Dynamics of two populations of phase oscillators with different frequency distributions

A large variety of rhythms are observed in nature. Rhythms such as electroencephalogram signals in the brain can often be regarded as interacting. In this study, we investigate the dynamical properties of rhythmic systems in two populations of phase oscillators with different frequency distributions. We assume that the average frequency ratio between two populations closely approximates some small integer. Most importantly, we adopt a specific coupling function derived from phase reduction theory. Under some additional assumptions, the system of two populations of coupled phase oscillators reduces to a low-dimensional system in the continuum limit. Consequently, we find chimera states in which clustering and incoherent states coexist. Finally, we confirm consistent behaviors of the derived low-dimensional model and the original model.

Figure 1

Top: Natural frequency distributions of the phase oscillators in the two populations whose mean frequencies satisfy the resonant relation $k:1$. Bottom: The coupling strengths is $\mu =(1+A)/2$ within the same population and $\nu =(1-A)/2$ between the two populations.

Figure 2

Time evolutions of the order parameters of the populations in the reduced system for (a) $A=0.9$, (b) $A=0.1$, (c) $A=-0.1$, and (d) $A=-0.9$.

Figure 3

Time evolutions of the order parameters of the populations in the modified systems; (a) $A=0.9$, (b) $A=0.1$, (c) $A=-0.1$, and (d) $A=-0.9$. Right panels show corresponding snapshots of the steady-state phase distributions (Up, fast population; Down, slow population).

Figure 4

Comparisons between the average (a) and the standard deviation (b) of the steady-state order parameters in the modified system (with $N={10}^4$) and the reduced system.

Figure 5

(a) Time evolutions of the order parameters in the modified and original systems with $N={10}^4$ for $A=0$. Bottom panels are snapshots of the steady-state phase distributions in the modified (b) and original (c) systems for $A=0$.

Figure 6

Comparison between averages (a) and standard deviations (b) of the steady-state order parameters in the reduced and original systems $\left(N={10}^4\right)$.

Figure 7

(a) Time evolutions of the order parameters in the modified and original models with $N={10}^4$ for $A=0$. Bottom panels are snapshots of the steady-state phase distributions in (b) the modified system and (c) the original system for $A=0.2$.

Figure 8

(Left) Averages and (Right) standard deviations of the steady-state order parameters in the reduced and original systems with $N={10}^4$. Top to bottom: (a) $k=3$, (b) $k=5$, (c) $k=7$, and (d) $k=11$.