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

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Necessary and sufficient conditions are provided for phase synchronization of systems of identical nonlinear oscillators. The oscillators considered are described by the set of equationsdot{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 isdot{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)whereg_1,g_2 in C^2(R)andg_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.

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Circuits and Systems, IEEE Transactions on  (Volume:24 ,  Issue: 4 )