Equivariant Degree, Synchrony and Birfurcations

• Applicant: Professor Dr. Reiner Lauterbach
• Subject Area: Mathematics
• Term: from 2009 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 157575227

Networks of coupled cells are represented by a system of ordinary differential equations where the coupling is described by a graph. Finding common features of networks is discussed in a few papers, the influence of the network architecture on the dynamics is not well understood (compare Strogatz [32], Wang [33] and the references therein). In [21] we studied local bifurcations in networks and noticed that they behave like equivariant systems (although the concepts are entirely different): colorings give a combinatorial description of a network, spectral properties of the adjacency matrix allow to define synchrony types, which are similar to isotropy types. We aim at networks with simple internal dynamics, since we want to focus on the dynamics caused by the network’s structure. Note, however, in [21] we have seen that genericity properties depend on the internal dynamics. Equivariant degree is a tool to study fixed points, periodic orbits and their bifurcations. We want to develop degree methods for networks based on the stratification of the total space by synchrony types. Besides this extension of degree theory, we want to work within equivariant bifurcation theory on the so called Ize conjecture: does the loss of stability through an absolutely irreducible representation imply bifurcation of steady states. We have shown in [27] that Fields approach [16] cannot answer this. Pursuing some new questions lets us hope to contribute to the solution of the Ize conjecture.

• DFG Programme: Research Grants