Reconstructing phase space from PDE simulations

Michael Kirby, Hans Armbruster

Research output: Contribution to journalArticle

31 Scopus citations

Abstract

We propose the Karhunen-Loève (K-L) decomposition as a tool to analyze complex spatio-temporal structures in PDE simulations in terms of concepts from dynamical systems theory. Taking the Kuramoto-Sivashinsky equation as a model problem we discuss the K-L decomposition for 4 different values of its bifurcation parameter α. We distinguish two modes of using the K-L decomposition: As an analytic and synthetic tool respectively. Using the analytic mode we find unstable fixed points and stable and unstable manifolds in a parameter regime with structurally stable homoclinic orbits (α=17.75). Choosing the data for a K-L analysis carefully by restricting them to certain burst events, we can analyze a more complicated intermittent regime at α=68. We establish that the spatially localized oscillations around a so called "strange" fixed point which are considered as fore-runners of spatially concentrated zones of turbulence are in fact created by a very specific limit cycle (α=83.75) which, for α=87, bifurcates into a modulated traveling wave. Using the K-L decomposition synthetically by determining an optimal Galerkin system, we present evidence that the K-L decomposition systematically destroys dissipation and leads to blow up solutions.

Original languageEnglish (US)
Pages (from-to)999-1022
Number of pages24
JournalZAMP Zeitschrift für angewandte Mathematik und Physik
Volume43
Issue number6
DOIs
StatePublished - Nov 1 1992

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics

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