The Lyapunov–Krasovskii theorem and a sufficient criterion for local stability of isochronal synchronization in networks of delay-coupled oscillators

Highlights: • We provide analytical conditions for stability of isochronal synchronization. • The criterion requires collectible parameters of the network and node dynamics. • The results simplify, relax, expand and generalize previous results. • The theoretical results help enhance knowledge for technological applications. This paper presents a self-contained framework for the stability assessment of isochronal synchronization in networks of chaotic and limit-cycle oscillators. The results were based on the Lyapunov–Krasovskii theorem and they establish a sufficient condition for local synchronization stability of as a function of the system and network parameters. With this in mind, a network of mutually delay-coupled oscillators subject to direct self-coupling is considered and then the resulting error equations are block-diagonalized for the purpose of studying their stability. These error equations are evaluated by means of analytical stability results derived from the Lyapunov–Krasovskii theorem. The proposed approach is shown to be a feasible option for the investigation of local stability of isochronal synchronization for a variety of oscillators coupled through linear functions of the state variables under a given undirected graph structure. This ultimately permits the systematic identification of stability regions within the high-dimensionality of the network parameter space. Examples of applications of the results to a number of networks of delay-coupled chaotic and limit-cycle oscillators are provided, such as Lorenz, Rössler, Cubic Chua's circuit, Van der Pol oscillator and the Hindmarsh–Rose neuron.