@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{8918,
  abstract     = {{The paper deals with cyclic periodic structures modelling bladed disk assemblies of blades with friction elements for vibration damping. These elements placed between adjacent blades reduce the vibration amplitudes by means of dry friction resulting from centrifugal forces acting on the elements and relative displacements of the blades. However, the application of these friction elements results in an additional dynamical coupling which together with mistuning of some system parameters (e.g., blade eigenfrequency or contact parameters) may cause localization of vibration. In the present paper a linear approximation of such a system is investigated. The structure composed of cyclic periodic cells modelled each as a clamped-free beam interacting with each other by means of viscoelastic elements of complex stiffness is applied for dynamic system analysis. In case of free vibrations as well as in case of steady-state dynamic response to a harmonic pressure field, a perfect periodic structure and the structures with periodically mistuned parameters (blade eigenfrequencies and contact parameters) are studied. Some regularities in the dynamic response of the systems with mistuning have been noticed. Despite the fact that only a linear approximation has been used, the results and conclusions can be applied for models which describe the blade interaction in a nonlinear way.}},
  author       = {{Krzyzynski, Tomasz and Popp, Karl and Sextro, Walter}},
  issn         = {{0960-0779}},
  journal      = {{Chaos, Solitons \& Fractals}},
  number       = {{10}},
  pages        = {{1597 -- 1609}},
  title        = {{{On some regularities in dynamic response of cyclic periodic structures}}},
  doi          = {{10.1016/S0960-0779(99)00080-6}},
  volume       = {{11}},
  year         = {{2000}},
}

@article{16614,
  author       = {{Guder, Rabbijah and Dellnitz, Michael and Kreuzer, Edwin}},
  issn         = {{0960-0779}},
  journal      = {{Chaos, Solitons & Fractals}},
  pages        = {{525--534}},
  title        = {{{An adaptive method for the approximation of the generalized cell mapping}}},
  doi          = {{10.1016/s0960-0779(96)00118-x}},
  year         = {{1997}},
}