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

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.

💡 Research Summary

This paper investigates two widely used pinning control strategies for synchronizing networks of weakly coupled oscillators: additive pinning, where an external input is injected into a selected subset of nodes, and parametric pinning, where the intrinsic parameters (specifically the natural frequency) of the pinned nodes are temporarily altered. The authors focus on systems that exhibit periodic limit‑cycle behavior and are weakly coupled, allowing the application of phase reduction techniques.

Model and Phase Reduction
The authors consider a network of identical Stuart‑Landau (SL) oscillators, each described by the planar equations
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