Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions

We discuss the desynchronization transition in networks of globally coupled identical oscillators with attractive and repulsive interactions. We show that, if attractive and repulsive groups act in antiphase or close to that, a solitary state emerges with a single repulsive oscillator split up from the others fully synchronized. With further increase of the repulsing strength, the synchronized cluster becomes fuzzy and the dynamics is given by a variety of stationary states with zero common forcing. Intriguingly, solitary states represent the natural link between coherence and incoherence. The phenomenon is described analytically for phase oscillators with sine coupling and demonstrated numerically for more general amplitude models.

Figure 1

Desynchronization transition for $N_{1,2}=5$, $\alpha =\beta =0$. (a) Bifurcation diagram. Dashed vertical line at ${\varepsilon }_{\text{cr}}=0$ shows the border between the full synchrony and the solitary state, and the solid hyperbolic curve shows where the latter loses its stability, i.e., ${\varepsilon }_{\text{sol}}^{+}$. The inset (b) presents an example of the solitary state, here for $\varepsilon =0.25$: There exists a cluster of nine elements and one oscillator is exactly in antiphase to it, hence, ${\rho }_1=1$, ${\rho }_2=0.6$ [cf. panel (e)]. Insets (c) and (d) present two examples for $\varepsilon =1>{\varepsilon }_{\text{sol}}^{+}$; (c) is a state with a fuzzy cluster, where all phases but two are very close though not identical (“mercedes state”), while (d) is a distributed state without fuzzy clusters. (e) Order parameters ${\rho }_{1,2}$ and the amplitude of the common forcing $h$ (blue circles, red pluses, and black diamonds, respectively) as functions of $\varepsilon$, for 100 runs; in each run the initial phases are chosen from the uniform random distribution in $0,2\pi$. For full synchrony and for the solitary state the dependence $h\left(\varepsilon \right)$ perfectly agrees with Eq. (3) which yields for these states $h=-\varepsilon /2$ and $h=0.2-0.3\varepsilon$, respectively. (f) The ratio $r$ of nonsolitary states (blue circles), i.e., of states with $h=0$, obtained in ${10}^4$ runs with random initial conditions; it is well approximated by $r\sim {\varepsilon }^{10/3}$ (the slope of the solid red line in the inset is $10/3$). (g) Histogram of ${\rho }_1$ for $\varepsilon =1$, obtained from ${10}^4$ runs with random initial conditions, demonstrates that the fuzzy clusters with ${\rho }_1\lesssim 1$ are dominating. Thus, the desynchronized states $h=0$ typically correspond to coherence of subpopulations.

Figure 2

(a) Domains of solitary state for $N_1=N_2$, $\alpha =0.05$, and for different $\beta$. The dashed vertical line marks ${\varepsilon }_{\text{cr}}$, where the full synchrony becomes unstable. The bold line 1 marks the right border ${\varepsilon }_{\text{sol}}^{+}$ of the domain for $\alpha =\beta =0.05$ [notice that it coincides with the curve $4{(N-4)}^{-1}$ shown in Fig. 1 for the case $\alpha =\beta =0$]. Curves 2–6 show the domains for $\beta =0.1$, $0.15$, $0.2$, $0.25$, and $0.35$, respectively; these domains shrink with the increase of $|\alpha -\beta |$. Interestingly, for $\alpha \ne \beta$ the full synchrony and the solitary states are separated by an interval of incoherent dynamics. (b) Solitary angle ${\eta }_{\text{sol}}$ as a function of the coupling parameter $\varepsilon$, for $N_1=N_2=5$, $\alpha =0.05$, and $\beta =-0.2$, 0, $0.05$, $0.15$, and $0.2$ (curves 1–5), dashed lines; the intervals where the solitary state is stable are shown by bold red lines. Bold horizontal dashed lines show ${lim}_{\varepsilon \to \infty }{\eta }_{\text{sol}}$; here it is $\approx 3.275$. For $\beta =0.0832$, ${\eta }_{\text{sol}}$ equals this value for all $\varepsilon >{\varepsilon }_{\text{cr}}$.

Figure 3

(a) An example of coherence exchange for $N_{1,2}=5$, $\alpha =2\pi /3$, $\beta =0$. Here ${\rho }_{1,2}$ and $h$ are shown by blue circles, red pluses, and black diamonds, respectively. For $\varepsilon <{\varepsilon }_{\text{cr}}=-1.5$ all oscillators are synchronized. For $\varepsilon >{\varepsilon }_{\text{cr}}$ the repulsive units form a fuzzy cluster, ${\rho }_2\lesssim 1$, while the attractive oscillators first desynchronize, so that $\rho \approx 0$ for $\varepsilon =-1$ and then synchronize again. When $\varepsilon$ becomes positive, the attractive oscillators form a fuzzy cluster, while the repulsive desynchronize. Initial phases for both groups are placed on two arcs of arbitrary length, shifted by $\pi$. Panels (b)–(d) are snapshots for $\varepsilon =-1$, $\varepsilon =-0.5$, and $\varepsilon =0.5$.

Figure 4

Solitary state in the ensemble of van der Pol oscillators (a) and of chaotic Rössler oscillators (b). Blue circles and red diamonds show the elements of the attractive and the repulsive group, respectively. Gray line shows the limit cycle of a single oscillator (a) and projection of the strange attractor of an autonomous Rössler system (b). In (a) all oscillators except one are in the cluster at $x_j\approx -1$; the solitary element is at $x_j\approx 1$, i.e., nearly in antiphase. In (b) all oscillators except for one repulsive element group with approximately the same phase (fuzzy cluster), as is typical for chaotic oscillators, where phase locking is generally not perfect.