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Synchrony-breaking Bifurcations in Coupled Cell Networks: A topological approach
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Coupled cell networks are networks of cells (systems of ordinary differential equations), which are coupled together through interactions. A coupled cell network can be represented by a graph, whose nodes or vertices represent cells and edges or arrows describe couplings. The theory of coupled cell networks was introduced for the study of synchronization properties in general networks using a framework of coupled ordinary differential equations and their associated bifurcations. A synchrony-breaking bifurcation is a special type of local bifurcation, where a fully synchronized state loses its coherence and breaks up into multiple clusters of synchronized cells, characterized by patterns of synchrony. All robust patterns of synchrony organize themselves as a complete lattice by set-inclusion. In this talk, we define a lattice degree and show that how the synchrony-breaking bifurcations can be studied using the lattice degree. As an example, we analyze a synchrony-breaking Hopf-bifurcation around a fully synchronous equilibrium in a regular $10$-dimensional $5$-cell network and obtain a classification of the bifurcating branches of oscillating solutions according to their synchrony types.