Abstract
Linear stability of synchronized states in networks of delay-coupled oscillators depends on the type of interaction, the network, and oscillator properties. For inert oscillator response, found ubiquitously from biology to engineering, states with time-dependent frequencies can arise. These generate side bands in the frequency spectrum or lead to chaotic dynamics. The time delay introduces multistability of synchronized states and an exponential term in the characteristic equation. Stability analysis using the resulting transcendental characteristic equation is a difficult task and is usually carried out numerically. We derive criteria and conditions that enable fast and robust analytical linear stability analysis based on the system parameters. These apply to arbitrary network topologies, identical oscillators, and delays.
Highlights
Self-organized synchronization can be observed in chemical oscillators, embryonic development, circadian clocks, ranging to power grids and the orchestration of mobile communications and microelectronic and mechanical systems [1,2,3,4,5,6,7,8,9,10]
We discuss how linear stability depends on physical properties such as time delay, inertia, damping or dissipation, interaction strength, and network topology
For Kuramoto models with time delay and inertia it has been shown that this criterion cannot sufficiently predict linear stability [34]
Summary
Self-organized synchronization can be observed in chemical oscillators, embryonic development, circadian clocks, ranging to power grids and the orchestration of mobile communications and microelectronic and mechanical systems [1,2,3,4,5,6,7,8,9,10] This type of synchronization has been considered for electronic networks since the 1980s due to its robustness and as its properties scale advantageously with growing system size [11,12]. We discuss how linear stability depends on physical properties such as time delay, inertia, damping or dissipation, interaction strength, and network topology These generic concepts can be related to application specific concepts like, e.g., the loop gain and bandwidth of electronic oscillators or the dissipation coefficients in power grids [29,30,31,32].
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