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Perspektivenwechsel. Bildinformationen anzeigen

Perspektivenwechsel.

Foto: Universität Paderborn

Raphael Gerlach, M.Sc.

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Publikationen
 Raphael Gerlach, M.Sc.

Institut für Industriemathematik

Geschäftsführer - Wissenschaftlicher Mitarbeiter

Lehrstuhl für Angewandte Mathematik

Mitglied - Wissenschaftlicher Mitarbeiter

Telefon:
+49 5251 60-5022
Fax:
+49 5251 60-4216
Büro:
TP21.1.25
Besucher:
Technologiepark 21
33100 Paderborn

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2020

A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors

R. Gerlach, A. Ziessler, B. Eckhardt, M. Dellnitz, SIAM Journal on Applied Dynamical Systems (2020), pp. 705-723

In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes .


    The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems

    R. Gerlach, A. Ziessler, in: Advances in Dynamics, Optimization and Computation, Springer, 2020, pp. 55-85

    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.


      2019

      The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques

      A. Ziessler, M. Dellnitz, R. Gerlach, SIAM Journal on Applied Dynamical Systems (2019), pp. 1265-1292

      In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler 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 Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo, and Ziessler. 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.


        Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems

        R. Gerlach, P. Koltai, M. Dellnitz, in: arXiv:1902.08824, 2019

        Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.


          On the equivariance properties of self-adjoint matrices

          M. Dellnitz, B. Gebken, R. Gerlach, S. Klus, Dynamical Systems (2019), pp. 1-19


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