Computing (un)stable manifolds with validated error bounds: non-resonant and resonant spectra

G.J.B. van den Berg, J. Mireles-James, C.P. Reinhardt

Research output: Contribution to JournalArticleAcademicpeer-review


We develop techniques for computing the (un)stable manifold at a hyperbolic equilibrium of an analytic vector field. Our approach is based on the so-called parametrization method for invariant manifolds. A feature of this approach is that it leads to a posteriori analysis of truncation errors which, when combined with careful management of round off errors, yields a mathematically rigorous enclosure of the manifold. The main novelty of the present work is that, by conjugating the dynamics on the manifold to a polynomial rather than a linear vector field, the computer-assisted analysis is successful even in the case when the eigenvalues fail to satisfy non-resonance conditions. This generically occurs in parametrized families of vector fields. As an example, we use the method as a crucial ingredient in a computational existence proof of a connecting orbit in an amplitude equation related to a pattern formation model that features eigenvalue resonances.
Original languageEnglish
Pages (from-to)1055-1095
JournalJournal of nonlinear science
Publication statusPublished - 2016


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