@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}},
}

@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{32518,
  abstract     = {{This study investigates age-related changes and dyadic-specific differences in adult child–parent
relationships. Using an individuation framework, two German samples of 224 and 105 participants
aged between 21 and 47 years were administered the Network of Relationships Inventory, the
Emotional Autonomy Scale and the Authority Reciprocity Questionnaire. Factor analyses resulted
in a measurement model valid for adult children, their mothers and fathers. The model includes
connectedness (with emotional and cognitive aspects) as well as individuality (assessed as power
symmetry). Connectedness decreased with age. Symmetry in father–child relationships increased over
time, while mother–child relationships were perceived to be symmetrical by early adulthood.
Child–mother relationships were more connected than child–father relationships. Sons described
themselves as more powerful than did daughters.}},
  author       = {{Buhl, Heike M.}},
  journal      = {{International Journal of Behavioral Development}},
  keywords     = {{adult child–parent relationships, adulthood, connectedness, Germany, individuation, symmetry}},
  number       = {{5}},
  pages        = {{381 -- 389}},
  title        = {{{Development of a model describing individuated adult child-parent relationships}}},
  volume       = {{32}},
  year         = {{2008}},
}