Homogeneous delays in the Kuramoto model with time-variable parameters

The Kuramoto model with time-varying parameters has been extended to consider the effect of delay in couplings. A collective dynamics arises from the interplay between the time scales of the original system, the external forcing, and the delays. This complex low-dimensional dynamics is described, uncovering an echo effect near the synchronization threshold. Hence, the delayed couplings substantially alter the dynamics of what is an open system and should be taken into consideration, depending on the ensemble's evolution time scale. We also introduce a first-harmonic approximation for the evolution of the mean field under harmonic forcing, valid for any delays and forcing of a coherent population.

Figure 1

Model A with cosine forcing. (a) The evolution of the order parameter, Eq. (3a) (solid gray), the low-dimensional dynamics, Eq. (8) (dashed black), and the approximation (13) (dot-dashed brown) for $\tau =0.5T,\Omega =0.1$. (b) A comparison between the magnitude of the mean-field oscillations $\Delta$ obtained using the approximative Eq. (15) (solid gray), the low-dimensional dynamics [Eq. (3a), discontinuous curves], and the high-frequency limit (16) (solid black). The dashed green, dotted black, and dot-dashed brown lines correspond to the time delays $\tau /T\in${0.05,0.5,0.75}, respectively $\left(\varepsilon =0.35\right)$. (c) The mean-field magnitude $\Delta$ as a function of the ratio of $\tau /T$ and the forcing frequency $\Omega$.

Figure 2

The echo effect in model A. Numerical simulations of the real dynamics, Eq. (3a) (light gray lines), confirm the low-dimensional evolution (8) for cosine forcing with $\Omega =0.1$ and $\varkappa =0.95$. Results correspond to echolike evolution $(\tau =T/2$, solid brown lines) and regular mode $(\tau =0.2T$, dashed black lines). (a) Low-dimensional behavior and simulations for these time delays. (b) Frequency spectra of the signals. (c) Mean-field limit cycles. The limit cycle for the half-period delay has a self-intersection point. (d) Limit cycle $\dot{r}(r\left(t\right),r\left(t-\tau \right))$. Incorporation of the retarded mean field effectively extends the system's dimension, thereby removing the self-crossing of the limit cycles.

Figure 3

The mean-field dynamics in model A. (a) Orbits of the complex order parameter $z\left(t\right)$, corresponding to a time delay $\tau =0.3T$ and mean frequencies $\widehat{\omega }=20$ (black) and $\widehat{\omega }=0.628$ (gray). (b) Average values $\overline{r}$ of the order parameter. Brown circles show the averages calculated from Eq. (3a) at a final simulation time of $t=500T$. Green cross signs are for Gaussian $g\left(\omega \right)$ as described in the text. Gaps correspond to stable incoherent states in agreement with the condition (12) (black dots). Parameters: $\widehat{\omega }=20,\varkappa =2,\varepsilon =0.2,\Omega =6.28$.

Figure 4

Model B with cosine forcing. The mean-field dynamics of the full system Eq. (3b) (light gray) is compared with the low-dimensional evolution (9) (dashed black) and the approximation (13) (dot-dashed brown). Parameters: $\Omega =0.3,\tau =0.65T,\varkappa =2.5,\varepsilon =0.25,\widehat{\omega }=0$.

Figure 5

The echo effect in model B with harmonic forcing near the incoherence threshold. (a) The full system (3b) (light gray curves) and the low-dimensional mean-field dynamics (9) for time lags $0.22T$ (dashed black) and $T/2$ (solid brown). (b) Limit-cycle self-intersection appears in the 2D phase space for these delays (designations are identical). Parameters: $\varkappa =1.7,\varepsilon =0.9,\Omega =0.08,\widehat{\omega }=0$.

Figure 6

The influence of chaotic forcing $(x$ component of the Lorentz attractor). (a) The adiabatic approximation (20a) (dashed black) in comparison with the mean-field responses (8) in the case of small $(\tau =3$ s, solid green) and large $(\tau =50$ s, dot-dashed brown) time delays for model A, (3a) (light solid gray). Parameters: $\gamma =1{\mathrm{s}}^{-1},K=4.2{\mathrm{s}}^{-1},\varepsilon =0.6,\widehat{\omega }=0$. (b) Similar comparison for model B: Equation (3b) (light solid gray) between the approximation (20b) (dashed black) and the TD mean field Eq. (9) for short $(\tau =2$ s, solid green) and for large $(\tau =70$ s, dot-dashed brown) time lags $\left(\gamma =1{\mathrm{s}}^{-1},K=5{\mathrm{s}}^{-1},\varepsilon =0.25,\widehat{\omega }=0\right)$.