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 .There is a connection through the Hamiltonian between the locally coupled Kuramoto model in non-linear dynamics and the 2-D x-y model  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.
 D. Witthaut and M. Timme, Kuramoto dynamics in Hamiltonian systems Physical Review E 90,032917(2014)
 V. Flovik, F. Macia, and E. Wahlstrom, Describing synchronization and topological excitations in array of magnetic spin torque oscillators through the Kuramoto model Scienti_c Reports 6:32528 (2016)
 N. Gupte and S. R. Shenoy. Physical interpretation of variables on the dual lattice. Physical Review B , Volume 31, Number 5 , 1 March 1985.
 R. Savit. Duality in _eld theory and statistical systems Rev. Mod. Phys. Vol.52, No. 2, Part I, April 1980
 Nigel Goldenfeld. Lectures on Phase Transition and The Renormalization Group. Advanced Book Program , 1992 , Massachusetts.