Synchronous states in time-delay coupled periodic oscillators: A stability criterion

There are several works showing that nonzero time delay between nodes in an oscillator network can be responsible for several kinds of behavior as synchronization and chaos. Here, by using the Lyapunov linearizing method, in a system of two coupled oscillators derived as a particular case of the full connected network, it is shown that the time delay parameter has two sets of values: one that destabilizes the whole system and other that implies stability. Besides, there is a set of time delay values responsible for chaotic behaviors, even in a simple coupled oscillators system.

► Coupled oscillators are modeled as two node cells from full-connected networks.
► First-order coupled cells present stable synchronous operation for any propagation delay.
► Second-order coupled cells present families of delays related to different behaviors.
► Bifurcation diagrams represent regions of stability and instability of the synchronous state.