Â鶹ӰԺ

Mean field theory of assortative networks of phase oscillators

Applied Mathematics,Ìý

Date and time:Ìý

Thursday, September 11, 2014 - 2:00pm

³¢´Ç³¦²¹³Ù¾±´Ç²Ô:Ìý

ECCR 257

´¡²ú²õ³Ù°ù²¹³¦³Ù:Ìý

In this talk I present and illustrate a new analytical technique to study synchronization in large networks of coupled oscillators. Employing the Kuramoto model of oscillator synchronization as an illustrative example, it is shown how a mean field approximation can be applied to large networks of phase oscillators with assortativity (the tendency of nodes with the same properties to connect with each other). The ansatz of Ott and Antonsen is then used to reduce the mean field kinetic equations describing the macroscopic behavior of the oscillators to a system of ordinary differential equations. The resulting formulation is illustrated by its application to a network Kuramoto problem with degree assortativity and correlation between the node degrees and the natural oscillation frequencies. Good agreement is found between the solutions of the reduced set of ordinary differential equations and full simulations of the system. One interesting result is that degree assortativity can induce transitions from a steady macroscopic state to a temporally oscillating macroscopic state through both (presumed) Hopf and SNIPER (saddle-node, infinite period) bifurcations. Possible use of these techniques to a broad class of phase oscillator network problems is discussed