Multistability, relaxation oscillations, and chaos in time-delayed optoelectronic oscillators with direct laser modulation.

We investigate the nonlinear dynamics of an optoelectronic oscillator that is implemented with a laser diode (LD) with time-delayed feedback. In this system, electrical-to-optical conversion is directly implemented using the direct modulation of the laser diode itself, instead of an electrooptical modulator as in conventional architectures. Moreover, we consider the cubic nonlinear saturation of the characteristic laser power-intensity (P-I) transfer function far above threshold, instead of its simplified piecewise linear counterpart. We perform the stability analysis of the oscillator, and we show that it displays a rich dynamics that includes quasi-harmonic, relaxation oscillations, and chaos. We also show that the oscillator is strongly hysteretic and displays a wide variety of multistable behaviors, including the rare case of bistability between chaotic attractors. Our analytical and numerical results are found to be in good agreement with the experimental measurements.

Dispersion of pollutants in a porous medium with finite thickness and variable dispersion coefficients.

Groundwater is subject to the intrusion of pollutants of various types. These pollutants can have natural or anthropogenic sources. Their consumption can therefore affect human health, but also affects the development of vegetation. The objective of this article is to analyze the effect of the dispersion coefficient parameters on the spatio-temporal distribution of pollutants in a saturated porous medium with a finite thickness. This layer is subjected to two types of conditions at the outlet of the aquifer where the dispersion coefficient is a function of depth. The partial differential equations were solved using an implicit finite difference technique. The results of the analysis suggest that the behavior of the concentration profile was influenced by the different output boundary conditions. On the other hand, the parameter, b of the dispersion coefficient depending on the distance has a significant impact on the movement of the solute in a saturated porous medium compared to the parameter, a. In other words, it has more effect on the dispersion of pollutants in aquifers. This study highlights the need to bring insight on the transports parameters while modeling the transport of solute in a porous media.

Numerical solution of the Burgers equation associated with the phenomena of longitudinal dispersion depending on time.

In this study, the Burgers equation governing the time-dependent dispersion phenomena is solved numerically using the finite difference scheme and the Runge-Kutta 4 algorithm with appropriate initial and boundary conditions. Two time-dependent dispersion functions have been implemented to analyze the spatio-temporal variation in the domain. For the values of KL and KA < 1.2 years, a significant retention of the mass of solute is observed when the dispersion function is asymptotic. The results obtained show that the concentration profiles are similar when the values of KL and KA ≥ 1.2 years. These results demonstrate the importance of the nature of the dispersion function on the retention capacity of solutes in the porous medium.

Long-range interaction effects on coupled excitable nodes: traveling waves and chimera state.

In this paper, analytical and numerical studies of the influence of the long-range interaction parameter on the excitability threshold in a ring of FitzHugh-Nagumo (FHN) system are investigated. The long-range interaction is introduced to the network to model regulation of the Gap junctions or hemichannels activity at the connexins level, which provides links between pre-synaptic and post-synaptic neurons. Results show that the long-range coupling enhances the range of the threshold parameter. We also investigate the long-range effects on the network dynamics, which induces enlargement of the oscillatory zone before the excitable regime. When considering bidirectional coupling, the long-range interaction induces traveling patterns such as traveling waves, while when considering unidirectional coupling, the long-range interaction induces multi-chimera states. We also studied the difference between the dynamics of coupled oscillators and coupled excitable neurons. We found that, for the coupled system, the oscillation period decreases with the increasing of the coupling parameter. For the same values of the coupling parameter, the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics. The analytical approximation shows good agreement with the numerical results.

Nonlinear dynamics in an optoelectronic feedback delay oscillator with piecewise linear transfer functions from the laser diode and photodiode.

We investigate the nonlinear dynamics of a recent architecture of an optoelectronic oscillator, where the emitting laser and the receiving diode are connected in a head-to-tail configuration via an optical fiber delay line. The resulting nonlinear transfer function is a piecewise linear profile, and its interplay with the delay leads to many complex behaviors such as relaxation oscillations and deterministic chaos. This system belongs to a recent class of optoelectronic oscillators where the nonlinearity does not originate from the sinusoidal transfer function of an imbalanced interferometer, and, in particular, it is a simple optoelectronic oscillator configuration that is capable of displaying a chaotic behavior. The results of the analytic study are confirmed by numerical simulations and experimental measurements.

Study of the effect of the offset phase in time-delay electro-optical systems.

We show that the effect of the offset phase on the dynamics of the time-delay optoelectronic oscillators that is observed experimentally can be explained in terms of switching between the subcritical and supercritical Hopf bifurcations. The domains of the offset phase for which the system functions are determined analytically. We also show that the width of these domains exceptionally depends on the interplay between the three time scales of the system. Our theoretical results fit with the experimental measurements.

A normal form method for the determination of oscillations characteristics near the primary Hopf bifurcation in bandpass optoelectronic oscillators: Theory and experiment.

We propose a framework for the analysis of the integro-differential delay Ikeda equations ruling the dynamics of bandpass optoelectronic oscillators (OEOs). Our framework is based on the normal form reduction of OEOs and helps in the determination of the amplitude and the frequency of the primary Hopf limit-cycles as a function of the time delay and other parameters. The study is carried for both the negative and the positive slopes of the sinusoidal transfer function, and our analytical results are confirmed by the numerical and experimental data.

Dynamics of three unidirectionally coupled autonomous Duffing oscillators and application to inchworm piezoelectric motors: Effects of the coupling coefficient and delay.

This work deals with the dynamics of three unidirectionally coupled Duffing oscillators and that of three coupled piezoelectric actuators, considering the special case of inchworm motors. Two configurations of the network are studied: ring configuration and chain configuration. The effects of the coupling coefficient and the time delay are analyzed through different bifurcation diagrams and phase difference variation. It is shown that varying the coupling coefficient and the time delay leads to the appearance of different dynamical behaviors: steady states, periodic and quasiperiodic oscillations, chaos, and phase synchronization.

Mixed-mode oscillations in slow-fast delayed optoelectronic systems.

In this article, we investigate the dynamical behavior of breathers in optoelectronic oscillators from the standpoint of mixed-mode oscillations. In the phase space, these breathers are composite oscillations that are damped to the attractive branches of an invariant manifold. Our study shows that the emergence of breather dynamics is linked to the apparition of inflection points in the phase space, and we develop an analytical framework based on the Liénard reduction form in order to provide an analytical insight into this phenomenology. Our theoretical results are in excellent agreement with experimental measurements.

Dynamics of coupled simplest chaotic two-component electronic circuits and its potential application to random bit generation.

We numerically investigate the possibility of using a coupling to increase the complexity in simplest chaotic two-component electronic circuits operating at high frequency. We subsequently show that complex behaviors generated in such coupled systems, together with the post-processing are suitable for generating bit-streams which pass all the NIST tests for randomness. The electronic circuit is built up by unidirectionally coupling three two-component (one active and one passive) oscillators in a ring configuration through resistances. It turns out that, with such a coupling, high chaotic signals can be obtained. By extracting points at fixed interval of 10 ns (corresponding to a bit rate of 100 Mb/s) on such chaotic signals, each point being simultaneously converted in 16-bits (or 8-bits), we find that the binary sequence constructed by including the 10(or 2) least significant bits pass statistical tests of randomness, meaning that bit-streams with random properties can be achieved with an overall bit rate up to 10×100 Mb/s = 1 Gbit/s (or 2×100 Mb/s =200 Megabit/s). Moreover, by varying the bias voltages, we also investigate the parameter range for which more complex signals can be obtained. Besides being simple to implement, the two-component electronic circuit setup is very cheap as compared to optical and electro-optical systems.

Semiconductor lasers driven by self-sustained chaotic electronic oscillators and applications to optical chaos cryptography.

In this work, we numerically study the dynamics of vertical cavity surface emitting laser (VCSEL) firstly when it is driven by Chua's oscillator, secondly in case where it is driven by a broad frequency spectral bandwidth chaotic oscillator developed by Nana et al. [Commun. Nonlinear Sci. Numer. Simul. 14, 2266 (2009)]. We demonstrated that the VCSEL generated robust chaotic dynamics compared to the ones found in VCSEL subject to a sinusoidally modulated current and therefore it is more suitable for chaos encryption techniques. The synchronization characteristics and the communication performances of unidirectional coupled VCSEL driven by the broad frequency spectral bandwidth chaotic oscillators are investigated numerically. The results show that high-quality synchronization and transmission of messages can be realized for suitable system parameters. Chaos shift keying method is successfully applied to encrypt a message at a high bitrate.

Electrical dark compacton generator: theory and simulations.

A modified Colpitts oscillator (MCO) associated with a nonlinear transmission line (NLTL) with intersite nonlinearity is introduced as a self-sustained generator of a train of modulated dark signals with compact shape. Equations of state describing the dynamics of the MCO part are derived and the stationary state is obtained. Using the Routh-Hurwitz criterion, the result of a stability analysis indicates the existence of a limit cycle in certain parameter regimes and there the oscillation of the circuit delivers pulselike electrical signals. The train of generated signals is then transformed into a train of compact modulated dark voltage solitons by the NLTL. The exactness of this analytical analysis is confirmed by numerical simulations performed on the circuit equations. Finally, simulations show the capacity of this circuit to work as a generator of compactlike dark voltage solitons. The performance of the generator, namely, the pulse width and the repetition rate, is controlled by the magnitude of the characteristic parameters of the electronic components of the device.

The design of a reflectionless arterial prosthesis.

We propose a new technique to characterize a reflectionless arterial prosthesis. The corresponding transmission and reflection coefficients are determined from the geometric and the elastic properties of the arterial wall, and the interaction between the latter and the prosthesis are studied accordingly.

Synchronization: stability and duration time.

We consider the problem of stability and duration of the synchronization process between self-excited oscillators, both in their regular and chaotic states. Making use of the properties of Hill equation describing the deviation between the slave and the master, we derive the stability conditions and expressions of the synchronization time. A fairly good agreement is obtained between the analytical and numerical results.