Controlling and enhancing synchronization through adaptive phase lags

We compare two methods for controlling synchronization in the Kuramoto model on an undirected network. The first is by driving selected oscillators at a desired frequency by coupling to an external driver, and the second is by including adaptive lags—or dynamical frustrations—within the Kuramoto interactions, with the lags evolving according to a dynamics as a function of the reference frequency with an associated time constant. Performing numerical simulations with random regular graphs, we find that above a certain connectivity driving via adaptive lags allows for stronger alignment to the external frequency at a lower value of the time constant compared to the corresponding coupling strength for the externally driven model. Numerical results are supported by equilibrium analysis based on a fixed-point ansatz for frequency synchronized clusters where we solve the spectrum of the associated Jacobian matrix. We find that at low connectivity the external driving mechanism is successful down to lower densities of controlled oscillators where the adaptive lag approach is Lyapunov unstable at all densities. As connectivity increases, however, the adaptive lag mechanism shows stability over similar ranges of density to the external driving and proves superior in terms of tighter splays of oscillators. In particular, the threshold for instability for the adaptive lag model shows robustness against variations in the associated time constant down to lower densities of controlled oscillators. A simple intuitive model emerges based on the interaction between splayed clusters close to a critical point.

Figure 1

Phase diagrams for random regular graph of $N=40$, showing the fraction of 100 instances achieving full synchronization to the frequency $\mathrm{\Omega }$ across the parameter space of $\rho$ vs $\tau$ for the externally driven reference model (a, c, e) and $\rho$ vs $\eta$ for the adaptive lag model (b, d, f), with different connectivities $k=6,8,12$, respectively, top, middle, and bottom rows.

Figure 2

Example of clustering for $N=40,\rho =0.8$ where gray are the controlled ${\theta }_i-\mathrm{\Omega }t$, cyan are the uncontrolled ${\theta }_i-\mathrm{\Omega }t$, and black are the ${\alpha }_i$ derived from Eq. (11).

Figure 3

Plots of $\alpha =\frac{1}{N}{\sum }_i{\alpha }_i$ (dashed) and $\mu =\frac{1}{N}{\sum }_i{\mu }_i$ (solid) from Eqs. (10) and (11) in the adaptive lag model for $N=40$ random regular $k_i=8$ with $\sigma =\tau =1,\mathrm{\Omega }=2.5$, averaged over 300 instances, for different values of $\rho$. Note that these are independent of $\tau$. Inset: Plot of $\frac{1}{N}{\sum }_{i}cos({\mu }_i+{\alpha }_i)$ averaged over the same instances. Error bars here and in subsequent plots represent the standard error.

Figure 4

Plots of (a) $\mu =\frac{1}{N}{\sum }_i{\mu }_i$, and (b) $\alpha =\frac{1}{N}{\sum }_i{\alpha }_i$ (bottom) in the externally driven model from Eqs. (20) and (19) for $N=40$ random regular $k_i=8$ with $\sigma =\tau =1,\mathrm{\Omega }=2.5$, averaged over 300 instances, as a function of $\rho$ for different values of $\eta$.

Figure 5

(a) Plot of the average value of the real part of the lowest (in real part) eigenvalue of the super-Laplacian ${\mathrm{\Lambda }}^{\mathrm{AdLg}}$ for $N=40$ random regular $k_i=8$ with $\sigma =1,\mathrm{\Omega }=2.5$, averaged over 300 instances, as a function of the density of controlled nodes $\rho$ and for different $\tau$. Inset: The real part on logarithmic scale. (b) Plot of the imaginary part of the eigenvalue.

Figure 6

Plots of the average value of the (real part of the) lowest eigenvalue of the super-Laplacian for $N=40$ random regular graphs with $\sigma =1,\mathrm{\Omega }=2.5$ as a function of the density of controlled nodes $\rho$, (a, c, e, g, i) for the reference model ${\mathrm{\Lambda }}^{\mathrm{Ref}}$ for different $\eta$ and (b, d, f, h, j) for the adaptive lag model ${\mathrm{\Lambda }}^{\mathrm{AdLg}}$ for different $\tau$, varying the connectivity as $k=4,5,8,12,14$ vertically down.

Figure 7

(a) Plot of the average value of the real part of the lowest (in real part) eigenvalue of the super-Laplacian ${\mathrm{\Lambda }}^{\mathrm{AdLg}}$ for the adaptive lags for $N=40$ random regular $k_i=8$ with $\mathrm{\Omega }=2.5,\rho =0.4$, averaged over 300 instances, as a function of the coupling strength $\sigma$ and for different $\tau$. (b) The average value of the real part of the lowest (in real part) eigenvalue of the external driven model, again for $N=40$ random regular $k_i=8$ with $\mathrm{\Omega }=2.5,\rho =0.4$, averaged over 300 instances, as a function of the coupling strength $\sigma$ but for different $\eta$. Error bars in both cases are the size of the data points.