Clustering in globally coupled oscillators near a Hopf bifurcation: Theory and experiments

A theoretical analysis is presented to show the general occurrence of phase clusters in weakly, globally coupled oscillators close to a Hopf bifurcation. Through a reductive perturbation method, we derive the amplitude equation with a higher-order correction term valid near a Hopf bifurcation point. This amplitude equation allows us to calculate analytically the phase coupling function from given limit-cycle oscillator models. Moreover, using the phase coupling function, the stability of phase clusters can be analyzed. We demonstrate our theory with the Brusselator model. Experiments are carried out to confirm the presence of phase clusters close to Hopf bifurcations with electrochemical oscillators.

Figure 1

Phase diagram of cluster states in the Brusselator system for (a) positive ($\kappa =0.001$) and (b) negative coupling ($\kappa =-0.001$). On the solid and dashed lines, $b_1$ and $b_2$ change their signs, respectively. The regions $b_1>0$ and $b_2<0$ are right of the lines. Numerical data are obtained by direct numerical simulations of Eqs. (35a) and (35b) with $N=4$, $B=(1+{\epsilon }^2)B_{\mathrm{c}},\epsilon =r\sqrt{\frac{2+A^2}{A^2(1+A^2)}},r=0.1$.

Figure 2

Snapshots of cluster states in the Brusselator system. For a better presentation, displayed snapshots are those before complete convergence. (a) The one-cluster state is obtained. (b) The balanced two-cluster state is obtained. Parameter values: $N=12$, $\kappa =-0.001,d=0.4,B=(1+{\epsilon }^2)B_{\mathrm{c}},\epsilon =r\sqrt{\frac{2+A^2}{A^2(1+A^2}}$, $r=0.1$, (a) $A=2.8$, (b) $A=1.1$.

Figure 3

Clustering behavior in the Brusselator system farther from the Hopf bifurcation point. Each line indicates how many times the $n$ cluster state (not necessarily perfectly balanced) was obtained out of 100 different initial conditions. $N=24$, $A=1.0$, $d=0.7$, $\kappa =-0.01$. Initial values of $x_i$ and $y_i$ are random numbers taken from the uniform distribution in the range $[-0.05,0.05]$.

Figure 4

Experiments: Three-cluster state close to Hopf bifurcation with negative global coupling of 64 electrochemical oscillators. $K=-0.88$. (a) Current time series and the three cluster configuration at $V=1.05$ V (close to a Hopf bifurcation). Solid, dashed, and dotted curves represent currents from the three clusters. (b) Cluster configuration. (White, black, and gray circles represent the three clusters.) (c) Response function and waveform (inset) of electrode potential $E=V-I{R}_{\text{tot}}$, where $I$ is the current of the single oscillator. (d) Phase coupling function. (e) Stability analysis of the clusters with experiment-based phase models for $K=-1$: the maximum of the real parts of eigenvalues for each balanced cluster state.

Figure 5

Experiments: Multiple cluster states observed as parameters are moved away from Hopf bifurcation with negative global coupling for 64 electrochemical oscillators. Top row: four clusters, $V=1.09\text{V}$, $K=-0.88$. Middle row: five clusters. $V=1.11\text{V}$, $K=-0.88$. (a) Current time series of the four clusters. (b) Cluster configuration (17:14:15:18) for four-cluster state. (c) Current time series. (d) Cluster configuration (15:13:14:7:15) of five-cluster state. (e) Experimentally observed cluster states as a function of applied potential. (f) The most stable cluster state as a function of potential predicted by the experimentally obtained phase model.