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Synchronization in networks of general, weakly nonlinear oscillators

We present a general approach to the study of synchrony in networks of weakly nonlinear systems described by singularly perturbed equations of the type $x''+x+εf(x,x')=0$. By performing a perturbative calculation based on normal form theory we analytically obtain an $Ø(ε)$ approximation to the Floquet multipliers that determine the stability of the synchronous solution. The technique allows us to prove and generalize recent results obtained using heuristic approaches, as well as reveal the structure of the approximating equations. We illustrate the results in several examples, and discuss extensions to the analysis of stability of multisynchronous states in networks with complex architectures.