Abstract
In this paper, we investigate the onset of synchronous behavior on strongly connected directed networks of heterogeneous piecewise linear (PWL) oscillators. Each oscillator is composed of a linear part, which is assumed to be identical for all oscillators, and a commutation function, which is different for each oscillator, creating in this way the heterogeneity in the whole network. Due to this heterogeneity, the oscillators do not achieve perfect or complete synchronization but instead, practical synchronization emerges. The stability properties of the practical synchronous solution in the network are analyzed using standard Lyapunov theory for systems with nonvanishing perturbations and the concept of disagreement vectors. Our results provide an ultimate bound on the norm of the disagreement vector and also a maximal bound on the synchronization error, such that a relationship between the disagreement bound and practical synchronization is established. Additionally, we provide numerical simulations to illustrate the theoretical results.
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