@unpublished{64865, abstract = {{We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired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 $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ 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.}}, author = {{von der Gracht, Sören and Lohse, Alexander}}, booktitle = {{arXiv:2603.06157}}, title = {{{Design of Hierarchical Excitable Networks}}}, year = {{2026}}, } @article{64979, abstract = {{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.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, issn = {{0960-0779}}, journal = {{Chaos, Solitons & Fractals}}, keywords = {{Coupled cell systems, Network dynamics, Dimension reduction, Bifurcation theory, Symmetry, Monoid representation theory}}, publisher = {{Elsevier BV}}, title = {{{Homogeneous coupled cell systems with high-dimensional internal dynamics}}}, doi = {{10.1016/j.chaos.2026.118196}}, volume = {{208}}, year = {{2026}}, } @article{66057, author = {{Bullerjahn, Nils and Kovács, Balázs}}, journal = {{ArXiv}}, title = {{{Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces}}}, doi = {{10.48550/arXiv.2606.25934}}, year = {{2026}}, } @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}}, } @article{55459, author = {{Bullerjahn, Nils and Kovács, Balázs}}, journal = {{IMA Journal of Numerical Analysis}}, title = {{{Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions}}}, doi = {{10.1093/imanum/draf009}}, year = {{2025}}, } @article{53141, author = {{Edelmann, Dominik and Kovács, Balázs and Lubich, Christian}}, journal = {{IMA Journal of Numerical Analysis}}, number = {{5}}, pages = {{2581----2627}}, title = {{{Numerical analysis of an evolving bulk--surface model of tumour growth}}}, doi = {{10.1093/imanum/drae077}}, volume = {{45}}, year = {{2025}}, } @article{55781, abstract = {{In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods}}, author = {{Bai, Genming and Kovács, Balázs and Li, Buyang}}, journal = {{IMA Journal of Numerical Analysis}}, title = {{{Maximal regularity of evolving FEMs for parabolic equations on an evolving surface}}}, doi = {{10.1093/imanum/draf082.}}, 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}}, } @article{58532, author = {{Bullerjahn, Nils}}, journal = {{arXiv}}, title = {{{Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions}}}, doi = {{10.48550/ARXIV.2502.03847}}, 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}}, } @article{52726, abstract = {{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.}}, author = {{Bick, Christian and von der Gracht, Sören}}, issn = {{2051-1329}}, journal = {{Journal of Complex Networks}}, keywords = {{Applied Mathematics, Computational Mathematics, Control and Optimization, Management Science and Operations Research, Computer Networks and Communications}}, number = {{2}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Heteroclinic dynamics in network dynamical systems with higher-order interactions}}}, doi = {{10.1093/comnet/cnae009}}, volume = {{12}}, year = {{2024}}, } @article{45972, author = {{Kovács, Balázs}}, journal = {{SIAM Journal on Scientific Computing}}, number = {{2}}, pages = {{A645----A669}}, title = {{{Numerical surgery for mean curvature flow of surfaces}}}, doi = {{10.1137/22M1531919}}, volume = {{46}}, year = {{2024}}, } @unpublished{55078, abstract = {{This paper develops and discusses a residual-based a posteriori error estimate and a space--time adaptive algorithm for solving parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. A space--time adaptive algorithm is proposed using the derived error indicator. Numerical experiments illustrate and complement the theory.}}, author = {{Kovács, Balázs and Lantelme, Michael Frederik Raúl}}, booktitle = {{arXiv:2407.02101}}, title = {{{A posteriori error estimates for parabolic partial differential equations on stationary surfaces}}}, year = {{2024}}, } @article{59171, abstract = {{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.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, issn = {{1364-5021}}, journal = {{Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}}, keywords = {{higher-order interactions, synchrony breaking, network dynamics, coupled cell systems}}, number = {{2301}}, publisher = {{The Royal Society}}, title = {{{Higher-order interactions lead to ‘reluctant’ synchrony breaking}}}, doi = {{10.1098/rspa.2023.0945}}, volume = {{480}}, year = {{2024}}, } @article{49326, abstract = {{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.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, issn = {{0036-1399}}, journal = {{SIAM Journal on Applied Mathematics}}, keywords = {{Applied Mathematics}}, number = {{6}}, pages = {{2329--2353}}, publisher = {{Society for Industrial & Applied Mathematics (SIAM)}}, title = {{{Hypernetworks: Cluster Synchronization Is a Higher-Order Effect}}}, doi = {{10.1137/23m1561075}}, volume = {{83}}, year = {{2023}}, } @article{45971, abstract = {{Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.}}, author = {{Bartels, Sören and Kovács, Balázs and Wang, Zhangxian}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints}}}, doi = {{10.1093/imanum/drad037}}, year = {{2023}}, } @article{53140, abstract = {{We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.}}, author = {{Contri, Alessandro and Kovács, Balázs and Massing, André}}, journal = {{arXiv}}, title = {{{Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates}}}, doi = {{10.48550/ARXIV.2312.05511}}, year = {{2023}}, } @unpublished{45498, abstract = {{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.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, booktitle = {{arXiv:2306.03320}}, pages = {{29}}, title = {{{A parametrisation method for high-order phase reduction in coupled oscillator networks}}}, year = {{2023}}, } @article{33264, abstract = {{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.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, keywords = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{4}}, pages = {{2073--2120}}, publisher = {{IOP Publishing}}, title = {{{Amplified steady state bifurcations in feedforward networks}}}, doi = {{10.1088/1361-6544/ac5463}}, volume = {{35}}, year = {{2022}}, } @article{45970, abstract = {{ We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text]. }}, author = {{Garcke, Harald and Kovács, Balázs and Trautwein, Dennis}}, issn = {{0218-2025}}, journal = {{Mathematical Models and Methods in Applied Sciences}}, keywords = {{Applied Mathematics, Modeling and Simulation}}, number = {{13}}, pages = {{2673--2758}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{Viscoelastic Cahn–Hilliard models for tumor growth}}}, doi = {{10.1142/s0218202522500634}}, volume = {{32}}, year = {{2022}}, }