Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network

The existence of a non-trivial phase-locked periodic orbit is established for a system of delay differential equations describing the dynamics of networks of two identical saturating neurons. We discuss the instability and the unstable manifold of this phase-locked orbit. In particular, we give detailed information about the spectrum of the related monodromy operator and establish the connection between the unstable manifold of the phase-locked orbit and the boundary of the global extension of a three-dimensional C-1-submanifold of the origin. We obtain a smooth solid spindle, contained in the global attractor, separated by a disk bordered by the phase-locked orbit. Major technical tools include a discrete Lyapunov functional, invariant manifolds and other recently developed geometric theory of delay differential equations. (C) 1999 Elsevier Science B.V. All rights reserved.