Abstract
A system of coupled oscillators can exhibit a rich variety of dynamical behaviors. When we investigate the dynamical properties of the system, we first analyze individual oscillators and the microscopic interactions between them. However, the structure of a coupled oscillator system is often hierarchical, so that the collective behaviors of the system cannot be fully clarified by simply analyzing each element of the system. For example, we found that two weakly interacting groups of coupled oscillators can exhibit anti-phase collective synchronization between the groups even though all microscopic interactions are in-phase coupling. This counter-intuitive phenomenon can occur even when the number of oscillators belonging to each group is only two, that is, when the total number of oscillators is only four. In this paper, we clarify the mechanism underlying this counter-intuitive phenomenon for two weakly interacting groups of two oscillators with global sinusoidal coupling.
Highlights
When we investigate the dynamical properties of the system, we first analyze individual oscillators and the microscopic interactions between them
To study the phase synchronization between macroscopic rhythms, we recently formulated a theory for the collective phase description of macroscopic rhythms emerging from coupled phase oscillators for the following three representative cases: (A) phase coherent states in globally coupled noisy identical oscillators[37,38,39], (B) partially phase-locked states in globally coupled noiseless nonidentical oscillators[40], and (C) fully phase-locked states in networks of coupled noiseless nonidentical oscillators[41]
In this paper, using the collective phase description method developed in Ref. 41, we study the phase synchronization between collective rhythms of coupled oscillator groups for case (C)
Summary
We found that two weakly interacting groups of coupled oscillators can exhibit anti-phase collective synchronization between the groups even though all microscopic interactions are in-phase coupling. We clarify the mechanism underlying this counter-intuitive phenomenon for two weakly interacting groups of two oscillators with global sinusoidal coupling. The dynamical behaviors exhibited by interacting groups of globally coupled phase oscillators have been intensively investigated[18,19,20,21,22,23,24,25,26,27]. For both cases, we found counter-intuitive phenomena in which the groups can exhibit anti-phase collective synchronization in spite of microscopic in-phase external coupling and vice versa
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