Duality on Two-dimensional Kuramoto Model
ABSTRACT :
The Kuramoto model, a well known mathematical model in non-linear dynamics, consists of a population of coupled phase oscillators. This model is used to describe synchronization. It has been recently studied that a classical Hamiltonian system with 2N state variables in its action-angle representation yields Kuramoto dynamics on N-dimensional invariant manifolds [1].There is a connection through the Hamiltonian between the locally coupled Kuramoto model in non-linear dynamics and the 2-D x-y model [5] of statistical mechanics .Using this Hamiltonian we perform a duality transformation (similar to x-y model) [3-4] on the partition function of the Kuramoto model to obtain its high temperature and low temperature expansion.