Global attractors of delay differential systems arising from networks of neurons.

摘要

In this thesis, we describe some important properties of the global attractor of a system of differential equations with delays, which describes the dynamics of a network of two saturatory amplifiers (neurons) with delayed outputs. We obtain a 2-dimensional closed disk bordered by a phase-locked periodic orbit and describe completely the dynamics, including various heteroclinic connections, in a 3-dimensional submanifold (solid spindle) as the global forward extension of a local leading unstable manifold of the trivial solution. We obtain precise information on the Floquet multipliers, which implies that the periodic orbit is linearly unstable, and we describe the unstable set of the periodic orbit and some smoothness results of various invariant sets of the global attractor. Furthermore, we show that the considered system exhibits coexistence of both synchronized and phase-locked periodic orbits and we describe the connecting orbits from synchronized orbits to phase-locked orbits and the basins of attraction of these orbits. Finally, we study two classes of singularly perturbed problems and obtain, as limiting profiles of various periodic orbits, square waves, pulses of bounded amplitudes, and pulses of unbounded amplitudes whose multiplications with the singular parameter tend to sawtooth waves or diamond-like waves.