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The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems

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Advances in Dynamics, Optimization and Computation (SON 2020)

Abstract

In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.

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Acknowledgments

We would like to acknowledge Michael Dellnitz for developing the underlying ideas as well as the theoretical foundations of this work.

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Correspondence to Raphael Gerlach .

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Gerlach, R., Ziessler, A. (2020). The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems. In: Junge, O., Schütze, O., Froyland, G., Ober-Blöbaum, S., Padberg-Gehle, K. (eds) Advances in Dynamics, Optimization and Computation. SON 2020. Studies in Systems, Decision and Control, vol 304. Springer, Cham. https://doi.org/10.1007/978-3-030-51264-4_3

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  • DOI: https://doi.org/10.1007/978-3-030-51264-4_3

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