Equivalence of additive and parametric pinning control protocols for systems of weakly coupled oscillators

Abstract

Controlling the behavior of nonlinear systems on networks is a paramount task in control theory, in particular the control of synchronization, given its vast applicability. In this work, we focus on pinning control and we examine two different approaches: the first, more common in engineering applications, where the control is implemented through an external input (additive pinning); the other, where the parameters of the pinned nodes are varied (parametric pinning). By means of the phase reduction technique, we show that the two pinning approaches are equivalent for weakly coupled systems exhibiting periodic oscillatory behaviors. Through numerical simulations, we validate the claim for a system of coupled Stuart--Landau oscillators. Our results pave the way for further applications of pinning control in real-world systems.

Figures (2)

• Figure 1: In this setting, the phase reduction approach is valid and so is the equivalence of the two pinning protocols. The upper panels show the case of additive pinning, while the lower panels show the parametric pinning. Panels a) and d) show a snapshot of the $x_i$ variables for $50$ time units (t.u.); panels b) and e) show a snapshot of the $y_i$ variables for $50$ time units (t.u.); panels c) and f) show the time series of the $y_i$ variables. The network is a $1$-dimensional $4$-regular lattice of $60$ nodes; the parameters of the SL oscillators are $\alpha=3$ and $\omega=1$; the coupling is $\varepsilon=0.01$ and the order of the pinning perturbation is $\lambda=0.1$. The system is integrated for $50$ t.u. with the Runge-Kutta IV method with integration step $dt=0.01$; the pinning is applied to $n_p=20$ nodes for $t_p=10$ t.u.
• Figure 2: In this setting, the phase reduction approach is not anymore valid and, hence, the two pinning protocols are not equivalent. The upper panels show the case of additive pinning, while the lower panels show the parametric pinning. Panels a) and d) show a snapshot of the $x_i$ variables for $50$ time units (t.u.); panels b) and e) show a snapshot of the $y_i$ variables for $50$ time units (t.u.); panels c) and f) show the time series of the $y_i$ variables. The network is a $1$-dimensional $4$-regular lattice of $60$ nodes; the parameters of the SL oscillators are $\alpha=3$ and $\omega=1$; the coupling is $\varepsilon=0.2$ and the order of the pinning perturbation is $\lambda=0.6$. The system is integrated for $50$ t.u. with the Runge-Kutta IV method with integration step $dt=0.01$; the pinning is applied to $n_p=20$ nodes for $t_p=10$ t.u.