Homogeneous Coupled Cell Systems with High-dimensional Internal Dynamics
Sören von der Gracht, Eddie Nijholt, Bob Rink
TL;DR
This work reveals that homogeneous networks with high-dimensional internal dynamics admit a robust dimension-reduction framework grounded in monoid symmetry and fundamental networks. By representing the network dynamics via tensor products and indecomposable subrepresentations, the authors show that generic bifurcation behavior in $l$-parameter families is determined by the corresponding 1D structure, with minimal internal dimension $\overline{d}=l$ for steady-state bifurcations and $\overline{d}=2l$ for Hopf bifurcations. Consequently, many bifurcation phenomena in high-dimensional internal dynamics can be understood and predicted from the well-studied 1D case, including synchrony patterns and center-manifold reductions. The paper also applies these principles to feedforward networks, demonstrating an amplification effect persists under high-dimensional internal dynamics and providing a principled way to lift 1D bifurcation branches to DD. Overall, it delivers a symmetry-based, dimension-agnostic toolkit for analyzing complex network dynamics with large internal state spaces.
Abstract
The analysis of network dynamics is oftentimes restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the 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 case. We describe the implications of this observation on steady state and Hopf bifurcations in $l$-parameter families of network vector fields. The main results 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 understanding 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)). We prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.