Network Synchronization of Piecewise-Linear Oscillators with Mean-Field Commutation Function

In this work, we investigate the onset of synchronization in a network of Piecewise-Linear (PWL) oscillators. Instead of using a commutation function for each system, as usually assumed in the literature, here, we consider a common mean-field commutation function, which governs the oscillatory behavior of the whole network, where the oscillators are synchronized via diffusive coupling. Some of the advantages of the proposed approach are: a more tractable mathematical analysis of the stability of the synchronous solution and a reduction in the computational burden. We illustrate this by three examples: a network of planar systems driven by hysteresis, a network of multiscroll oscillators, and a network of PWL Rössler systems. In all cases, we provide analytic conditions that guarantee the asymptotic stability of the synchronous solution for different input–output combinations and also, we illustrate our findings by numerical simulations. Additionally, the effect of initial conditions on the boundedness of solutions in the network is numerically investigated. An interesting characteristic of the proposed mean-field commutation function approach is that the nodes in the network act as oscillators only if the whole network is synchronized; otherwise, the nodes behave as unstable systems. Thus, the proposed scheme rewards the cooperative motion but penalizes the divergent individual behavior.