Collective-phase description of coupled oscillators with general network structure

We develop a collective-phase description for a population of nonidentical limit-cycle oscillators with any network structure undergoing fully phase-locked collective oscillations. The whole network dynamics can be described by a single collective-phase variable. We derive a general formula for the collective-phase sensitivity, which quantifies the phase response of the whole network to weak external perturbations applied to the constituent oscillators. Moreover, we consider weakly interacting multiple networks and develop an effective phase coupling description for them. Several examples are given to illustrate our theory.

Figure 1

Examples of the weight vector in identical oscillators. Using Eq. (30), we find (a) $w_1:w_2=1:\kappa$ and (b) $w_1:w_2:w_3:w_4=2:4:3:1$.

Figure 2

(Color online) Examples of the weight vector in globally coupled nonidentical oscillators. (a) The weights $w_i$ for $N=2$, i.e., Eq. (39), as a function of $\alpha$ with fixed $K=2$. There is no phase-locked solution in the region of $\alpha$ where the weights are absent. (b) The natural frequencies ${\omega }_i$, (c) the relative phases ${\varphi }_i^0-{\varphi }_{50}^0$ in a fully phase-locked state, and (d) the weights multiplied by the system size, $N{w}_i$, for $N=100$, $K=1.5$, and $\alpha =0,\pm 0.7$.

Figure 3

(Color online) Results for a network of limit-cycle oscillators. (a) Network structure under consideration. (b) Wave form $x\left(\varphi \right)$ and phase sensitivity $Z\left(\varphi \right)$ of an individual oscillator. (c) Antisymmetric part of the coupling function, ${\Gamma }_{\text{a}}\left(\Delta \varphi \right)$. (d) Phase difference $\Delta {\varphi }^0$ between oscillators in each group. (e) Collective-phase sensitivity $\zeta \left(\Theta \right)$ defined for uniform forcing. For $K=0.007,0.002,0.0015$, $\left(w_1,w_2\right)$ is respectively about $\left(1.35,-0.35\right)$, $\left(2.87,-1.87\right)$, and $\left(2.89,-1.89\right)$. (f) Antisymmetric part of the collective coupling function, ${\gamma }_{\text{a}}\left(\Delta \Theta \right)$. (g) Phase difference $\Delta \Theta$ between the groups.