Necessary and sufficient conditions for synchronization of nonlinear oscillators with a given class of coupling

Necessary and sufficient conditions are provided for phase synchronization of systems of identical nonlinear oscillators. The oscillators considered are described by the set of equations
\dot{f}_i=f_1(x_i,y_i,\mu)
\dot{y}_i= f_2(x_i,y_i,\mu)i= 1,\cdots,n
. The type of coupling considered is
\dot{x}_i = f_1 (x_i,y_i,\mu) + \Sigma_{j=1}^{n} g_1(x_j- x_i)
\dot{y}_i = f_2(x_i,y_i,\mu) + \Sigma_{j=1}^{n} g_2(y_j-y_i)
where
g_1,g_2 \in C^2(R)
and
g_1(0)=g_2(0)=0
. Our approach is not based on the asymptotic methods of Mitropolsky and Bogoliubov and the coupling can be arbitrarily large. The conditions can be used successfully in several cases and some of the results can be extended to systems of slightly different oscillators.