Overdetermined ODEs and rigid periodic states in network dynamics

Abstract

We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010-12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, 'strong hyperbolicity', which is related to a network analogue of the Kupka-Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the H=K Theorem in equivariant dynamics. We prove the Rigidity Conjectures for all 1-and 2-colourings and all 2-and 3-node networks by showing that strong hyperbolicity is generic in these cases.

Symmetry and Network Topology in Neuronal Circuits: Complicity of Form and Function

“…This conjecture has been proved for a broad class of networks, including all homogeneous and fully inhomogeneous networks [Golubitsky et al, 2010]. In [Stewart, 2022] it is proved for all networks under a slightly stronger condition than hyperbolicity. There are analogous results for phase patterns [Golubitsky et al, 2012;Stewart, 2022].…”

Stable Synchronous Propagation of Signals by Feedforward Networks

“…We call such behaviour a phase pattern. Cyclic group symmetry is intimately involved in such patterns [38,94,96] and [40,Chapter 17]; see Section 7.1. In the context of a chain of successive nodes, such states can be viewed as travelling waves.…”

“…This topic originated in equivariant dynamics [43]; more recent network analogues are discussed comprehensively in [40,Chapter 17]. In particular, there are good reasons to suppose that, subject to some technical conditions, the quotient network by synchrony must have cyclic group symmetry to support a discrete rotating wave [38,94,96] in a structurally stable manner.…”

Stable Synchronous Propagation of Signals by Feedforward Networks

“…We call such behaviour a phase pattern. Cyclic group symmetry is intimately involved in such patterns [35,89,91] and [37,Chapter 17]; see Section 7.1. In the context of a chain of successive nodes, such states can be viewed as travelling waves.…”

“…This topic originated in equivariant dynamics [40]; more recent network analogues are discussed comprehensively in [37,Chapter 17]. In particular, there are good reasons to suppose that, subject to some technical conditions, the quotient network by synchrony must have cyclic group symmetry to support a discrete rotating wave [35,89,91] in a structurally stable manner.…”