Practical dynamic consensus of Stuart–Landau oscillators over heterogeneous networks

ABSTRACT

In Panteley and Loria (2017), a framework for the study of synchronisation and collective behaviour of networked heterogeneous systems was introduced. It was underlined that in such scenario an emergent collective behaviour arises, one that is inherent to the network and that is independent of the interconnection strength. Therefore, the natural way to make complete study of synchronisation is by investigating, on one hand, the stability of the emergent dynamical system and, on the other, by assessing the difference between the motion of each individual system and that of the emergent one. Thus, if all systems' motions approach that of the emergent dynamics, we say that they reach dynamic consensus. In this paper, we study dynamic consensus of a fairly general class of nonlinear heterogeneous oscillators, called Stuart–Landau. We establish that the emergent dynamics consists in that of an ‘averaged’ oscillator with a global attractor that consists in a limit-cycle and, moreover, we determine its frequency of oscillation. Then, we show that the heterogeneous oscillators achieve practical dynamic consensus, that is, their synchronisation errors measured relative to the collective motion are ultimately bounded.

KEYWORDS:

• Networked systems
• consensus
• nonlinear oscillators
• robust stability
• synchronisation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 That is, the poles of the linearised system have all positive real parts.

2 Solutions starting arbitrarily close to it, are repelled away.

3 We stress the difference between complex orthogonal matrices, which satisfy $T^{-1}=T^{\mathrm{\top }}$, and unitary matrices satisfying $T^{-1}=T^{\ast }$.

Additional information

Funding

This work was partially supported by the Department of STITS of University of Paris Saclay and by the Government of Russian Federation [grant number 08-08].