Arnold tongues in oscillator systems with nonuniform spatial driving

Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially coupled oscillators: the complex Ginzburg-Landau equation (CGLE) and a generalization of the CGLE in which oscillators are coupled through an external medium (emCGLE). We focus on external control drives that vary in both space and time. We find that the spatial distribution of the drive signal controls the frequency ranges over which oscillators synchronize to the drive and that boundary conditions strongly influence synchronization to external drives for the CGLE. Our calculations also show that the emCGLE has a low density regime in which a broad range of frequencies can be synchronized for low drive amplitudes. We study the bifurcation structure of these models and find that they are very similar to results for the driven Kuramoto model, a system with no spatial structure. We conclude by discussing qualitative implications of our results for controlling coupled oscillator systems such as the social amoebae Dictyostelium and populations of Belousov Zhabotinsky (BZ) catalytic particles using spatially structured external drives.

Figure 1

Changing the wave number of the drive shifts the Arnold tongue. (a) Plots of $\langle \left|\mathrm{\Omega }\right|\rangle =\langle |{\partial }_t\mathrm{\Phi }|\rangle$ (the spatially averaged time derivative of the phase) as a function of the drive parameters $\nu$, the detuning frequency (the difference between the drive frequency and the natural frequency), and $B$, the drive amplitude. The drive wave number $k_0$ increases from left to right, taking the values 0, 0.2, and 0.4, respectively. Red contours show the approximate Arnold tongue described by Eq. (9), which estimates the region in which the medium is synchronized to the drive. The dotted orange contours show the boundary of the region in which the ODEs describing a uniform system, Eqs. (6) and (7), are stably synchronized to the drive. The dashed green contour shows the region within which the ODE system has exactly three real solutions for $R_0^2$. (b) $\mathrm{\Phi }$ and $\mathrm{\Omega }$ as a function of space and/or time at the drive parameters indicated in (a), $\nu =1.06, B=0.2$. (c) Same as (b) but for the drive parameters marked by (c), $\nu =0.7, B=0.2$. Distances are measured in units of the drive wavelength: ${\lambda }_0=2\pi /k_0$. Parameters: $\alpha =0.5$ and $\beta =4$.

Figure 2

Zero-flux boundary conditions can cause desynchronization in directly coupled CGL systems even where analytical calculations predict synchronization. (a) Analytically calculated boundary of Arnold tongue for $k_0=0.4$ (red) using Eq. (9) and numerical simulations of $\mathrm{\Omega }=\langle {\partial }_t\mathrm{\Phi }\rangle$ as a function of the detuning frequency $\nu$ and driving amplitude $B$. (b) The phase $\mathrm{\Phi }$ for the point labeled “(b)” in (a), $\nu =1.06, B=0.2$. (c) The phase $\mathrm{\Phi }$ for the point labeled “(c)” in (a), $\nu =1.06, B=0.35$. (d) The amplitude $R$ as a function of spatial position $x$ at $t=100$ for the parameter set shown in (b). Parameters: $\alpha =0.5, \beta =4$, and $k_0=0.4$.

Figure 3

Zero-flux boundary conditions can lead to more complex spatial mode-locking patterns in regions: 4:3 spatial locking. (a) The phase $\mathrm{\Phi }$ as a function of space and time for zero-flux boundary conditions. (b) The winding number, defined in terms of the unwrapped phase ${\mathrm{\Phi }}_u$, as ${\mathrm{\Phi }}_u/2\pi$, as a function of $x$ to $t=900$, with the dashed line showing 1:1 spatial locking and the dotted line 4:3 spatial locking. (c) The amplitude $R$ as a function of spatial position $x$ at $t=900$. Parameters: $\alpha =0.5, \beta =4, k_0=0.4, B=0.25$, and $\nu =0.66$.

Figure 4

Zero-flux boundary conditions can lead to more complex spatial mode-locking patterns in regions: 2:1 spatial locking. (a) The phase $\mathrm{\Phi }$ as a function of space and time for zero-flux boundary conditions. (b) The winding number, defined in terms of the unwrapped phase ${\mathrm{\Phi }}_u$, as ${\mathrm{\Phi }}_u/2\pi$, as a function of $x$ to $t=2900$, with the dashed line showing 1:1 spatial locking and the dotted line 2:1 spatial locking. (c) The amplitude $R$ as a function of spatial position $x$ at $t=2900$. Parameters:$\alpha =0.5, \beta =4, k_0=0.4, B=0.25$, and $\nu =0.68$.

Figure 5

CGL model coupled through an external medium exhibits a low-density amplitude death regime in which very wide ranges of frequencies can be synchronized with small driving amplitudes. Top: $\langle \left|\mathrm{\Omega }\right|\rangle$; middle: $R$; and bottom: $r$ as a function of $\nu$ and $B$, with analytic boundaries of the approximate Arnold tongue, Eq. (22) (red solid contour), the existence of a stable, synchronized fixed point (orange dotted contour), and the area within which there are exactly three synchronized fixed points (green dashed contour). This indicates the same approximate bifurcation structure as in the driven Kuramoto model and the CGLE, with SNIPER, saddle-node, and Hopf bifurcations. As the density declines, the coupling to the medium happens at a slower and slower rate compared to the rates associated with medium turnover ($J$) or the drive frequency (${\omega }_0-\nu$). As this happens, the medium has effectively zero amplitude (bottom right) and no longer couples the individual oscillators. Then, either they oscillate along with the drive or they also decay to a low amplitude. In this limit, the width of the Arnold tongue increases and becomes proportional to $B^2/(D-1)$. Parameters: $\alpha =0.5, \beta =1, k_0=0.1, D=1.1, J=0.5, {\omega }_0=0.5$, and $\rho =2.0,1.5,1.0,0.5,0.15,0.1$ from left to right. Note that there are no orange lines in the rightmost plots since the prefactor in Eq. (22) is zero.