On controlling networks of limit-cycle oscillators

Abstract

The control of network-coupled nonlinear dynamical systems is an active area of research in the nonlinear science community. Coupled oscillator networks represent a particularly important family of nonlinear systems, with applications ranging from the power grid to cardiac excitation. Here we study the control of network-coupled limit cycle oscillators, extending previous work that focused on phase oscillators. Based on stabilizing a target fixed point, our method aims to attain complete frequency synchronization, i.e., consensus, by applying control to as few oscillators as possible. We develop two types of control. The first type directs oscillators towards to larger amplitudes, while the second does not. We present numerical examples of both control types and comment on the potential failures of the method. Collective rhythms in ensembles of interacting units generate novel phenomena in mathematics, physics, engineering, and biology 1,2 . Moreover, robust collective rhythms characterized by synchronization is vital to the functionality of systems ranging from power grids 3 and Josephson junction arrays 4 to cardiac tissue 5 and circadian rhythms 6 . This has motivated a need for control and optimization methods for coupled oscillator networks -specifically towards attaining consensus among the individual oscillators 7,8 . In a recent publication we developed a simple control mechanism for attaining consensus in networks of coupled phase-oscillators based on identifying and stabilizing a target synchronized state 9 . Here we extend this this method to the case of networks of nonlinear limit-cycle oscillators, where the state of each oscillator is characterized not only by a phase angle, but also an amplitude 10 . While the presence of an amplitude for each oscillator yields a richer set of dynamical states overall, we find that consensus can still be attained in this more complicated scenario.

Models of communication and control for brain networks: distinctions, convergence, and future outlook

“…Subsequent work focuses on expanding the form of the control input (Skardal & Arenas, 2016), and modifications to the coupling strength to a single node (Fan, Wang, Yang, & Wang, 2019). Hence, we observe that targeted modification to the dynamics of subsets of oscillators can indeed set their lead-lag relationships.…”

Models of communication and control for brain networks: distinctions, convergence, and future outlook

“…Subsequent work focuses on expanding the form of the control input [127], and modifications to the coupling strength to a single node [128]. Hence, we observe that targeted modification to the dynamics of subsets of oscillators can indeed set their lead-lag relationships.…”

Introduction to focus issue: Patterns of network synchronization

“… Nagao et al [73] experimentally study the use of delay-coupled networks of electrochemical reactions to show that changes in coupling can induce changes in behavior from oscillation death to anti-phase synchronization.  Skardal and Arenas [74] analyze two methods of control for Stuart-Landau coupled oscillators, with the goal of directing the system to a desired state with a minimal intervention.  Deng et al [75] study the optimization of coupling strengths to suppress amplitude death in a chain of coupled Stuart-Landau oscillators.…”