37G35 Attractors and their bifurcations
For a system of globally pulse-coupled phase-oscillators,
we derive conditions for stability of the completely synchronous
state and all possible two-cluster states and explain
how the different states are naturally connected via
bifurcations. The coupling is modeled using the phaseresponse-
curve (PRC), which measures the sensitivity of
each oscillator’s phase to perturbations. For large systems
with a PRC, which turns to zero at the spiking threshold,
we are able to find the parameter regions where multiple
stable two-cluster states coexist and illustrate this by an
example. In addition, we explain how a locally unstable
one-cluster state may form an attractor together will its
homoclinic connections. This leads to the phenomenon
of intermittent, asymptotic synchronization with abating
beats away from the perfect synchrony.
In this paper we study the destabilization mechanism in a ring of unidirectionally coupled oscillators. We derive an amplitude equation of Ginzburg-Landau type that describes the destabilization of the stationary state for systems with a large number of oscillators. Based on this amplitude equation, we are able to provide an explanation for the fast transition to chaos (or hyperchaos)
that can be observed in such systems. We show that the parameter interval, where the transition from a stable periodic state to chaos occurs, scales like the inverse
square of the number of oscillators in the ring. In particular, for a sufficiently large
number of oscillators a practically immediate transition to chaos can be observed.
The results are illustrated by a numerical study of a system of unidirectionally
coupled Duffing oscillators.