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Photo: Paderborn University

Dr. Raphael Gerlach

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Publications
Dr. Raphael Gerlach

Institut für Industriemathematik

Manager - Academic Councillor for a Limited Period

Chair of Applied Mathematics

Member - Academic Councillor for a Limited Period

Phone:
+49 5251 60-5022
Fax:
+49 5251 60-4216
Office:
TP21.1.25
Visitor:
Technologiepark 21
33100 Paderborn

Open list in Research Information System

2021

The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems

R. Gerlach, 2021

One central aspect in the study of dynamical systems is the analysis of its invariant sets such as the global attractor and (un)stable manifolds. In particular, when the underlying system depends on a parameter it is crucial to efficiently track those set with respect to this parameter. For the computation of invariant sets we rely on numerical algorithms for their approximation but typically those tools can only be applied to finite-dimensional dynamical systems. Thus, in thesis we present a numerical framework for the global dynamical analysis of infinite-dimensional systems. We will use embedding techniques for the definition of the core dynamical system (CDS) which is a dynamically equivalent finite-dimensional system. The CDS is then used for the approximation of related embedded invariant sets, i.e, one-to-one images, by set-oriented numerical methods. For the construction of the CDS it is crucial to choose an appropriate observation map and to design its corresponding inverse. Therefore, we will present suitable numerical realizations of the CDS for DDEs and PDEs. For a subsequent geometric analysis of the embedded invariant set we will consider a manifold learning technique called diffusion maps which reveals its intrinsic geometry and estimates its dimension. Finally, we apply our develop numerical tools on some well-known infinite-dimensional dynamical systems such as the Mackey-Glass equation, the Kuramoto-Sivashinsky equation and the Navier-Stokes equation.


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

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.


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.


On the equivariance properties of self-adjoint matrices

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

We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.


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