Design of Hierarchical Excitable Networks - von der Gracht, Sören; Lohse, Alexander (arXiv:2603.06157)
We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $G_1,\dotsc,G_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $G_1,\dotsc,G_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.

Homogeneous coupled cell systems with high-dimensional internal dynamics - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (Chaos, Solitons & Fractals)
We investigate homogeneous coupled cell systems with high-dimensional internal dynamics. In many studies on network dynamics, the analysis is restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between dynamical behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the underlying network topology is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink, J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal. 46.2 (2014)) can be expressed in terms of monoid representations, which uniquely decompose into indecomposable subrepresentations. In the high-dimensional internal dynamics case, these subrepresentations are isomorphic to multiple copies of those one computes in the one-dimensional internal dynamics case. This has interesting implications for possible center subspaces in bifurcation analysis. We describe the effect on steady state and Hopf bifurcations in l-parameter families of network vector fields. The main results in that regard are that (1) generic one-parameter steady state bifurcations are qualitatively independent of the dimension of the internal dynamics and that, (2) in order to observe all generic l-parameter bifurcations that may occur for internal dynamics of any dimension, the internal dynamics has to be at least l-dimensional for steady state bifurcations and 2l-dimensional for Hopf bifurcations. Furthermore, we illustrate how additional structure in the network can be exploited to obtain even greater understanding of bifurcation scenarios in the high-dimensional case beyond qualitative statements about the collective dynamics. One-parameter steady state bifurcations in feedforward networks exhibit an unusual amplification in the asymptotic growth rates of individual cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink. Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4 (2022)). As another main result, we prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.
Keywords: Coupled cell systems; Network dynamics; Dimension reduction; Bifurcation theory; Symmetry; Monoid representation theory

On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies - Gerlach, Raphael; von der Gracht, Sören; Dellnitz, Michael (Lecture Notes in Computer Science)

Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility - Gerlach, Raphael; von der Gracht, Sören; Hahn, Christopher; Harbig, Jonas; Kling, Peter (28th International Conference on Principles of Distributed Systems (OPODIS 2024))
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).
Keywords: Swarm Algorithm; Swarm Robots; Distributed Algorithm; Pattern Formation; Limited Visibility; Oblivious

Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical Systems - Gerlach, Raphael; von der Gracht, Sören (arXiv:2503.07576)
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.
Keywords: dynamical systems; coupled systems; distributed computing; robot swarms; autonomous mobile robots; symmetry; equivariant dynamics

Heteroclinic dynamics in network dynamical systems with higher-order interactions - Bick, Christian; von der Gracht, Sören (Journal of Complex Networks)
Heteroclinic structures organize global features of dynamical systems. We analyse whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analysed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design of heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.
Keywords: Applied Mathematics; Computational Mathematics; Control and Optimization; Management Science and Operations Research; Computer Networks and Communications

Higher-order interactions lead to ‘reluctant’ synchrony breaking - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)
To model dynamical systems on networks with higher-order (non-pairwise) interactions, we recently introduced a new class of ordinary differential equations (ODEs) on hypernetworks. Here, we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady-state branch ‘reluctant’ if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady-state branch.
Keywords: higher-order interactions; synchrony breaking; network dynamics; coupled cell systems

Hypernetworks: Cluster Synchronization Is a Higher-Order Effect - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (SIAM Journal on Applied Mathematics)
Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.
Keywords: Applied Mathematics

A parametrisation method for high-order phase reduction in coupled oscillator networks - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (arXiv:2306.03320)
We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators.

Amplified steady state bifurcations in feedforward networks - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (Nonlinearity)
We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients.
Keywords: Applied Mathematics; General Physics and Astronomy; Mathematical Physics; Statistical and Nonlinear Physics

Linear Algebra driven by Data Science - Schramm, Thomas; Gasser, Ingenuin; Schwenker, Sören; Seiler, Ruedi; Lohse, Alexander; Zobel, Kay
Dieses Lernangebot widmet sich der linearen Algebra als dem Teil der Mathematik, der neben der Optimierung und der Stochastik die Grundlage für praktisch alle Entwicklungen im Bereich Künstliche Intelligenz (KI) darstellt. Das Fach ist jedoch für Anfänger meist ungewohnt abstrakt und wird daher oft als besonders schwierig und unanschaulich empfunden. In diesem Kurs wird das Erlernen mathematischer Kenntnisse in linearer Algebra verknüpft mit dem aktuellen und faszinierenden Anwendungsfeld der künstlichen neuronalen Netze (KNN). Daraus ergeben sich in natürlicher Weise Anwendungsbeispiele, an denen die wesentlichen Konzepte der linearen Algebra erklärt werden können. Behandelte Themen sind: Der Vektorraum der reellen Zahlentupel, reelle Vektorräume allgemein Lineare Abbildungen Matrizen Koordinaten und darstellende Matrizen Lineare Gleichungssysteme, Gaußalgorithmus Determinante Ein Ausblick auf nichtlineare Techniken, die für neuronale Netzwerke relevant sind.

Generalised Symmetry in Network Dynamics - Nijholt, Eddie; Rink, Bob; Schwenker, Sören (DSWeb)

A new algorithm for computing idempotents of ℛ-trivial monoids - Nijholt, Eddie; Rink, Bob; Schwenker, Sören (Journal of Algebra and Its Applications)
The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for ZM, after which we prove that it also works for RM where R is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if R is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for RM is obtained from the one our algorithm yields for ZM in a straightforward manner. In other words, for any finite R-trivial monoid M our algorithm only has to be performed for ZM, after which a system of idempotents follows for any ring with a given system of idempotents.
Keywords: Applied Mathematics; Algebra and Number Theory

Quiver Representations and Dimension Reduction in Dynamical Systems - Nijholt, Eddie; Rink, Bob W.; Schwenker, Sören (SIAM Journal on Applied Dynamical Systems)
Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction.
Keywords: Modeling and Simulation; Analysis

Genericity in Network Dynamics - Schwenker, Sören
This thesis deals with the investigation of dynamical properties – in particular generic synchrony breaking bifurcations – that are inherent to the structure of a semigroup network as well the numerous algebraic structures that are related to these types of networks. Most notably we investigate the interplay between network dynamics and monoid representation theory as induced by the fundamental network construction in terms of hidden symmetry as introduced by RINK and SANDERS. After providing a brief survey of the field of network dynamics in Part I, we thoroughly introduce the formalism of semigroup networks, the customized dynamical systems theory, and the necessary background from monoid representation theory in Chapters 3 and 4. The remainder of Part II investigates generic synchrony breaking bifurcations and contains three major results. The first is Theorem 5.11, which shows that generic symmetry breaking steady state bifurcations in monoid equivariant dynamics occur along absolutely indecomposable subrepresentations – a natural generalization of the corresponding statement for group equivariant dynamics. Then Theorem 7.12 relates the decomposition of a representation given by a network with high-dimensional internal phase spaces to that induced by the same network with one-dimensional internal phase spaces. This result is used to show that there is a smallest dimension of internal dynamics in which all generic l-parameter bifurcations of a fundamental network can be observed (Theorem 7.24). In Part III, we employ the machinery that was summarized and further developed in Part II to feedforward networks. We propose a general definition of this structural feature of a network and show that it can equivalently be characterized in different algebraic notions in Theorem 8.35. These are then exploited to fully classify the corresponding monoid representation for any feedforward network and to classify generic synchrony breaking steady state bifurcations with one- or highdimensional internal dynamics.

Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems - Schwenker, Sören (SIAM Journal on Mathematical Analysis)
We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in [B. Rink and J. Sanders, SIAM J. Math. Anal., 46 (2014), pp. 1577--1609]. It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group---finite or compact Lie. Our generalization also includes noncompact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.
Keywords: Applied Mathematics; Computational Mathematics; Analysis

Equivariant bifurcations in four-dimensional fixed point spaces - Lauterbach, Reiner; Schwenker, Sören (Dynamical Systems)
In this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in previous papers we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behaviour is different from what we have seen in the known examples.
Keywords: Computer Science Applications; General Mathematics