Consensus in coupled van der Pol systems

In this paper, we focus on the consensus problem of coupled van der Pol oscillators under linear and nonlinear protocols respectively. For the linear protocol case, a matrix spectral radius condition guaranteeing consensus is derived by means of linearization method and Floquet theory. For the case of nonlinear protocols based on Morse potential, the effects of attractive and repulsive features of the protocols on consensus are discussed. Numerical simulations reveal interesting phenomena in nonlinear protocols case. One is group consensus, that is, the coupled oscillators split into groups and achieve consensus within each group, and the other is related to the loss of inherent relaxation oscillations in certain individual oscillators.