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Reconstructing phase space from PDE simulations. (English) Zbl 0764.35008

Summary: We propose the Karhunen-Loève decomposition (K-L) as a tool to analyze complex spatio-temporal structures in PDE simulations in terms of concepts from dynamical systems theory. Taking the Kuramoto-Sivashinky equation as a model problem we discuss the K-L decomposition for 4 different values of its bifurcation parameter \(\alpha\). 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 \((\alpha=17.75)\). 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 \(\alpha=68\). 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 \((\alpha=83.75)\) which, for \(\alpha=87\), 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.

MSC:

35B32 Bifurcations in context of PDEs
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
35Q35 PDEs in connection with fluid mechanics
35A35 Theoretical approximation in context of PDEs
58E09 Group-invariant bifurcation theory in infinite-dimensional spaces
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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