Chimera States on a Flat Torus

Chimera states are surprising spatiotemporal patterns in which regions of coherence and incoherence coexist. Initially observed numerically, these mathematical oddities were recently reproduced in a laboratory setting, sparking a flurry of interest in their properties. Here we use asymptotic methods to derive the conditions under which two-dimensional “spot” and “stripe” chimeras (similar to those observed in experiments) can exist in a periodic space. We also discover a previously unobserved asymmetric chimera state, whose existence plays a major role in determining when other chimera states are observable in experiment and simulation. Finally, we use numerical methods to verify theoretical predictions and determine which states are dynamically stable.

Figure 1

Stripe and symmetric spot chimeras. Each panel contains a 200 by 200 grid of oscillators. Each pixel corresponds to a single oscillator with color representing the phase. Panels (a) and (b) display a stripe chimera state with $a_1=0.2322$, $a_2=0$, and $\delta =0.1065$, while (c) and (d) display a symmetric chimera state with a coherent spot and with $a_1=0.1366$, $a_2=0.1366$, and $\delta =0.1494$. Numerical integration indicates that these states are stable.

Figure 2

Solutions to system (9). The inset displays the full set of solutions. The main panel shows the quadrant containing only $a_1$, $a_2\ge 0$. Loops correspond to symmetric spot chimeras ($a_2=a_1$) and two types of equivalent stripe chimeras ($a_1=0$ and $a_2=0$). Branches ($a_1\ne a_2$, both nonzero) correspond to two types of asymmetric spot chimeras: one with $\delta >0$ and one with $\delta <0$.

Figure 3

Cross sections of Fig. 2. The magenta (dash dotted) curves represent unstable chimeras, the blue (solid) curves represent stable chimeras, the green (dashed) curves correspond to modulated drift states, and the red (solid, $a_1=0$) curves correspond to uniform drift states. Panel (a) displays the stripe chimera loop ($a_2=0$). Panel (b) displays the symmetric spot loop ($a_1=a_2$). Note that curves and labeled points were determined analytically, while stability was evaluated numerically. Stability boundaries were observed to correspond to labeled points.

Figure 4

Asymmetric spot chimeras. Panels (a) and (b) display an asymmetric chimera from the lower branch with $a_1=0.07635$, $a_2=0.02929$, and $\delta =-0.02608$, while (c) and (d) display an asymmetric chimera state from the upper branch with $a_1=0.2089$, $a_2=0.04844$, and $\delta =0.1769$. Numerical integration reveals that these states are unstable.

Figure 5

Dynamics of the fraction drifting and asymmetry. Solid lines indicate stable states, while dashed lines indicate unstable states. Red curve, symmetric spot chimeras; blue curve, stripe chimeras; green curve, asymmetric spot chimeras ($\delta >0$); purple curve, asymmetric spot chimeras ($\delta <0$). Unstable states evolve along planes of fixed ${\beta }_1$ to nearby stable states (with a different fraction drifting and/or asymmetry; see the text for definition) as indicated by the arrows in panel (a). Panel (b) contains a two-dimensional projection of panel (a) showing the fraction drifting as a function of ${\beta }_1$ for the various chimeras. Note that fraction drifting was inferred analytically from Eq. (10), while stability and dynamical behavior were determined numerically.