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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 |

### Keywords:

propagation of large scale structures in weak turbulent flame fronts; structure of traveling wave solutions; chaotic bursting; homoclinic bursts; strange fixed point; Karhunen-Loève decomposition (K-L); complex spatio-temporal structures; dynamical systems theory; Kuramoto- Sivashinky equation; spatially localized oscillations; optimal Galerkin system; blow up solutions
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\textit{M. Kirby} and \textit{D. Armbruster}, Z. Angew. Math. Phys. 43, No. 6, 999--1022 (1992; Zbl 0764.35008)

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### References:

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