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

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, or, in the presence of symmetry, a pitchfork (if the action on the critical eigenspace is nontrivial). Network structure introduces constraints that may change the generic behaviour. 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. Assuming that the cells have 1-dimensional internal dynamics, we give conditions on the critical eigenvectors of the linearisation and its transpose that determine when a generic bifurcation is transcritical, pitchfork, or more degenerate. A characterisation of all smooth admissible maps permits a singularity-theoretic analysis based on Liapunov-Schmidt reduction. In bidirectional networks, generic bifurcation is transcritical or pitchfork, but the role of symmetry is minor. More generally, 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, culminating in a 6-cell network with a normal form that is determined only at order 6, and a path-connected 5-cell network with a normal form that is determined only at order 4. On the other hand, we prove that for all regular networks, generic synchrony-breaking steady-state bifurcation at a simple eigenvalue is finitely determined. We prove a sharper version of this theorem in the path-connected case. We also sketch a simple proof that two networks are ODE-equivalent if and only if they are linearly equivalent.