of Dynamics and Differential Equations
Issue number4
StatePublished - Dec 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis


  • Computer assisted proof
  • Lyapunov-Perron method
  • Parabolic partial differential equations
  • Parameterization method
  • Rigorous numerics
  • stable manifold


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