Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings

We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling wave and $\pi$-states, in addition to the classical fully synchronized and incoherent states. Bistability between all collective states is reported. Analytical results are generally in excellent agreement with simulations; for some parameters and initial conditions, however, we numerically detect chimera-like states which are not captured by the OA theory.

Figure 1

Schematic illustration of a finite system composed of two intertwined subpopulations. Oscillators belonging to subpopulation 1 (2) interact among themselves via couplings $K_1{G}_1\left(K_2{G}_2\right)$, while oscillators from different subpopulations interact asymmetrically with effective couplings $K_1{G}_2$ and $K_2{G}_1$, as depicted in the figure.

Figure 2

Long-time profile on the complex unit circle of the states observed for a finite system [Eq. (1)] with two intertwined subpopulations coupled as depicted in Fig. 1: (a) Incoherent state, (b) zero-lag sync, (c) $\pi$-state, (d) TW1, (e) blurred zero-lag sync, and (f) blurred $\pi$-state. Only in the TW1 state (d), the collective frequencies $\dot{\mathrm{\Theta }}$ [Eq. (18)], $\dot{\mathrm{\Phi }}$ [Eq. (13)] and $\mathrm{\Omega }$ [Eq. (14)] are different from zero, and the oscillators travel across all possible phase values in the corotating frame defined by the natural frequency. The configuration of the TW2 state is obtained by interchanging the subindexes and colors in panel (d).

Figure 3

Bifurcation diagram of the reduced system [Eqs. (12)] for (a, b) $\mathrm{\Delta }K=8$ and $\mathrm{\Delta }G=2$; and (c, d) $K_0=3$ and $G_0=2$. Zero-lag sync corresponds to the perfectly synchronized state ($r_{1,2}=1$ and $\delta =0$). “$\pi$-state” refers to the state in which $r_{1,2}=1$ and phase-lag separation $\delta =\pi$. Similarly, blurred zero-lag and blurred $\pi$-states denote the states with $r_{1,2}<1$ along with $\delta =0$ and $\delta =\pi$, respectively. Traveling wave states are characterized by $r_1=1, r_2<1$ (TW1), and $r_2=1, r_1<1$ (TW2). Both TW1 and TW2 exhibit $\mathrm{\Omega }\ne 0$ [Eq. (14)]. Solid lines are obtained from Eqs. (20, 21, 22, 23, 24) and Eqs. (27)–(29). Dashed line in panels (a) and (c) depict the condition $K_0\mathrm{\Delta }G-\mathrm{\Delta }K{G}_0=0$ for which the couplings are symmetric, i.e., when the network connections are undirected. Panels (b) and (d) show zoomed-in regions of panels (a) and (c), respectively.

Figure 4

Order parameters $r_{1,2}, r$, phase lag $\delta$, and collective frequencies $\dot{\mathrm{\Theta }}$ [Eq. (18)] and $\mathrm{\Omega }$ [Eq. (14)] for (a, d, g) $G_0=0, \mathrm{\Delta }K=8$, and $\mathrm{\Delta }G=2$; (b, e, h) $G_0=2, \mathrm{\Delta }K=8$, and $\mathrm{\Delta }G=2$; and (c, f, i) $G_0=2, \mathrm{\Delta }K=3$ and $\mathrm{\Delta }G=10$. Dots are obtained by numerically integrating the original system [Eq. (1)] using Heun's method with $N={10}^4$ oscillators. For each coupling value $K_0$, the quantities are averaged over $t\in [500,1500]$ with a time step $dt=0.005$. In all panels, initial conditions ${\theta }_i(t=0)\forall i$ are randomly distributed according to a uniform distribution between $[-\pi ,\pi ]$. Solid lines correspond to the analytic solutions obtained in Sec. 4.

Figure 5

Comparison between simulations (colormaps) and theory (solid lines). (a) Total order parameter $r$; (b) local order parameter $r_1$ of subpopulation 1; (c) phase lag $\delta$ measuring the separation between the two subpopulations; and (d) average frequency $\mathrm{\Omega }$ [Eq. (14)]. For each pair of couplings $(G_0,K_0)$, Eq. (1) is evolved numerically with the Heun's method considering $N={10}^4$ oscillators and with an integration time step $dt=0.005$. The long-time behavior of each parameter is quantified by averaging the trajectories over $t\in [500,1000]$. For all $(G_0,K_0)$, the initial phases ${\theta }_i(t=0)$ are drawn uniformly at random over the interval $[-\pi ,\pi ]$. Coupling mismatch parameters: $\mathrm{\Delta }K=8$ and $\mathrm{\Delta }G=2$. $G_0\times K_0$ grid resolution: $100\times 100$ couplings

Figure 6

Comparison between simulations (colormaps) and theory (solid lines) for different sets of initial conditions: (a) ${\theta }_i(t=0)$ randomly chosen from the uniform distribution $[-\pi ,\pi ]$ (same as in Fig. 5); (b) for each coupling pair ($G_0,K_0$) the initial phases of subpopulations 1 and 2 were drawn from Gaussian distributions with standard deviation $\sigma =2$, and means ${\theta }_1$ and ${\theta }_2$, respectively, which were chosen uniformly at random between $[-\pi ,\pi ]$. The “$+$” marks a point in the diagram for which the behavior observed in the simulation departs from the dynamics predicted by the theory. Figure 7 shows the temporal evolution of the collective variables at the “$+$” point depicted in panel (b). Other parameters: $N={10}^4, \mathrm{\Delta }K=8$ and $\mathrm{\Delta }G=2$. In both panels the resolution of the grid is $100\times 100$ couplings. Integration was performed with Heun's method using a time step $dt=0.005$.

Figure 7

Temporal evolution of (a) local order parameters $r_{1,2}$, (b) phase lag $\delta$, and (c) locking frequency $\dot{\mathrm{\Theta }}$ [Eq. (18)]. Solid lines are obtained from simulations, while dashed lines correspond to the results yielded by the numerical integration of the reduced system [Eq. (12)]. In panel (a) the solid and dashed lines of $r_2$ overlap each other at $r_2=1$. Average in- and out-coupling strengths are taken from the “$+$” point in Fig. 6, that is, $(G_0,K_0)=(-0.5,-5.95)$. Other parameters: $N={10}^4$ oscillators, $\mathrm{\Delta }K=8, \mathrm{\Delta }G=2$, and $dt=0.005$.

Figure 8

Comparison between simulations (colormaps) and theory (solid lines). (a) Total order parameter $r$; (b) local order parameter $r_1$ of subpopulation 1; (c) phase lag $\delta$ measuring the separation between the two subpopulations; and (d) average frequency $\mathrm{\Omega }$ [Eq. (14)]. For each pair of couplings $(G_0,K_0)$, Eq. (1) are integrated numerically with the Heun's method considering $N={10}^4$ oscillators and with an integration time step $dt=0.005$. The long-time behavior of each parameter is quantified by averaging the trajectories over $t\in [500,1000]$. For all coupling pairs $(G_0,K_0)$, Eq. (1) is initiated with the exact same configuration for ${\theta }_i(t=0)$: phases in subpopulation 1 were randomly chosen from a Gaussian distribution with mean ${\theta }_1\simeq 0.17$ and standard deviation ${\sigma }_1\simeq 0.85$, yielding $r_1(t=0)\simeq 0.69$; phases of subpopulation 2 were also Gaussian distributed, but with mean ${\theta }_2\simeq 1.09$ and standard deviation ${\sigma }_2\simeq 1.053$, yielding $r_2(t=0)\simeq 0.57$. Coupling mismatch parameters: $\mathrm{\Delta }K=8$ and $\mathrm{\Delta }G=2$. $G_0\times K_0$ grid resolution: $100\times 100$ couplings.