Summary: Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM J. APPLIED DYNAMICAL SYSTEMS c 2007 Society for Industrial and Applied Mathematics
Vol. 6, No. 4, pp. 728­758
Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators
Peter Ashwin, G´abor Orosz, John Wordsworth, and Stuart Townley
Abstract. Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic net-
works. We investigate such a heteroclinic network between partially synchronized states where the
phases cluster into three groups. For the coupling considered there exist 30 different three-cluster
states in the case of five oscillators. We study the structure of the heteroclinic network and demon-
strate that it is possible to navigate around the network by applying small impulsive inputs to the
oscillator phases. This paper shows that such navigation may be done reliably even in the presence
of noise and frequency detuning, as long as the input amplitude dominates the noise strength and
the detuning magnitude, and the time between the applied pulses is in a suitable range. Further-
more, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as
a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states
and replace the heteroclinic network by a network of excitable three-cluster states. The resulting
"excitable network" has the same structure as the heteroclinic network and navigation around the
excitable network is also possible by applying large impulsive inputs. We also discuss features that
have implications for related models of neural activity.
Key words. globally coupled oscillators, three-cluster state, heteroclinic connection/network, winnerless com-

Source: Awtar, Shorya - Department of Mechanical Engineering, University of Michigan

Collections: Engineering