Resonance clustering in globally coupled electrochemical oscillators with external forcing

Experiments are carried out with a globally coupled, externally forced population of limit-cycle electrochemical oscillators with an approximately unimodal distribution of heterogeneities. Global coupling induces mutually entrained (at frequency ${\omega }_1$) states; periodic forcing produces forced-entrained $\left({\omega }_F\right)$ states. As a result of the interaction of mutual and forced entrainment, resonant cluster states occur with equal spacing of frequencies that have discretized frequencies following a resonance rule ${\omega }_n\cong n{\omega }_1-\left(n-1\right){\omega }_F$. Resonance clustering requires an optimal, intermediate global coupling strength; at weak coupling the clusters have smaller sizes and do not strictly follow the resonance rule, while at strong coupling the population behaves similar to a single, giant oscillator.

Figure 1

Experiments: Dimensionless frequency histograms $\left(\omega =f/⟨f⟩-1\right)$ at different coupling strengths for a unimodal population. The mean natural frequency $⟨f⟩=0.47\text{Hz}$, standard deviation $\sigma =8.5\text{mHz}$. (a) $K=0$, (b) $K=0.026$, (c) $K=0.031$, (d) $K=0.042$.

Figure 2

Experiments: Periodic forcing on a unimodal population with $K=0.042$. $V_0=1.09\text{V}$. Mean frequency before forcing $⟨f⟩=0.4497\text{Hz}$, forcing frequency $f_F=0.47\text{Hz}$, $f_F-⟨f⟩>2\sigma$. Top row: Frequency histograms at different forcing amplitudes. Middle row: Frequencies in presence of coupling and forcing versus the corresponding natural frequencies. Upper and lower dashed line are $f_F$ and $⟨f⟩$, respectively. Bottom row: Time series of order $r$. (a) $b=3.3\text{mV}$, (b) $b=4.0\text{mV}$, (c) $b=4.6\text{mV}$, (d) $b=5.3\text{mV}$.

Figure 3

Periodic forcing on a unimodal population with weak coupling strengths. Frequencies in presence of coupling and forcing versus the corresponding natural frequencies. Top row: $K=0.031$, $⟨f⟩=0.466\text{Hz}$, $f_F=0.486\text{Hz}$. (a) $b=3.3\text{mV}$, (b) $b=4.0\text{mV}$, (c) $b=4.6\text{mV}$, (d) $b=5.3\text{mV}$. Bottom row: $K=0.026$, $⟨f⟩=0.4571\text{Hz}$, $f_F=0.48\text{Hz}$. (e) $b=3.3\text{mV}$, (f) $b=4.95\text{mV}$, (g) $b=5.3\text{mV}$, (h) $b=5.95\text{mV}$. Upper and lower dashed line are $f_F$ and $⟨f⟩$, respectively.

Figure 4

(Color online) Numerical simulations: Mutual and forced entrainment in a model of 64 globally coupled oscillators with external forcing. (a) Natural (inherent) frequency distribution. (b) Mutual entrainment with global coupling only. Number of mutually entrained oscillators vs coupling strength $K$. (c) Forced entrainment with external forcing only. Number of oscillators locked to external forcing signal vs forcing amplitude $b$. (d) Phase diagram showing forced and mutually entrained regions in the coupling strength—forcing amplitude parameter space. $F$, forced entrainment. $M$, mutual entrainment. $D$, desynchronized state. $f_F=0.0745$.

Figure 5

Numerical simulations: frequency distributions and order parameters at $K=0.67$. Top row: frequency distribution of oscillators. Middle row: frequency of oscillators vs inherent frequency (dashed line denotes the forcedly entrained cluster). Bottom row: order parameter vs time. (a) Mutual entrainment dominated state with weak forcing, $b=0.428$. (b) Resonance clustering at intermediate forcing strength, $b=0.443$. (c) Entrainment to strong forcing, $b=0.452$.

Figure 6

Numerical simulations: frequencies of oscillators (with coupling and forcing) vs inherent frequencies at weaker interactions $K=0.54$. (a) Weak forcing, $b=0.385$. (b) Intermediate forcing strength, $b=0.398$. (c) Strong forcing, $b=0.410$.

Figure 7

Numerical simulations: population of 512 oscillators exhibiting resonance clustering at $K=0.67$ and $b=0.47$. (a) Frequency distribution of oscillators. (For $f=0.075$ there is a large peak with $N=325$, therefore, a zoom is shown.) (b) Frequency of oscillators vs inherent frequency (dashed line denotes the forcing frequency).