Properties of stationary states of delay equations with large delay and applications to laser dynamics

We consider properties of periodic solutions of the differential-delay system, which models a laser with optical feedback. In particular, we describe a set of multipliers for these solutions in the limit of large delay. As a preliminary result, we obtain conditions for stability of an equilibrium of a generic differential-delay system with fixed large delay $\tau$. We also show a connection between characteristic roots of the equilibrium and multipliers of the mapping obtained via the formal limit $\tau\to\infty$

Discretization of frequencies in delay coupled chaotic oscillators

We study the dynamics of two mutually coupled oscillators with a time delayed coupling. Due to the delay, the allowed frequencies of the oscillators are shown to be discretized. The phenomenon is observed in the case when the delay is much larger than the characteristic period of the solitary uncoupled oscillator

Delay-induced patterns in a two-dimensional lattice of coupled oscillators

We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. A "hybrid dispersion relation" is introduced, which allows studying the stability of time-periodic patterns analytically in the limit of large delay. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators

Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence

Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms