Oscillatory networks: insights from piecewise-linear modeling (Q6636579)

The study of coupled oscillator networks is a large area of interest, with much focus being on the existence and stability of various synchronized or partially synchronized states. Established tools include phase reductions, phase--amplitude reductions, and the master stability function. However, most studies have considered smooth dynamical systems. Here the authors consider networks of piecewise-linear systems, generalizing the techniques mentioned above. Piecewise-linear systems have the advantage of being explicitly solvable, and periodic solutions can be found by piecing together solutions in different parts of phase space. However, the study of the stability of periodic solutions in networks of piecewise-linear systems is complicated by the need to properly describe the behavior of perturbations of a solution as boundaries in phase space are crossed. The authors give a detailed presentation of how to use ``saltation operators'' to describe this process.\N\NA variety of models are used to demonstrate the techniques discussed in this review, ranging from neuron models to impact oscillators and herds of cows.