Chimera states in networks of phase oscillators: The case of two small populations

Chimera states are dynamical patterns in networks of coupled oscillators in which regions of synchronous and asynchronous oscillation coexist. Although these states are typically observed in large ensembles of oscillators and analyzed in the continuum limit, chimeras may also occur in systems with finite (and small) numbers of oscillators. Focusing on networks of $2N$ phase oscillators that are organized in two groups, we find that chimera states, corresponding to attracting periodic orbits, appear with as few as two oscillators per group and demonstrate that for $N>2$ the bifurcations that create them are analogous to those observed in the continuum limit. These findings suggest that chimeras, which bear striking similarities to dynamical patterns in nature, are observable and robust in small networks that are relevant to a variety of real-world systems.

Figure 1

Schematic diagram showing a network of $2N$ oscillators in two groups of $N$ coupled with strength $\mu$ within group and $\nu$ between groups.

Figure 2

Bifurcations of chimeras in the continuum limit $N=\infty$ of (2) and (3). (A similar figure appears in Ref. [17].) The regions between the red (dashed) and blue (solid) curves and between the green (dash-dotted) and red (dashed) curves correspond to the respective regions of existence for stationary and breathing chimeras.

Figure 3

Trajectories of the order parameter in the asynchronous group for various values of $N$. Parameters: $A=0.1,\beta =0.025$.

Figure 4

Trajectories of the order parameter in the asynchronous group for various values of $N$. Parameters: $A=0.4,\beta =0.025$.

Figure 5

Poincaré section $\mathrm{\Psi }=\pi$ for Eqs. (11, 12, 13). Top: The value of $\rho$ (positive) and $\mathrm{\Delta }$ (negative). Bottom: Period of periodic orbits. (Further continuation of the higher-period branch was not possible for numerical reasons.) Solid: Stable; dashed: unstable. Parameters: $\beta =0.1,N=4$.

Figure 6

Values of $\rho$ and $\mathrm{\Delta }$ on the Poincaré section $\mathrm{\Psi }=\pi$ for $A=0.28$ to $A=0.35$ (inner to outer) in steps of $0.01$. Parameters: $\beta =0.1,N=4$.

Figure 7

Saddle-node and Hopf bifurcation curves for $N=4$. (See Fig. 5.)

Figure 8

Poincaré section ${\psi }_3=\pi$. Top: The value of ${\psi }_1,{\psi }_2$. Bottom: Period of periodic orbits. (Further continuation of the higher-period branch was not possible for numerical reasons.) Solid: Stable; dashed: unstable. Parameters: $N=3,\beta =0.1$.

Figure 9

Times between successive crossings of the Poincaré section ${\psi }_3=\pi$. Vertical axis: Times between successive crossings for a family of attractors corresponding to chimeras. Parameters: $N=3,\beta =0.1$.

Figure 10

Saddle-node and Hopf bifurcation curves for $N=3$ (see Fig. 8).

Figure 11

Schematic of equilibria for Eq. (16). As $A$ increases, equilibria $p_1, p_2, p_1^{\prime }$, and $p_2^{\prime }$ move along the curve (blue dashed) in the direction of the displayed arrows. Ultimately, both pairs of equilibria collide and cease to exist in a saddle-node bifurcation.

Figure 12

Trajectories for Eq. (16) with $A=0.1$ and varying $\beta$. Top: For $\beta =0$, all trajectories (blue) remain on level curves of the invariant $L$ (19). The gray-shaded regions indicate neutrally stable chimera periodic orbits while the diagonal red line is a line of neutrally stable fixed points. Bottom: For $\beta =0.15$ fixed points are indicated by black circles. Trajectories within the gray band are stable chimeras.

Figure 13

Phase plots for Eq. (16). Each panel contains a phase plot for $\beta =0.15$ and for various values of $A$ (a) $A=0.03$, (b) $A=0.0662$, (c) $A=0.1$, (d) $A=0.194$, (e) $A=0.25$, and (f) $A=0.71$. Black arrows indicate the direction of flow in the plane. Fixed points are marked with black circles. Curves indicate selected stable (blue dashed) and unstable (magenta solid) manifolds of the saddle points.

Figure 14

Foliation of the phase plane. Left and right panels depict the phase plane for $A=0.25$ and $\beta =0.15$. The white region exhibits no oscillation. The colored regions contain chimera states with finite lifetimes. The number of cycles (increasing from the outermost regions to the innermost region) before a fixed point is reached is indicated by the color.

Figure 15

Poincaré section ${\psi }_1=\pi$. Top: The value of ${\psi }_2$. Bottom: Period of periodic orbits. (Continuation to higher periods was not possible for numerical reasons.) Solid: Stable; dashed: unstable. Parameters: $\beta =0.1$.

Figure 16

Bifurcations for the $N=2$ case. Black (dotted): Saddle-node of fixed points; blue (dash-dotted): pitchfork of chimeras; red (dashed): heteroclinic connection between $p_1$ and $p_1^{\prime }$ (with one variable increased by $2\pi$); green (solid): homoclinic connection between $p_1$ and $p_1$ (with both variables increased by $2\pi$). Stable chimeras exist in the shaded regions bounded by the green and red curves, and the line $\beta =0$. The blue circle ($A_0\approx 0.145898$) and blue square ($A_1\approx 0.285714$) with $\beta =0$ indicate analytical predictions for the bifurcation points.