Synchronized Hopf Bifurcation Analysis in a Delay-Coupled Semiconductor Lasers System

The dynamics of a system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Hopf bifurcation properties. Some numerical simulations are carried out to illustrate the analysis results

Measuring the universal synchronization properties of coupled oscillators across the Hopf instability

When a driven oscillator loses phase-locking to a master oscillator via a Hopf bifurcation, it enters a bounded-phase regime in which its average frequency is still equal to the master frequency, but its phase displays temporal oscillations. Here we characterize these two synchronization regimes in a laser experiment, by measuring the spectrum of the phase fluctuations across the bifurcation. We find experimentally, and confirm numerically, that the low frequency phase noise of the driven oscillator is strongly suppressed in both regimes in the same way. Thus the long-term phase stability of the master oscillator is transferred to the driven one, even in the absence of phase-locking. The numerical study of a generic, minimal model suggests that such behavior is universal for any periodically driven oscillator near a Hopf bifurcation point.Comment: 5 pages, 5 figure

Excitable-like chaotic pulses in the bounded-phase regime of an opto-radiofrequency oscillator

We report theoretical and experimental evidence of chaotic pulses with excitable-like properties in an opto-radiofrequency oscillator based on a self-injected dual-frequency laser. The chaotic attractor involved in the dynamics produces pulses that, albeit chaotic, are quite regular: They all have similar amplitudes, and are almost periodic in time. Thanks to these features, the system displays properties that are similar to those of excitable systems. In particular, the pulses exhibit a threshold-like response, of well-defined amplitude, to perturbations, and it appears possible to define a refractory time. At variance with excitability in injected lasers, here the excitable-like pulses are not accompanied by phase slips.Comment: 2nd versio

High Speed Chaos in Optical Feedback System with Flexible Timescales

We describe a new opto-electronic device with time-delayed feedback that uses a Mach-Zehnder interferometer as passive nonlinearity and a semiconductor laser as a current-to-optical-frequency converter. Bandlimited feedback allows tuning of the characteristic time scales of both the periodic and high dimensional chaotic oscillations that can be generated with the device. Our implementation of the device produces oscillations in the frequency range of tens to hundreds of MHz. We develop a model and use it to explore the experimentally observed Andronov-Hopf bifurcation of the steady state and to estimate the dimension of the chaotic attractor.Comment: 7 pages, 6 figures, to be published in IEEE J. Quantum Electro

Dynamics of two mutually coupled semi conductor lasers: Instantaneous coupling limit

We consider two semiconductor lasers coupled face to face under the assumption that the delay time of the injection is small. The model under consideration consists of two coupled rate equations, which approximate the coupled Lang-Kobayashi system as the delay becomes small. We perform a detailed study of the synchronized and antisynchronized solutions for the case of identical systems and compare results from two models: with the delay and with instantaneous coupling. The bifurcation analysis of systems with detuning reveals that self-pulsations appear via bifurcations of stationary (i.e. continuous wave) solutions. We discover the connection between stationary solutions in systems with detuning and synchronous (also antisynchronous) solutions of coupled identical systems. We also identify a codimension two bifurcation point as an organizing center for the emergence of chaotic behavior

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart–Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding selffeedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behaviorWe would like to thank T. Erneux, E. Schöll, S. Yanchuk, and P. Perlikowski for helpful discussions. O.D. acknowledges the Research Foundation Flanders FWO-Vlaanderen for a fellowship and for project support. This work was partially supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under Grant No. IAP VI-10 “photonics@be,” by MICINN Spain under project DeCoDicA Grant No. TEC2009- 14101 ,, and by the project PHOCUS EU FET Open Grant No. 240763 .Peer reviewe

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior