Obtaining certificates for complete synchronisation of coupled oscillators
A linearly coupled system of identical oscillators has the form \[ \frac{dx}{dt} = F(x) + \kappa (C\otimes D)x, \] where \(x=[x_1,\dots,x_N]^T\), \(x_j\in\mathbb{R}^n\), \(F(x)=[f(x_1),\dots,f(x_N)]\) \(f: \mathbb{R}^n \to \mathbb{R}^n\), \(C\) and \(D\) are \(N\times N\) and \(n\times n\) coupling matrices, respectively. Previously, sufficient conditions have been obtained for the global synchronization of this system, i.e. for the stability of the invariant subspace \(S = \{x: x_1=\dots=x_N\}\). These sufficient conditions are based on finding some quadratic form, which plays the role of the Lyapunov function and guarantees the global stability of the subspace \(S\). In this paper, the authors suggest a numerical method for finding and optimizing such a quadratic form using the semidefinite programming technique. Moreover, the authors also establish new conditions for synchronization using Bendixson's criterion for higher-dimensional systems. The results are illustrated on Lorenz and van der Pol systems.
Bendixson's criterion for higher dimensions
scientific article; zbMATH DE number 5871267
scientific article