On the efficiency of pairwise Hamiltonian control to desynchronize the higher-order Kuramoto model

Abstract

Synchronization of coupled oscillators is observed in many natural and engineered systems and emerges due to the interactions within the system. It can be both beneficial, e.g., in power grids, and harmful, e.g., in epileptic seizures. In the latter case, efficient control methods to desynchronize the systems are crucial. Recent studies have shown that interactions are not always pairwise, but higher-order, i.e., many-body, and this greatly affects the dynamics. For instance, higher-order interactions increase the linear stability of synchronized states but simultaneously shrink their attraction basin, with potentially opposite effects on control methods. Here, we use a minimally invasive pairwise control based on Hamiltonian control theory, and investigate its efficiency on phase oscillators with higher-order interactions. We show that, if the initial phases are close to the synchronized state, higher-order interactions make desynchronization more difficult to achieve. Otherwise, a non-monotonic effect appears: intermediate strengths of higher-order interactions impede desynchronization while larger ones facilitate it. In all cases, the control can desynchronize the system with a sufficient number of controlled nodes and intensity.

Figures (12)

• Figure 1: Desynchronizing phase oscillators on a hyperring. We show (a) the level of phase synchronization $\hat{R}$, and (b) of frequency synchronization $\hat{R}_{\dot\theta}$ as functions of triadic coupling strength $K_2$ and control strength $\mu$. For each value of ($\mu$,$K_2$), we show the value averaged over 50 frequency distributions $\bm{\omega} \sim \mathcal{N}(0,0.01)$. Insets zoom in on the low $\mu$ region. Panel (c) is obtained by merging the information from panels (a) and (b) and shows the three regimes of the controlled system---phase synchronization (blue, $\hat{R}\geq 0.4$), only frequency synchronization (yellow, $\hat{R}<0.4$ and $\hat{R}_{\dot\theta}\geq 0.5$), and no synchronization (red, otherwise)---and associated to the couples $(\mu,K_2)$. Panels (e)-(g) show example snapshots of these regimes as identified with the symbols: plus, cross and star. Parameters are set to $N=100$, $K_1=1$ and $r=2$.
• Figure 2: Minimum control strength $\mu_c$ required to desynchronize $\hat{R}$ < 0.4 increases with triadic coupling strength $K_2$ on hyperring. We show $\mu_c$ averaged (solid line) over 50 frequency distributions $\bm{\omega}$, and one standard deviation away (dashed lines). Parameters are set to $N=100$, $K_1=1$ and $r=2$, as in \ref{['fig:K2_v_mu']}.
• Figure 3: Desynchronizing phase oscillators on random hypergraphs. We show (a) the division in three zones: phases synchronization ($\hat{R}\geq 0.4$, blue), frequency synchronization only ($\hat{R}<0.4$ and $\hat{R}_{\dot\theta}\geq 0.5$, yellow), and no synchronization (otherwise, red). The inset displays details of the low $\mu$ region. Panels (b)-(d) show example snapshots of these regimes with $\mu = 0.001$, $\mu=0.01$ and $\mu=0.1$, respectively; (e) shows the boxplots of the associated effective frequency distributions, and that of the natural frequencies (left). Parameters are set to $N=100$, $K_1=1$, $\langle k^{(1)} \rangle=2r$, $\langle k^{(2)} \rangle=2r(r-1)$ and $r=2$.
• Figure 4: Minimum control strength $\mu_c$ required to desynchronize $\hat{R}$ < 0.4 increases with triadic coupling strength $K_2$ on random hypergraphs. We show $\mu_c$ averaged (solid line) over 50 frequency distributions $\bm{\omega}$, and one standard deviation away (dashed lines). Parameters are, as in \ref{['fig:K2_v_mu_r_2_random']}, set to $N=100$, $K_1=1$, $\langle k^{(1)} \rangle=2r$, $\langle k^{(2)} \rangle=2r(r-1)$ and $r=2$.
• Figure 5: Desynchronizing phase oscillators on a hyperring, with partial control. We show the level of phase synchronization $\hat{R}$ against triadic coupling strength $K_2$, by controlling (a)-(d) $M=0$, $5$, $10$, and $20$ nodes, respectively. For each value of $M$, we show three values of initial distance to basin center $\epsilon=0$, 3, and 5. Other parameters are set to $N=100$, $r=2$ and $K_1=1$. Order parameter $\hat{R}$ is averaged over 10 random realizations of frequencies $\bm{\omega}$ and 50 of initial conditions $\theta_0$.