Time-delayed systems are known to exhibit symmetric square waves oscillating with a period close to twice the delay. Here, we show that strongly asymmetric square waves of a period close to one delay are possible. The plateau lengths can be tuned by changing a control parameter. The problem is investigated experimentally and numerically using a simple bandpass optoelectronic delay oscillator modeled by nonlinear delay integrodifferential equations. An asymptotic approximation of the square-wave periodic solution valid in the large delay limit allows an analytical description of its main properties (extrema and square pulse durations). A detailed numerical study of the bifurcation diagram indicates that the asymmetric square waves emerge from a Hopf bifurcation. In the past three years, intensified efforts have been made to explore the physical effects of time delays [1][2][3][4][5]. In biology, time delays are called maturation periods, control reflexes, and cellular transport times, and they appear in various areas including neural networks, where propagation delays of the electrical signals connecting different neurons are taken into account [3]. In the industry, machine tool chatter is a persistent problem in metal cutting applications, where a delay term is typically involved in the applied force [6]. Optical feedback from a distant reflector is another example where a time delay is causing erratic intensity oscillations in lasers used in optical communication systems [7]. Time-delayed feedbacks are also used for practical applications. They might be deliberately implemented to control neural disturbances, e.g., to suppress undesired synchrony of firing neurons in Parkinson's disease [8]. They are also designed for producing new effects with optoelectronic oscillators (OEOs). An OEO typically incorporates a nonlinear (intensity) modulator, an optical fiber delay line, and an optical detector, followed by a bandpass filter, in a closed-loop resonating configuration. This hybrid microwave source is capable of generating, within the same optoelectronic cavity, either an ultralow-jitter (low-phase-noise) single tone microwave oscillation (narrow band filtering), as used in radar applications [9], or a wide band chaotic carrier (broadband filter) typically intended for physical encryption in high bit rate optical communications [10].Because of the large variety of applications, there is an increasing interest for problems modeled by delay differential equations (DDEs) and their specific dynamical phenomena. A fundamental property of nonlinear dynamical systems controlled by a delayed feedback is their tendency to exhibit square-wave oscillations if the delay τ D is sufficiently large. These oscillations typically consist of 2τ D -periodic transitions between two or more flat plateaus. The following scalar DDE,has been studied intensively for its square-wave solutions (the prime stands for the derivative with respect to the time s). Equation (1) The two fixed points of the period-2 solution of Eq. (2) . Close to ...

We theoretically investigate a weakly birefringent all-fiber cavity subject to linearly polarized optical injection. We describe the propagation of light inside the cavity using, for each linear polarization component of the electric field, the Lugiato-Lefever model. These two components are coupled by cross-phase modulation. We show that, for a wide range of parameters, there is a coexistence between a homogeneous steady state and two different types of temporal vector cavity solitons, which can be hosted in the same system. They differ by their polarization state and peak intensity. We construct their bifurcation diagram and show that they are connected through a saddle-node bifurcation. Finally, we show that vector cavity solitons exhibit multistability involving different polarization states with different energies.

International audienceSquare-wave oscillations exhibiting different plateau lengths have been observed experimentally by investigating an electro-optic oscillator. In a previous study, we analysed the model delay differential equations and determined an asymptotic approximation of the two plateaus. In this paper, we concentrate on the fast transition layers between plateaus and show how they contribute to the total period. We also investigate the bifurcation diagram of all possible stable solutions. We show that the square waves emerge from the first Hopf bifurcation of the basic steady state and that they may coexist with stable low-frequency periodic oscillations for the same value of the control parameter

The rate equations for a laser with a polarization rotated optical feedback are investigated both numerically and analytically. The frequency detuning between the polarization modes is now taken into account and we review all earlier studies in order to motivate the range of values of the fixed parameters. We find that two basic Hopf bifurcations leading to either stable sustained relaxation or square-wave oscillations appear in the detuning versus feedback rate diagram. We also identify two key parameters describing the differences between the total gains of the two polarization modes and discuss their effects on the periodic square-waves.

The rate equations for a laser diode subject to a filtered phase-conjugate optical feedback are studied both analytically and numerically. We determine the Hopf bifurcation conditions, which we explore by using asymptotic methods. Numerical simulations of the laser rate equations indicate that different pulsating intensity regimes observed for a wide filter progressively disappear as the filter width increases. We explain this phenomenon by studying the coalescence of Hopf bifurcation points as the filter width increases. Specifically, we observe a restabilization of the steady-state solution for moderate width of the filter. Above a critical width, an isolated bubble of time-periodic intensity solutions bounded by two successive Hopf bifurcation points appears in the bifurcation diagram. In the limit of a narrow filter, we then demonstrate that only two Hopf bifurcations from a stable steady state are possible. These two Hopf bifurcations are the Hopf bifurcations of a laser subject to an injected signal and for zero detuning.

We experimentally report the sequence of bifurcations destabilizing and restabilizing a laser diode with phase-conjugate feedback when the feedback rate is increased. Specifically, we successively observe the initial steady state, undamped relaxation oscillations, quasi-periodicity, chaos, and oscillating solutions at harmonics up to 13 times the external cavity frequency but also the restabilization to a steady state. The experimental results are qualitatively well reproduced by a model that accounts for the time the light takes to penetrate the phase-conjugate mirror. The theory points out that the system restabilizes through a Hopf bifurcation whose frequency is a harmonic of the external cavity frequency.

We analyze the FitzHugh-Nagumo equations subject to time-delayed self-feedback in the activator variable. Parameters are chosen such that the steady state is stable independent of the feedback gain and delay τ . We demonstrate that stable large-amplitude τ -periodic oscillations can, however, coexist with a stable steady state even for small delays, which is mathematically counterintuitive. In order to explore how these solutions appear in the bifurcation diagram, we propose three different strategies. We first analyze the emergence of periodic solutions from Hopf bifurcation points for τ small and show that a subcritical Hopf bifurcation allows the coexistence of stable τ -periodic and stable steady-state solutions. Second, we construct a τ -periodic solution by using singular perturbation techniques appropriate for slow-fast systems. The theory assumes τ = O(1) and its validity as τ → 0 is investigated numerically by integrating the original equations. Third, we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable. The theory is applied to the FitzHugh-Nagumo equations with threshold nonlinearity, and we show that the branch of τ -periodic solutions emerges from a limit point of limit cycles.

Light polarization is an inherent property of the coherent laser output that finds applications, for example, in vision, imaging, spectroscopy, cosmology, and communications. We report here on light polarization dynamics that repeatedly switches between a stationary state of polarization and an irregularly pulsating polarization. The reported dynamics is found to result from the onset of chimeras. Chimeras in nonlinear science refer to the counterintuitive coexistence of coherent and incoherent dynamics in an initially homogeneous network of coupled nonlinear oscillators. The existence of chimera states has been evidenced only recently in carefully designed experiments using either mechanical, optomechanical, electrical, or optical oscillators. Interestingly, the chimeras reported here originate from the inherent coherent properties of a commercial laser diode. The spatial and temporal properties of the chimeras found in light polarization are controlled by the laser diode and feedback parameters, leading, e.g., to multistability between chimeras with multiple heads and to turbulent chimeras.