From phase to amplitude oscillators

Abstract

We analyze an intermediate collective regime where amplitude oscillators distribute themselves along a closed, smooth, time-dependent curve $\mathcal{C}$, thereby maintaining the typical ordering of (identical) phase oscillators. This is achieved by developing a general formalism based on two partial differential equations, which describe the evolution of the probability density along $\mathcal{C}$ and of the shape of $\mathcal{C}$ itself. The formalism is specifically developed for Stuart-Landau oscillators, but it is general enough to be applied to other classes of amplitude oscillators. The main achievements consist in: (i) identification and characterization of a new transition to self-consistent partial synchrony (SCPS), which confirms the crucial role played by higher Fourier hamonics in the coupling function; (ii) an analytical treatment of SCPS, including a detailed stability analysis; (iii) the discovery of a new form of collective chaos, which can be seen as a generalization of SCPS and characterized by a multifractal probability density. Finally, we are able to describe given dynamical regimes both at the macroscopic as well as the microscopic level, thereby shedding further light on the relationship between the two different levels of description.

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Cited by in corpus (6)

• Phase reduction beyond the first order: the case of the mean-field complex Ginzburg-Landau equation
• Enlarged Kuramoto Model: Secondary Instability and Transition to Collective Chaos
• Phase and amplitude dynamics of coupled oscillator systems on complex networks
• Irregular collective dynamics in a Kuramoto-Daido system
• Quasi phase reduction of all-to-all strongly coupled $\lambda -\omega$ oscillators near incoherent states
• Chimeras with uniformly distributed heterogeneity: two coupled populations