On the study of cellular nonlinear networks via amplitude and phase dynamics

The purpose of this manuscript is to propose a method for investigating the global dynamics of nonlinear oscillatory networks, with arbitrary couplings. The procedure is based on the assumption that each oscillator can be accurately described via its time-varying amplitude and phase variables. The proposed method allows one to derive a set of coupled nonlinear ordinary differential equations governing this couple of variables. By analyzing the evolution of amplitudes and phases, one can investigate the stability properties of the limit cycles for the whole system in a simpler way with respect to the latest available methodologies. Furthermore, it is proved that this technique also works for weakly connected oscillatory networks. Finally, as a case study, a chain of third-order oscillators (Chua's circuits) is considered and the results are compared to those obtained via a numerical technique, entirely based on the harmonic balance (HB) approach.