Bistable chimera attractors on a triangular network of oscillator populations

We study a triangular network of three populations of coupled phase oscillators with identical frequencies. The populations interact nonlocally, in the sense that all oscillators are coupled to one another, but more weakly to those in neighboring populations than to those in their own population. This triangular network is the simplest discretization of a continuous ring of oscillators. Yet it displays an unexpectedly different behavior: in contrast to the lone stable chimera observed in continuous rings of oscillators, we find that this system exhibits two coexisting stable chimeras. Both chimeras are, as usual, born through a saddle-node bifurcation. As the coupling becomes increasingly local in nature they lose stability through a Hopf bifurcation, giving rise to breathing chimeras, which in turn get destroyed through a homoclinic bifurcation. Remarkably, one of the chimeras reemerges by a reversal of this scenario as we further increase the locality of the coupling, until it is annihilated through another saddle-node bifurcation.

Figure 1

Networks of three populations of oscillators. The gray disks symbolize the populations of oscillators, populated by individual oscillators symbolized by black dots. Their bidirectional coupling is represented by black arrows. (a) Chainlike general case with structural tuning parameter $c$. (b) Triangular network structure, corresponding to $c=1$. Each population has a self-coupling of unit strength 1, and is coupled to the neighboring populations with strength $1-A$.

Figure 2

(Color online) Phase portraits for the $SDS$ chimera, based on the polar coordinates $\rho$ and $\psi$ given by Eqs. (13) ($x$ and $y$ denote real and imaginary components, respectively). $A$ is increased while keeping $\beta$ constant. The unit circle is displayed in gray (coinciding with trajectories for all four cases). Stable fixed points are shown as solid (red) dots, unstable fixed points as empty (green) circles, and saddles as half-filled (green) circles, respectively. The limit cycle in panel c) is emphasized in blue color. Stable and unstable manifolds of the saddle are shown as red (solid) and green (dashed) trajectories, respectively. Additional trajectories in black (dash-dotted) are shown for visualization of the flow field. The point in $\left(\rho ,\psi \right)=\left(1,0\right)$ is a nodal sink. The position of the nodal source depends on $\beta$ and moves in clockwise direction with growing values of $\beta$.

Figure 3

(Color online) Bifurcation diagram for the $SDS$ (a) and $DSD$ (b) chimera. The displayed curves are: the saddle-node curve (solid red), the Hopf curve (dashed blue), and the homoclinic curve (dotted black). Dots mark the bifurcation points obtained by inspection of the phase plane. The homoclinic curves (dotted) are interpolations based on these points, whereas all the solid curves were obtained analytically.

Figure 4

(Color online) Phase portraits for the $DSD$ chimeras, based on the polar coordinates $\rho$ and $\psi$ given by Eqs. (14). $A$ is increased while keeping $\beta$ constant. The unit circle is displayed in gray (the unstable green dashed manifold of the saddle coincides with it). Stable fixed points are shown as solid (red) dots, unstable fixed points as empty (green) circles, and saddles as half-filled (green) circles, respectively. Limit cycles are emphasized in blue color. Stable and unstable manifolds are shown as red (solid) and green (dashed) trajectories, respectively. The point in $\left(\rho ,\psi \right)=\left(1,0\right)$ is a nodal sink. The position of the saddle depends on $\beta$ and moves in clockwise direction with increasing $\beta$.

Figure 5

Bifurcation diagram obtained from numerical simulation, for states $SDS$, shown in (a), and $DSD$ in (b). Light dots and dark circles correspond to local maxima and minima, respectively, that are detected by the algorithm. The dashed curve represents the analytical result for the continuum case $\left(N\to \infty \right)$. The computations were performed with $N_{\sigma }=40$ oscillators per population for a simulation time of $T=100$. The question mark indicates that we could not conclusively confirm the existence of the second DSD state. (The kink in the lower left dashed branch is an artifact from the limit cycle reaching into the left hand side quadrants, as $\rho$ is not measured relative to the limit cycle center but to the origin.)