Synchrony-Breaking Bifurcation at a Simple Real Eigenvalue for Regular Networks 1: 1-Dimensional Cells

Abstract

We study synchrony-breaking local steady-state bifurcation in networks of dynamical systems when the critical eigenvalue is real and simple, using singularity theory to transform the bifurcation into normal form. In a general dynamical system, a generic steady-state local bifurcation from a trivial state is transcritical. In the presence of symmetry, a pitchfork is also possible generically. Network structure introduces constraints that may change the generic behavior. We consider regular networks, in which all cells have the same type and all arrows have the same type, and every cell receives inputs from the same number of arrows. A characterization of all smooth admissible maps permits a singularity-theoretic analysis based on Liapunov–Schmidt reduction. Assuming that the cells have 1-dimensional internal dynamics, we give conditions on the critical eigenvectors of the linearization and its transpose that determine when a generic bifurcation is transcritical, pitchfork, or more degenerate. We prove that for all regular n-cell networks, such bifurcations are generically n-determined. In the path-connected case, this is improved to $(n-1)$-determined. In bidirectional networks, generic bifurcation is transcritical or pitchfork, but the role of symmetry is minor. In the general case, degenerate cases can occur: the network must have at least 4 cells (5 in the path-connected case). We give examples of networks for which generic bifurcations are degenerate, including a 6-cell network with a normal form that is determined only at degree 6 and a path-connected 5-cell network with a normal form that is determined only at degree 4.