@inbook{60048, author = {{Gerlach, Raphael and von der Gracht, Sören and Dellnitz, Michael}}, booktitle = {{Lecture Notes in Computer Science}}, isbn = {{9783031917356}}, issn = {{0302-9743}}, publisher = {{Springer Nature Switzerland}}, title = {{{On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies}}}, doi = {{10.1007/978-3-031-91736-3_19}}, year = {{2025}}, } @inproceedings{56298, abstract = {{In the general pattern formation (GPF) problem, a swarm of simple autonomous, disoriented robots must form a given pattern. The robots' simplicity imply a strong limitation: When the initial configuration is rotationally symmetric, only patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS 2010]. The only known algorithm to form large patterns with limited visibility and without memory requires the robots to start in a near-gathering (a swarm of constant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know any near-gathering algorithm guaranteed to preserve symmetry but most natural gathering strategies trivially increase symmetries [Castenow et al.; OPODIS 2022]. Thus, we study near-gathering without changing the swarm's rotational symmetry for disoriented, oblivious robots with limited visibility (the OBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on the theory of dynamical systems to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation. Until now, it was unknown whether the considered OBLOT-model allows for any non-trivial algorithm that always preserves symmetry. Our first result shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters. Our second result is a symmetry-preserving near-gathering algorithm that works on swarms with a convex boundary (the outer boundary of the unit disc graph) and without holes (circles of diameter 1 inside the boundary without any robots).}}, author = {{Gerlach, Raphael and von der Gracht, Sören and Hahn, Christopher and Harbig, Jonas and Kling, Peter}}, booktitle = {{28th International Conference on Principles of Distributed Systems (OPODIS 2024)}}, editor = {{Bonomi, Silvia and Galletta, Letterio and Rivière, Etienne and Schiavoni, Valerio}}, isbn = {{978-3-95977-360-7}}, issn = {{1868-8969}}, keywords = {{Swarm Algorithm, Swarm Robots, Distributed Algorithm, Pattern Formation, Limited Visibility, Oblivious}}, location = {{Lucca, Italy}}, publisher = {{Schloss Dagstuhl -- Leibniz-Zentrum für Informatik}}, title = {{{Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility}}}, doi = {{10.4230/LIPIcs.OPODIS.2024.13}}, volume = {{324}}, year = {{2025}}, } @unpublished{58953, abstract = {{In this article, we investigate symmetry properties of distributed systems of mobile robots. We consider a swarm of n robots in the OBLOT model and analyze their collective Fsync dynamics using of equivariant dynamical systems theory. To this end, we show that the corresponding evolution function commutes with rotational and reflective transformations of R^2. These form a group that is isomorphic to O(2) x S_n, the product group of the orthogonal group and the permutation on n elements. The theory of equivariant dynamical systems is used to deduce a hierarchy along which symmetries of a robot swarm can potentially increase following an arbitrary protocol. By decoupling the Look phase from the Compute and Move phases in the mathematical description of an LCM cycle, this hierarchy can be characterized in terms of automorphisms of connectivity graphs. In particular, we find all possible types of symmetry increase, if the decoupled Compute and Move phase is invertible. Finally, we apply our results to protocols which induce state-dependent linear dynamics, where the reduced system consisting of only the Compute and Move phase is linear.}}, author = {{Gerlach, Raphael and von der Gracht, Sören}}, booktitle = {{arXiv:2503.07576}}, keywords = {{dynamical systems, coupled systems, distributed computing, robot swarms, autonomous mobile robots, symmetry, equivariant dynamics}}, pages = {{23}}, title = {{{Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical Systems}}}, year = {{2025}}, } @phdthesis{32057, abstract = {{Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung.}}, author = {{Gerlach, Raphael}}, title = {{{The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}}}, doi = {{10.17619/UNIPB/1-1278}}, year = {{2021}}, } @inbook{17994, 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.}}, author = {{Gerlach, Raphael and Ziessler, Adrian}}, booktitle = {{Advances in Dynamics, Optimization and Computation}}, editor = {{Junge, Oliver and Schütze, Oliver and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}}, isbn = {{9783030512637}}, issn = {{2198-4182}}, pages = {{66--85}}, publisher = {{Springer International Publishing}}, title = {{{The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems}}}, doi = {{10.1007/978-3-030-51264-4_3}}, volume = {{304}}, year = {{2020}}, } @article{16712, abstract = {{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.}}, author = {{Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan}}, issn = {{1468-9367}}, journal = {{Dynamical Systems}}, number = {{2}}, pages = {{197--215}}, title = {{{On the equivariance properties of self-adjoint matrices}}}, doi = {{10.1080/14689367.2019.1661355}}, volume = {{35}}, year = {{2020}}, } @article{16710, abstract = {{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 . }}, author = {{Gerlach, Raphael and Ziessler, Adrian and Eckhardt, Bruno and Dellnitz, Michael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, pages = {{705--723}}, title = {{{A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors}}}, doi = {{10.1137/19m1247139}}, year = {{2020}}, } @article{16708, abstract = {{ 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. }}, author = {{Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{1265--1292}}, title = {{{The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}}}, doi = {{10.1137/18m1204395}}, volume = {{18}}, year = {{2019}}, } @unpublished{16711, abstract = {{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.}}, author = {{Gerlach, Raphael and Koltai, Péter and Dellnitz, Michael}}, booktitle = {{arXiv:1902.08824}}, title = {{{Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems}}}, year = {{2019}}, }