Bipartite networks of oscillators with distributed delays: Synchronization branches and multistability

We study synchronization in bipartite networks of phase oscillators with general nonlinear coupling and distributed time delays. Phase-locked solutions are shown to arise, where the oscillators in each partition are perfectly synchronized among themselves but can have a phase difference with the other partition, with the phase difference necessarily being either zero or $\pi$ radians. Analytical conditions for the stability of both types of solutions are obtained and solution branches are explicitly calculated, revealing that the network can have several coexisting stable solutions. With increasing value of the mean delay, the system exhibits hysteresis, phase flips, final state sensitivity, and an extreme form of multistability where the numbers of stable in-phase and antiphase synchronous solutions with distinct frequencies grow without bound. The theory is applied to networks of Landau-Stuart and Rössler oscillators and shown to accurately predict both in-phase and antiphase synchronous behavior in appropriate parameter ranges.

Figure 1

(a) In bipartite networks, the nodes separate into two sets, here marked in red (light gray) and blue (dark gray), such that connections are only between differently colored nodes. (b) Simple example of a bipartite relay network with bidirectional coupling of strength $\varepsilon$ and delay $\tau$ in the interaction. (c) General coupling scheme in a bipartite topology: In partitions $A$ and $B$ there are $N_A$ and $N_B$ oscillators. The links may be directed. (d) Complete bipartite network with $N_A=N_B=3$ that has been used in the numerical simulations presented in this paper.

Figure 2

(a) Frequencies of stable phase-locked solutions of coupled Kuramoto oscillators (17), as given by Eq. (20) as a function of the mean delay $\tau$ (we set $\varepsilon =0.05$). (b) Corresponding phase difference between the partitions, which is either zero or $\pi$. The blue (dark gray) and red (light gray) curves correspond to in-phase ${\Omega }_0$ and antiphase ${\Omega }_{\pi }$ solutions, respectively. Their overlaps, shown by dashed boxes in (a) and (b), indicate regions of multistability. Simulation results from the system (17), shown by green triangles, for frequencies and phase differences between the two partitions are matched with analytical results in (a) and (b), respectively. (c) Variation of the frequencies with the coupling strength, calculated from Eq. (20), for $\varepsilon$ = 0.05 (dots), 0.25 (triangles), and 0.5 (crosses).

Figure 3

Dynamical phase diagram as a function of the parameters $\varepsilon$ and $\tau$. In the blue (dark gray) regions, only in-phase synchronized solutions are stable, whereas in the red (medium gray) regions only antiphase solutions are stable. In the gray (light gray) region there is multistability with several in-phase and antiphase solutions coexisting.

Figure 4

Variation of the number of stable in-phase and antiphase solutions as a function of (a) coupling strength $\varepsilon$ and (b) mean delay $\tau$.

Figure 5

Variation of the number of stable solutions as a function of the time delay, at a fixed coupling strength $\varepsilon =0.5$. In (a), the blue dots indicate in-phase and the red crosses antiphase solutions. The range of solutions remain bounded in the range from $1-\varepsilon$ to $1+\varepsilon$, while the number of solutions increases with the delay as shown by solid green line in (b). The number of solutions (shown as red triangles) are calculated by counting the intervals satisfying Eqs. (26) and (27) for delay values $\tau =n\pi /\omega$, $n=1,2,\cdots$ (details in Sec. 4).

Figure 6

Numerically computed frequencies (green triangles) of the Kuramoto oscillators plotted as a function of delay $\tau$ at coupling strength $\varepsilon =1.0$. We take a complete bipartite network with $N_A=N_B=3$. The analytically computed frequencies (in-phase and antiphase) are represented by the dots. A large number of stable frequencies have values very close to each other and depending on initial conditions the system settles into one of these in the range $[1-\varepsilon ,1+\varepsilon ]=[0,2]$.

Figure 7

Hysteresis curves in the multistability regime. (a) Collective frequency $\Omega$ and (b) phase difference $\varphi$ when $\tau$ is varied in steps of $\delta \tau$ from 0 to 4 and in steps of $-\delta \tau$ from 4 to 0 ($\delta \tau =0.01$, $\varepsilon =0.05$).

Figure 8

Collective synchronous frequency $\Omega$ (vertical axis) from Eqs. (24) and (25) plotted against the mean delay $\tau$ (horizontal axis). Stable solution branches are indicated by $B_n$, where even and odd values of $n$ indicate in-phase and antiphase branches, respectively. The dashed boxes are the region of multistability and the dotted boxes, indicated by the arrows, show the highest number of overlaps, three overlapping branches in this case.

Figure 9

Variation of frequencies with mean delay in coupled Landau-Stuart oscillators on a bipartite network ($N_A=N_B=3$). Blue and red lines (indicated by ${\Omega }_0$ and ${\Omega }_{\pi }$) are, respectively, the in-phase and antiphase frequencies calculated from the transcendental equations (20) at $\varepsilon =1.0$. Green triangles are the numerically calculated frequencies from the system (31).

Figure 10

Behavior of coupled Landau-Stuart oscillators on a bipartite network ($N_A=N_B=3$) for coupling strength $\varepsilon =0.05$. After removing transients, the variable $x\left(t\right)$ for oscillators $i=1\in A$ (solid red line) and $i=4\in B$ (dashed green line) is plotted as a function of time. The behavior of the system is identical to that of coupled Kuramoto oscillators at the same parameter values. (a) At $\tau =3.0$, the oscillators from different partitions are antiphase synchronized. (b) At $\tau =5.0$, the oscillators from different partitions are in-phase synchronized. (c) Oscillators show multistable behavior at $\tau =17.0$. Here two different initial conditions lead to (c1) in-phase and (c2) antiphase states.

Figure 11

Rössler oscillators on a bipartite network (again with $N_A=N_B=3$ and $a=0.2$, $f=0.2$, $c=1.0$, and $\omega =1.0$ taken for the numerics). The green triangles are the numerically calculated frequencies from the system (35) at coupling strength $\varepsilon =2.0$. The blue and red lines (indicated by ${\Omega }_0$ and ${\Omega }_{\pi }$) are, respectively, the in-phase and antiphase frequencies for the systems calculated from the transcendental equation (40) at equivalent coupling strength $K=\varepsilon /2=1$.

Figure 12

Identical Rössler oscillators in the chaotic regime $a=0.2,f=0.2,c=9.0,\omega =1.0$ for coupling strength $\varepsilon =2$. The attractor in the $x\text{−}y$ plane for one of the oscillators is plotted for delays (a) $\tau =3.0$ and (c) $\tau =13.0$. The variable $x\left(t\right)$ for oscillators $i=1\in A$ (solid red) and $i=4\in B$ (dashed green) are plotted in (b) and (d) as a function of time. (b) At $\tau =3.0$, the oscillators from different partitions are in antiphase synchrony and (d) at $\tau =13.0$, oscillators are in-phase synchronized. There are three oscillators in each partition, $N_A=N_B=3$.