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The numerical computation of unstable manifolds for infinite dimensional dynamical systems by embedding techniques. (English) Zbl 1435.37103

Summary: In this work we extend the novel framework developed by A. Ziessler et al. [J. Comput. Dyn. 3, No. 1, 93–112 (2016; Zbl 1358.34081)] to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by M. Dellnitz and A. Hohmann [Numer. Math. 75, No. 3, 293–317 (1997; Zbl 0883.65060)] for the computation of such objects of finite dimensional systems with the results obtained in the work of A. Ziessler et al.. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto-Sivashinsky equation as well as for the Mackey-Glass delay differential equation.

MSC:

37M21 Computational methods for invariant manifolds of dynamical systems
35B42 Inertial manifolds
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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