Time delay in the Kuramoto model with bimodal frequency distribution

We investigate the effects of a time-delayed all-to-all coupling scheme in a large population of oscillators with natural frequencies following a bimodal distribution. The regions of parameter space corresponding to synchronized and incoherent solutions are obtained both numerically and analytically for particular frequency distributions. In particular, we find that bimodality introduces a new time scale that results in a quasiperiodic disposition of the regions of incoherence.

Figure 1

(Color online) Frequency $\Omega$, phase difference $\beta$, and leading Lyapunov exponents $\lambda$ (thick red line), $\mu$ (thin blue line), for the stable synchronized solutions (3) as a function of $K$ for $\mathrm{\Delta }\omega =\pi ∕15$, $\overline{\omega }=\pi ∕2$ (dotted line, upper panels).

Figure 2

(Color online) Stability boundaries for Synchronization and Incoherence. Theoretical boundaries for (full) synchronization are obtained from Eqs. (6, 7). Disintegration $(\lambda \to 0^{-})$ and unlocking $(\mu \to 0^{-})$ are indicated with red-dashed and blue-solid lines, respectively. Black-thin curves: Boundaries of stable incoherence [Eq. (12)]. Shaded areas: Incoherence regions obtained by numerical integration of Eqs. (1), using a fifth-order Adams-Bashforth-Moulton scheme with $dt=\tau ∕20$, except for $\tau <1$, where $dt=0.05$ ($N=24$, $\overline{\omega }=\pi ∕2$). Dark-shaded regions: bistability between incoherence and synchronization.

Figure 3

(Color online) $\mathrm{\Delta }\omega \text{−}K$ diagram showing the regions of stable incoherence for a bi-Lorentzian frequency distribution centered at $\overline{\omega }=\pi ∕2$ and $\Gamma =0.1$. Solid-black $(\Gamma =0.1)$ and brown-dashed $(\Gamma =0)$ curves: analytical boundaries obtained from Eqs. (15, 12), respectively. Shaded areas: Regions of incoherence obtained by numerical integration of Eqs. (1), $\gamma =0.1,D=0,N=2000$. (a) Green curves $(\tau =0.1)$: symmetry breaking that results in two Hopf bifurcations (the solid green curve becomes the new boundary). To the right of the red-dashed line at $\mathrm{\Delta }\omega =2\gamma ∕\sqrt{3}$ [only shown in panel (a)], the bi-Lorentzian distribution is bimodal.

Figure 4

(Color online) $\tau \text{−}K$ diagram with the stable regions of the incoherent state for the bi-Lorentzian frequency distribution for $\Gamma =0$ [dashed brown, from Eq. (15)] and [from Eq. (12)] for $\Gamma =0.05$ (solid green) and $\Gamma =0.1$ (solid black). Shaded area: Incoherence regions obtained by numerical integration of Eq. (1) $(\gamma =0.1,D=0,N=2000)$.