Using Hamiltonian control to desynchronize Kuramoto oscillators

Many coordination phenomena are based on a synchronization process, whose global behavior emerges from the interactions among the individual parts. Often in nature, such self-organized mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are, however, cases where synchronization acts against the stability of the system; for instance in some neurodegenerative diseases or epilepsy or the famous case of Millennium Bridge where the crowd synchronization of the pedestrians seriously endangered the stability of the structure. In this paper we propose an innovative control method to tackle the synchronization process based on the application of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronization. We present our results on a generalized class of the paradigmatic Kuramoto model.

Figure 1

Time evolution of action-angle variables for the Hamiltonian KM and of the original KM composed of three coupled oscillators. The angles drift once the actions remain close to the Kuramoto torus [(d)], as one can appreciate from (a) where we report the phase difference between the first and the second oscillator for both the Hamiltonian KM and the original one, the curves are extremely close and thus one cannot differentiate between the two models. On the contrary, partial phase-locked regime–horizontal plateau—is present in (b) (Hamiltonian model in blue and the original KM in red) corresponding to instability of the actions near the Kuramoto torus (e); let us observe that now the angle variables behave differently in the Hamiltonian model and in the original KM. Instead, if the KM is controlled the angles drift again [(c)] and the action turn stable [(f)]. Once again the dynamics of the two systems are extremely close and thus one cannot differentiate between the two models. Initial conditions have been randomly chosen as a small perturbation of order ${10}^{-4}$ near the Kuramoto torus $I_i=1/2$, angles are uniformly random distributed in $[0,2\pi ]$, the natural frequencies are set $({\omega }_1,{\omega }_2,{\omega }_3)=(0.8,1,1.2)$ and the coupling constant is set to $K=0.1$ in (a), (d) and to $K=0.3$ in (b), (c), (e), (f).

Figure 2

The order parameter $r$ as a function of the coupling coefficient $K$ for 20 coupled oscillators averaging over a time interval of 250 unities. In the controlled model (red triangles) the coupled oscillators are prevented from synchronization even for ranges of $K$ where the original model (blue squares) do synchronize. The remaining parameters are set as in Fig. 1.