Locking and regularization of chimeras by periodic forcing

We study how a chimera state in a one-dimensional medium of nonlocally coupled oscillators responds to a homogeneous in space periodic in time external force. On a macroscopic level, where a chimera can be considered as an oscillating object, forcing leads to entrainment of the chimera's basic frequency inside an Arnold tongue. On a mesoscopic level, where a chimera can be viewed as an inhomogeneous, stationary, or nonstationary pattern, strong forcing can lead to regularization of an unstationary chimera. On a microscopic level of the dynamics of individual oscillators, forcing outside of the Arnold tongue leads to a multiplateau state with nontrivial locking properties.

Figure 1

(a) Dependence of the length $L\left(\mathrm{\Theta }\right)$ of a chimera pattern on the phase shift parameter $\theta \left(0\right)=\mathrm{\Theta }$ for $\mathrm{\Omega }=-0.8$ and $\varepsilon =0.025$. The horizontal dashed straight line corresponds to $L=\overline{L}=4.41$. The vertical dashed lines depict values ${\mathrm{\Theta }}_1=-0.92$ and ${\mathrm{\Theta }}_2=2.35$. (b, c) Direct numerical simulations of the set of $N=4096$ oscillators, performed within the phase model (1). Spatiotemporal plots show the phases in the reference frame rotating with an angular velocity $\mathrm{\Omega }$ for (b) $\mathrm{\Theta }={\mathrm{\Theta }}_1$ (stable phase shift) and (c) $\mathrm{\Theta }={\mathrm{\Theta }}_2$ (unstable phase shift).

Figure 2

Dependencies $L_{max}\left(\mathrm{\Omega }\right)$ (filled red circles) and $L_{min}\left(\mathrm{\Omega }\right)$ (open blue circles), numerically calculated for $\varepsilon =0.025$. The upward-oriented triangular markers depict the values of $L_{min}$ and $L_{max}$ for $\mathrm{\Omega }=\overline{\mathrm{\Omega }}=-0.8$. The downward-oriented triangular markers show the values of ${\mathrm{\Omega }}_{min}$ and ${\mathrm{\Omega }}_{max}$ for a fixed length $L=\overline{L}=4.41$ of the medium. The range between ${\mathrm{\Omega }}_{max}$ and ${\mathrm{\Omega }}_{min}$ determines the width of the Arnold tongue for $\varepsilon =0.025$ in the case where $L=\overline{L}=4.41$.

Figure 3

Existence domains of the locked chimera patterns for (a) $L=4.41$, (b) $L=6.854$, and (c) $L=7.332$. The corresponding natural frequencies of autonomous stationary chimeras are (a) ${\mathrm{\Omega }}_0=-0.8$ (stable), (b) ${\mathrm{\Omega }}_0=-0.69$ (unstable, evolves to a breather), and (c) ${\mathrm{\Omega }}_0=-0.68$ (unstable, evolves to a turbulent state); cf., Fig. 4. Inside of each presented Arnold tongue, in domain $A$ (blue color) one of the two existing locked chimeras is stable. In regions $B$ and $C$ (red and yellow colors), all stationary chimeras are unstable, and the observed state is either turbulent in domain $B$ or time-periodic (breathing chimera) in region $C$.

Figure 4

Dynamics of the system (1) in the autonomous case $\varepsilon =0$. Panels (a)–(c) show a space-time plot of the phases in the reference frame rotating with angular velocity ${\mathrm{\Omega }}_0$. Panels (d)–(f) depict profiles $\left|z\right(x\left)\right|$ (solid line) and $\left|h\right(x\left)\right|$ (dotted line), found with the method of Sec. 3, and determining the initial phase distribution for a numerical simulation. Panels (g)–(i) demonstrate the spectrum $\lambda$ of linear perturbations of solutions (7) of Eqs. (4) and (5). Violet circles are the eigenvalues of the finite-size matrices, not corresponding to the discrete part of the spectrum. Because of a finite discretization, this set consists not only of purely real negative and purely imaginary numbers, although it has the characteristic $T$ shape. Red diamonds are eigenvalues belonging to the point spectrum ${\lambda }_p$ responsible for instability. (a, d, g) Stable chimera for ${\mathrm{\Omega }}_0=-0.8$, $L=4.41$. (b, e, h) Breathing chimera state for ${\mathrm{\Omega }}_0=-0.69$, $L=6.854$. (c, f, i) Turbulent regime for ${\mathrm{\Omega }}_0=-0.68$, $L=7.332$.

Figure 5

Locking of a stable chimera. Direct numerical simulations of the set of $N=4096$ oscillators performed within the phase model (1) with parameters $\varepsilon =0.025$ and different driving frequencies: (a, d) $\mathrm{\Omega }=-0.86$, (b, e) $\mathrm{\Omega }=-0.83$, and (c, f) $\mathrm{\Omega }=-0.725$. Panels (a)–(c) show spatiotemporal plots of the phases in the reference frame rotating with the driving frequency $\mathrm{\Omega }$. Panels (d)–(f) demonstrate average frequencies of the elements (blue lines) together with the forcing frequency $\mathrm{\Omega }$ (gray doted line); insets in panels (d) and (f) enlarge plateaus on frequency profiles. In each case, the initial state was close to the autonomous chimera state at the length $L=4.41$ and the corresponding natural frequency ${\mathrm{\Omega }}_0=-0.8$. For situations depicted in panels (a) and (c), the external force was presented during the full simulation time. In the case shown in panel (b), the force was switched on abruptly at the time instant $t_0=500$ (black dashed straight line in panel (b)). A red dashed curve in panel (e) shows a profile of average frequencies of the oscillators for a free-running chimera state.

Figure 6

Phase (in the reference frame rotating with the force frequency $\mathrm{\Omega }$) vs time for the oscillator belonging to the subgroup entrained by the external force. The diagram shows the dynamical behavior of the element located at $x=0.64$ in the numerical simulation corresponding to Figs. 5 and 5.

Figure 7

Synchronization of a breathing chimera. Panels (a) and (b) show a space-time plot of the phases in the reference frame rotating with the driving frequency $\mathrm{\Omega }$ for $L=6.854$. Panels (c) and (d) depict profiles $\left|z\right(x\left)\right|$ (solid line) and $\left|h\right(x\left)\right|$ (dotted line), found with the method of Sec. 3, and determining the initial phase distributions for numerical simulations. Panels (e) and (f) demonstrate the spectrum $\lambda$ of linear perturbations for the corresponding solutions. Violet circles are the eigenvalues of discrete matrices not corresponding to the discrete part of the spectrum. Red diamonds are eigenvalues belonging to the point spectrum ${\lambda }_p$ responsible for instability. (a, c, e) Stable chimera for $\mathrm{\Omega }=-0.74$, $\varepsilon =0.05$. (b, d, f) Turbulent regime for $\mathrm{\Omega }=-0.63$, $\varepsilon =0.05$.

Figure 8

Synchronization of breathing chimera for $L=6.854$. (a) Spatial distribution of the phases in the reference frame rotating with the driving frequency $\mathrm{\Omega }$. (b) Absolute value $\left|Z(x,t)\right|$ of the complex order parameter. Numerical simulations of the set of $N=8192$ oscillators were performed within the framework of the phase model (1) with $\varepsilon =0.06$, $\mathrm{\Omega }=-0.69$. The initial conditions were chosen in the form of an unforced breathing chimera state. The force was switched on abruptly at the time instant $t_0=1000$, as marked by a black dashed straight line. The coarse-grained order parameter $Z(x,t)$ was calculated via local averaging with a Gaussian kernel $exp\left(-x^2/2{\varsigma }^2.\right)$, with $\varsigma =0.1$.

Figure 9

(a, b) Dependencies of the frequency difference ${\mathrm{\Omega }}_{\mathrm{obs}}-\mathrm{\Omega }$ (filled circles) and the standard deviation of the global real mean field $\overline{\mathcal{R}}$ fluctuations (triangles) for the case of a forced breathing chimera of Fig. 3, in a lattice of $N=4096$ units. (a) $\varepsilon =0.03$, (b) $\varepsilon =0.05$. Dashed horizontal line shows zero level; dashed vertical lines are borders of existence of the locked solution [bold black lines in Fig. 3]. One can see that transition to turbulence (at the right side of the panels) is rather abrupt and immediately leads to a deviation of the observed frequency from the forced one. (c) Time series of the global real mean field $\overline{\mathcal{R}}\left(t\right)$ for a stationary chimera (dotted blue line), for a periodic chimera (solid brown line) and for a turbulent chimera (dashed red line).

Figure 10

Synchronization of turbulent chimera for $L=7.332$. The dynamics of amplitude of the local complex order parameter $\left|Z(x,t)\right|$ calculated as in Fig. 8 in direct numerical simulations of the set of $N=8192$ oscillators within the phase model (1) with the parameters (a) $\varepsilon =0.03$, $\mathrm{\Omega }=-0.68$, and (b) $\varepsilon =0.06$, $\mathrm{\Omega }=-0.68$. The initial conditions were chosen in the form of a stationary chimera solution, which develops due to the instability into a turbulent state. The force was switched on abruptly at the time instant $t_0=1500$, which is marked by a black dashed straight line. One can see that a large enough forcing can stabilize and regularize the behavior of the system, which evolves to a standard chimera regime (see panel b).