Existence and stability of traveling-wave states in a ring of nonlocally coupled phase oscillators with propagation delays

We investigate the existence and stability of traveling-wave solutions in a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. A comprehensive stability diagram in the parametric space of the system is presented that shows a rich structure of multistable regions and illuminates the relative influences of time delay, the nonlocality parameter and the intrinsic oscillator frequency on the dynamics of these states. A decrease in the intrinsic oscillator frequency leads to a break-up of the stability domains of the traveling waves into disconnected regions in the parametric space. These regions exhibit a tongue structure for high connectivity, whereas they submerge into the stable region of the synchronous state for low connectivity. One finding is the existence of forbidden regions in the parametric space where no phase-locked solutions are possible. We also discover a new class of nonstationary breather states for this model system that are characterized by periodic oscillations of the complex order parameter.

Figure 1

Synchronous frequency $\Omega$ for various wave mode numbers $k$ for fixed values of $\kappa$ and $\omega$. $k=0,1,2$, and 3 are shown in blue, red, green, and magenta, respectively, and are also marked with the corresponding $k$ values.

Figure 2

Plots of $\Omega -\omega$ vs. $\Omega \overline{\tau }$ for various phase-locked states ($k=0,1,2,3$) at three different values of $\kappa$. The shapes of all the curves are invariant to $\omega$, but the stable region is $\omega$ dependent. The solid black segments denote stable states for $\omega =1$. The thin segments shown in red denote the unstable regions. The insets highlight the width of stable regimes for different $k$ values.

Figure 3

Stability diagrams in the parameter space of $\Omega \overline{\tau }$ vs. $\kappa$ for mode numbers $k=0,1,2$, and 3, shown in blue, red, green, and magenta, respectively. The solid black curves in each panel represent the condition $H^{\prime }=0$. Solid color areas represent stability regions for $\omega =1$ and the lighter-shaded areas show the expansion of the stability region when $\omega =10$. Note that the stability region in panel $\left(a\right)$ for $k=0$ is independent of $\omega$.

Figure 4

(a) The superimposed stability diagrams of $k=1$ and 2 for $\omega =1$ shows a forbidden region where no phase-locked states are possible. The inset gives an enlarged view of the forbidden region. Similar forbidden regions are shown for $\omega =0.8$ in (b) and (c) and for $\omega =0.5$ in (d). The color coding is the same as described in the legend of Fig. 3.

Figure 5

Spatiotemporal characteristics of a nonstationary breather state. Panel (a) shows the snapshot of the spatial pattern of phase $\phi$ after the transients are over. The corresponding inset shows the spatial variations of phase velocity ($\dot{\phi }$) and the amplitude $R$ of the complex order parameter ($\mathbb{Z}$). Panels (b) and (c) exhibit temporal behavior of $R$ and $\dot{\phi }$, respectively. The parameter values are: $\omega =1$, $\kappa =2$, and ${\tau }_m=6.4$.

Figure 6

The same as in Fig. 5 but with parameter values: $\omega =1$, $\kappa =10$, and ${\tau }_m=12.8$.