43 research outputs found
Small-world networks of Kuramoto oscillators
The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is
analyzed in this work. When the number of oscillators in the network goes to
infinity, the model acquires a family of steady state solutions of degree q,
called q-twisted states. We show that this class of solutions plays an
important role in the formation of spatial patterns in the Kuramoto model on SW
graphs. In particular, the analysis of q-twisted elucidates the role of
long-range random connections in shaping the attractors in this model.
We develop two complementary approaches for studying q-twisted states in the
coupled oscillator model on SW graphs: the linear stability analysis and the
numerical continuation. The former approach shows that long-range random
connections in the SW graphs promote synchronization and yields the estimate of
the synchronization rate as a function of the SW randomization parameter. The
continuation shows that the increase of the long-range connections results in
patterns consisting of one or several plateaus separated by sharp interfaces.
These results elucidate the pattern formation mechanisms in nonlocally
coupled dynamical systems on random graphs
The nonlinear heat equation on W-random graphs
For systems of coupled differential equations on a sequence of W-random
graphs, we derive the continuum limit in the form of an evolution integral
equation. We prove that solutions of the initial value problems (IVPs) for the
discrete model converge to the solution of the IVP for its continuum limit.
These results combined with the analysis of nonlocally coupled deterministic
networks in [9] justify the continuum (thermodynamic) limit for a large class
of coupled dynamical systems on convergent families of graphs