Optimal cost tuning of frustration: Achieving desired states in the Kuramoto-Sakaguchi model

There are numerous examples of studied real-world systems that can be described as dynamical systems characterized by individual phases and coupled in a networklike structure. Within the framework of oscillatory models, much attention has been devoted to the Kuramoto model, which considers a collection of oscillators interacting through a sinus function of the phase differences. In this paper, we draw on an extension of the Kuramoto model, called the Kuramoto-Sakaguchi model, which adds a phase lag parameter to each node. We construct a general formalism that allows us to compute the set of lag parameters that may lead to any phase configuration within a linear approximation. In particular, we devote special attention to the cases of full synchronization and symmetric configurations. We show that the set of natural frequencies, phase lag parameters, and phases at the steady state is coupled by an equation and a continuous spectra of solutions is feasible. In order to quantify the system's strain to achieve that particular configuration, we define a cost function and compute the optimal set of parameters that minimizes it. Despite considering a linear approximation of the model, we show that the obtained tuned parameters for the case of full synchronization enhance frequency synchronization in the nonlinear model as well.

Figure 1

Network of seven nodes.

Figure 2

Implied cost to achieve the phase configuration in Eq. (17) as a function of the chosen control node, $C$, for the network in Fig. 1 and considering five different values of ${\alpha }_C$. Notice that the minimum cost is given, in this case, by ${\alpha }_C=0$ and $C=1$ or $C=5$.

Figure 3

Implied cost to achieve the symmetric configuration as a function of the degree corresponding to different choices of the control node, $s_C$, for a network of 8 nodes (upper panels) and 9 nodes (lower panels). The distinct colors and markers correspond to different values of ${\alpha }_C$. The symmetric configuration is generated by a value of ${\alpha }_h=0.1$.

Figure 4

Implied cost to achieve the fully synchronized configuration as a function of the chosen control node, $C$ (upper panel) and natural frequencies of nodes (bottom panel) for the network in Fig. 3. Five different values of ${\alpha }_C$ are considered (marked colored lines). Natural frequencies are set as the example in Eq. (43).

Figure 5

Average order parameter, $〈r〉$, as a function of the coupling strength, $K$, for the linear [in panel (a)] and the nonlinear [in panel (b)] Kuramoto-Sakaguchi (KS) model on regular random graphs with homogeneous node degree $s_i=4$ and $N=100$ nodes. Natural frequencies are obtained from a uniform random distribution in the range ${\omega }_i\in [-1,1]$. Each data point represents an average over ten optimized configurations. We compare three types of tuning for the set of frustration parameters, $\left\{{\alpha }_i\right\}$: the original Kuramoto dynamics or ${\alpha }_i=0\forall i$ (spotted continuous red line); type II [21] KS dynamics with the frustration parameters set to $sin\left({\alpha }_i\right)=-{\omega }_i/\left(K{s}_i\right)$ if $|{\omega }_i|<K{s}_i$ and $sin\left({\alpha }_i\right)=\pm 1$ otherwise (squared dashed green line); and KS dynamics with the frustration parameters determined by the derived linear approximation in Eq. (39) (squared discontinuous purple line).

Figure 6

For the network in Fig 1, phases obtained after tuning the set of frustration parameters to the symmetric configuration (four distinct symmetries).

Figure 7

Three examples of the general function $f\left(x\right)=|\frac{a-x}{a}|$, with $a=2, a=3$ and $a=4$, and the resulting sum of them.

Figure 8

Derivative of the function $f\left(x\right)=|\frac{2-x}{2}|+|\frac{3-x}{3}|+|\frac{4-x}{4}|$ defined in Fig. 7. Red dashed line at $y=0$.