Chaotic breathers in delayed electro-optical systems

Complex dynamics in nonlinear small time-delayed optoelectronic oscillator and application in fast reservoir computing and pulse generation

We investigated a time-delayed optoelectronic oscillator (OEO) that displays a wide range of complex dynamic behavior under small time delay. The phase-space trajectory distributions in different dynamic regimes were compared which brings a new perspective on the underlying mechanism of the transition process. It was found that bifurcation is always possible no matter how small the time delay is even if the universal adiabatic approximation model is invalid. Hereby we proposed a versatile simple oscillator which has a potential capacity as memory carrier and high-dimensional state spatial mapping ability that brings 1000 times computing-efficiency improvements of reservoir computing over the large time delay one. Furthermore, we demonstrated a new approach for a tunable optoelectronic pulse generator (repetition rate at 0.2 MHz and 0.25 GHz) which depends critically on time-delayed input electrical pulse. The proposed oscillator is also a promising system for the applications of fast chaos-based communication.

Controllable non-uniformly distributed spiking cluster generation in broadband optoelectronic oscillator

An approach to achieve controllable non-uniformly distributed spiking cluster generation is proposed and demonstrated based on an externally-triggered broadband optoelectronic oscillator (OEO). The theory of controlling the distribution of the spiking pulses in a spiking cluster is established. Based on the theory, the dynamic and the distribution characteristics are analyzed and revealed in the stable spiking oscillation state under different externally-injected trigger signal voltages. The peak-voltage envelop of the cluster and the interval of the spiking pulses are demonstrated to have an approximate negative linearity relationship with the externally-injected trigger signal voltage in both the numerical simulation and the experiment, where a square waveform, a trapezoidal waveform, a parabola waveform, and a half-sinusoidal waveform are used as the externally-injected trigger signals. The results indicate that the spiking pulse distribution in the generated spiking cluster can be well controlled through tuning the externally-injected trigger signal voltage. The proposed scheme can be utilized in spiking encoding and reservoir computing.

Hybrid integrated optical chaos circuits with optoelectronic feedback

A chip-scale chaotic laser system with optoelectronic delayed feedback is proposed and analyzed by numerical simulation. This chip eliminates the need for bulky delay components such as long optical fibers, free propagation and external cavities, relying solely on internal devices and waveguides to achieve feedback delay. This approach simplifies integration, maintaining a compact chip size. According to the results, the chip-scale system exhibits rich dynamics, including periodicity, quasi-periodicity, and chaotic states. Chaos resembling Gaussian white noise is achieved with picosecond-level delay time, highlighting the complexity of chip-scale signals. Furthermore, time delay signature (TDS) concealment is enhanced with a short delay comparable to the inverse bandwidth τ, albeit at a cost of sacrificing chaotic signal complexity. Applying the photonic integrated circuits to practical applications, 1 Gbps back-to-back communication transmission is feasible. Results demonstrate low bit error rates (BERs) for authorizers (<10-6) and high BERs for eavesdroppers (>10-2), ensuring communication confidentiality and chaotic synchronization. Lastly, preliminary experiments validate the feasibility. Our theoretical work has demonstrated the feasibility of hybrid integrated optical chaos circuits with optoelectronic feedback based on photonic wire bonding, which can provide a stable and flexible integrated chaos source.

Simulation of optoelectronic oscillator injection locking, pulling & spiking phenomena

Complex envelope and reduced phase simulation models describing the dynamical behaviour of an optoelectronic oscillator (OEO) under injection by an external source are described. The models build on the foundations of a previously reported delay integral/differential equation (DDE) theory of injection locking of time delay oscillators (TDO) such as the OEO. The DDE formulation is particularly amenable to high precision simulation using the Simulink™ block diagram environment. The correspondence between the blocks and the oscillator components offers intuition and considerable freedom to explore different circuit architectures and design variations with minimal coding effort. The simulations facilitate the study of the profound effect the multimode nature of a TDO has on its dynamical behavior. The reduced phase models that make use of the Leeson approximation are generally successful in reproducing the results of complex envelope models for established oscillations except for spiking phenomena for which the Leeson approximation fails. Simulation results demonstrating phenomena not captured by classical injection theory are presented, including multimode oscillation, the appearance of sidemodes in the RF and phase noise spectrum, and persistent spike trains redolent of recent experimental observations of [Formula: see text] phase pulse trains in a broadband OEO under injection.

Neuromorphic regenerative memory optoelectronic oscillator

Neuromorphic spiking information processing based on neuron-like excitable effect has achieved rapid development in recent years due to its advantages such as ultra-high operation speed, programming-free implementation and low power consumption. However, the current physical platforms lack building blocks like compilers, logic gates, and more importantly, data memory. These factors become the shackles to construct a full-physical layer neural network. In this paper, a neuromorphic regenerative memory scheme is proposed based on a time-delayed broadband nonlinear optoelectronic oscillator (OEO), which enables reshaping and regenerating on-off keying encoding sequences. Through biasing the dual-drive Mach-Zehnder electro-optic modulator in the OEO cavity near its minimum transmission point, the OEO can work in excitable regime, where localized states are maintained for robust nonlinear spiking response. Both simulation and experiment are carried out to demonstrate the proposed scheme, where the simulation results and the experimental results fit in with each other. The proposed OEO-based neuromorphic regenerative memory scheme exhibits long-term response ability for short-term excitation, which shows an enormous application potential for high-speed neuromorphic information buffering, optoelectronic interconnection and computing.

Numerical and experimental investigation of a dispersive optoelectronic oscillator for chaotic time-delay signature suppression

Chaos generation from a novel single-loop dispersive optoelectronic oscillator (OEO) with a broadband chirped fiber Bragg grating (CFBG) is numerically and experimentally investigated. The CFBG has much broader bandwidth than the chaotic dynamics such that its dispersion effect rather than filtering effect dominates the reflection. The proposed dispersive OEO exhibits chaotic dynamics when sufficient feedback strength is guaranteed. Suppression of chaotic time-delay signature (TDS) is observed as the feedback strength increases. The TDS can be further suppressed as the amount of grating dispersion increases. Without compromising bandwidth performance, our proposed system extends the parameter space of chaos, enhances the robustness to modulator bias variation, and improves TDS suppression by at least five times comparing to the classical OEO. Experimental results qualitatively agree well with numerical simulations. In addition, the advantage of dispersive OEO is further verified by experimentally demonstrating random bit generation with tunable rate up to 160 Gbps.

Modeling of a multi-parameter chaotic optoelectronic oscillator based on the Fourier neural operator

A model construction scheme of chaotic optoelectronic oscillator (OEO) based on the Fourier neural operator (FNO) is proposed. Different from the conventional methods, we learn the nonlinear dynamics of OEO (actual components) in a data-driven way, expecting to obtain a multi-parameter OEO model for generating chaotic carrier with high-efficiency and low-cost. FNO is a deep learning architecture which utilizes neural network as a parameter structure to learn the trajectory of the family of equations from training data. With the assistance of FNO, the nonlinear dynamics of OEO characterized by differential delay equation can be modeled easily. In this work, the maximal Lyapunov exponent is applied to judge whether these time series have chaotic behavior, and the Pearson correlation coefficient (PCC) is introduced to evaluate the modeling performance. Compare with long and short-term memory (LSTM), FNO is not only superior to LSTM in modeling accuracy, but also requires less training data. Subsequently, we analyze the modeling performance of FNO under different feedback gains and time delays. Both numerical and experimental results show that the PCC can be greater than 0.99 in the case of low feedback gain. Next, we further analyze the influence of different system oscillation frequencies, and the generalization ability of FNO is also analyzed.

Farey tree and devil's staircase of frequency-locked breathers in ultrafast lasers

Nonlinear systems with two competing frequencies show locking or resonances. In lasers, the two interacting frequencies can be the cavity repetition rate and a frequency externally applied to the system. Conversely, the excitation of breather oscillations in lasers naturally triggers a second characteristic frequency in the system, therefore showing competition between the cavity repetition rate and the breathing frequency. Yet, the link between breathing solitons and frequency locking is missing. Here we demonstrate frequency locking at Farey fractions of a breather laser. The winding numbers exhibit the hierarchy of the Farey tree and the structure of a devil's staircase. Numerical simulations of a discrete laser model confirm the experimental findings. The breather laser may therefore serve as a simple test bed to explore ubiquitous synchronization dynamics of nonlinear systems. The locked breathing frequencies feature a high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications.

Square-wave generation in vertical external-cavity Kerr-Gires-Tournois interferometers

We study theoretically the mechanisms of square-wave (SW) formation in vertical external-cavity Kerr-Gires-Tournois interferometers in the presence of anti-resonant injection. We provide simple analytical approximations for their plateau intensities and for the conditions of their emergence. We demonstrate that SWs may appear via a homoclinic snaking scenario, leading to the formation of complex-shaped multistable SW solutions. The resulting SWs can host localized structures and robust bound states.

Experimental demonstration of a broadband optoelectronic chaos system based on highly nonlinear configuration of IQ modulator

We experimentally investigated a novel broadband optoelectronic chaos generation scheme. The proposed system is achieved by adopting the highly nonlinear operation of an electro-optical exclusive-NOR (XNOR) logic gate and two delayed feedback loops that refer to the Boolean chaos model. The XNOR gate is established by a commercial use inphase and quadrature-phase (IQ) modulator that works at a specific bias point. The resulting power spectrum of the broadband chaos signal extends from DC to 29.1 GHz (10 dB bandwidth), and the probability density distribution is Gaussian distribution like. Owing to the strong nonlinearity of XNOR operation, the conditions to enter the chaos region are more relaxed compared to traditional optoelectronic oscillator (OEO) chaos systems, and the time delay signature (TDS) of the feedback loop is also suppressed. Moreover, to further enhance the performance of the generated chaos signal, we injected the optoelectronic chaotic signal into a semiconductor laser. Experimental results indicate that after the cascade optical injection, the bandwidth of the output chaos signal is extended to 38.4 GHz and the TDS is completely concealed; meanwhile, a perfect Gaussian distribution can be obtained.

Time delay estimation from the time series for optical chaos systems using deep learning

We propose a model-free time delay signature (TDS) extraction method for optical chaos systems. The TDS can be identified from time series without prior knowledge of the actual physical processes. In optical chaos secure communication systems, the chaos carrier is usually generated by a laser diode subject to opto-electronic/all-optical time delayed feedback. One of the most important factors to security considerations is the concealment of the TDS. So far, statistical analysis methods such as autocorrelation function (ACF) and delayed mutual information (DMI) are usually used to unveil the TDS. However, the effectiveness of these methods will be reduced when increasing the nonlinearity of chaos systems. Meanwhile, certain TDS concealment strategies have been designed against statistical analysis. In our previous work, convolutional neural network shows its effectiveness on TDS extraction of chaos systems with high loop nonlinearity. However, this method relies on the knowledge of detailed structure of the chaos systems. In this work, we formulate a blind identification method based on long short-term memory neural network (LSTM-NN) model. The method is validated against the two major types of optical chaos systems, i.e. opto-electronic oscillator (OEO) chaos system and laser chaos system based on internal nonlinearity. Moreover, some security enhanced chaotic systems are also studied. The results show that the proposed method has high tolerance to additive noise. Meanwhile, the data amount needed is less than existing methods.

Electro-optic chaos system with time delay signature concealment based on XOR operation and multi-bit PRBS

A novel chaos system with XOR operations and multi-bit PRBS is proposed to improve the sequence complexity and the security of the classic electro-optic intensity chaos system. Through the bifurcation diagram and permutation entropy analysis, the PE can be increased to 0.99. The key space is enlarged because additional DSP parameters and PRBS are introduced. The impacts of ADC/DAC characteristics and PRBS characteristics are analyzed in detail. The simulation results show that the time delay signature can be concealed with the appropriate DSP parameters.

Pulse-timing symmetry breaking in an excitable optical system with delay

Excitable systems with delayed feedback are important in areas from biology to neuroscience and optics. They sustain multistable pulsing regimes with different numbers of equidistant pulses in the feedback loop. Experimentally and theoretically, we report on the pulse-timing symmetry breaking of these regimes in an optical system. A bifurcation analysis unveils that this originates in a resonance phenomenon and that symmetry-broken states are stable in large regions of the parameter space. These results have impact in photonics for, e.g., optical computing and versatile sources of optical pulses.

Time-delay signature characteristics of the chaotic output from an optoelectronic oscillator by introducing an optical feedback

In this work, via autocorrelation function (ACF) and permutation entropy (PE) methods, we numerically investigate the time-delay signature (TDS) characteristics of the chaotic signal output from an optoelectronic oscillator (OEO) after introducing an extra optical feedback loop. The results demonstrate that, for such a chaotic system, both the optoelectronic feedback with a delay time of T₁ and the optical feedback with a delay time of T₂ contribute to the TDS of generated chaos. The TDS of the chaotic signal should be evaluated within a large time window including T₁ and T₂ by the strongest peak in the ACF curve of the chaotic signal, and the strongest peak may locate at near T₁ or T₂. Through mapping the evolution of the TDS in the parameter space of the optical feedback strength and time, certain optimized parameter regions for achieving a chaotic signal with a relatively weak TDS can be determined.

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.

Unveil the time delay signature of optical chaos systems with a convolutional neural network

We propose a time delay signature extraction method for optical chaos systems based on a convolutional neural network. Through transforming the time delay signature of a one-dimensional time series into two-dimensional image features, the excellent ability of convolutional neural networks for image feature recognition is fully utilized. The effectiveness of the method is verified on chaos systems with opto-electronic feedback and all optical feedback. The recognition accuracy of the method is 100% under normal conditions. For the system with extremely strong nonlinearity, the accuracy can be 93.25%, and the amount of data required is less than traditional methods. Moreover, it is verified that the proposed method possesses a strong ability to withstand the effects of noise.

Laminar Chaos in Experiments: Nonlinear Systems with Time-Varying Delays and Noise

A new type of dynamics called laminar chaos was recently discovered through a theoretical analysis of a scalar delay differential equation with time-varying delay. Laminar chaos is a low-dimensional dynamics characterized by laminar phases of nearly constant intensity with periodic durations and a chaotic variation of the intensity from one laminar phase to the next laminar phase. This is in stark contrast to the typically observed higher-dimensional turbulent chaos, which is characterized by strong fluctuations. In this Letter we provide the first experimental observation of laminar chaos by studying an optoelectronic feedback loop with time-varying delay. The noise inherent in the experiment requires the development of a nonlinear Langevin equation with variable delay. The results show that laminar chaos can be observed in higher-order systems, and that the phenomenon is robust to noise and a digital implementation of the variable time delay.

Delay-induced switched states in a slow-fast system

We consider the two-component delay system εx'(t) = - x(t) - y(t) + f(x(t - 1)), y'(t) = ηx(t) with small para- meters ε, η and positive feedback function f. Previously, such systems have been reported to model switching in optoelectronic experiments, where each switching induces another one after approximately one delay time, related to one round trip of the signal. In this paper, we study these delay-induced switched states. We provide conditions for their existence and show how the formal limits ε → 0 and/or η → 0 facilitate our understanding of this phenomenon. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Delayed dynamical systems: networks, chimeras and reservoir computing

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays. This architecture allows the design of appropriate filters and multiple time delays, and greatly extends the possibilities for exploring synchronization patterns in arbitrary network topologies. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

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.

Bistatic radar scheme based on the digital-analog hybrid chaos system

Bistatic radar with the separate transmitter and receiver has some attractive merits, thus obtaining many attentions. However, in a traditional bistatic radar system, there are still many problems that hinder it from practical applications. Here we introduce a bistatic radar scheme using a well-designed optical chaos system, from which the analog chaos signal could be determined by generated random binary sequence. The broad bandwidth analog signal is used as surveillance signal and the digital signal are transmitted to the receiver by optical fibers. Finally, high spatial and velocity resolutions of radar system could be obtained by using the analog chaos signal. A high-quality regeneration of the reference signal at different locations can be established by transmitting the digital sequence. Moreover, the mutual interferences could be concealed since the analog surveillance signal and the transmitted digital sequence are delivered by different paths. These could be advantageous for radar applications.

Experiments with arbitrary networks in time-multiplexed delay systems

We report a new experimental approach using an optoelectronic feedback loop to investigate the dynamics of oscillators coupled on large complex networks with arbitrary topology. Our implementation is based on a single optoelectronic feedback loop with time delays. We use the space-time interpretation of systems with time delay to create large networks of coupled maps. Others have performed similar experiments using high-pass filters to implement the coupling; this restricts the network topology to the coupling of only a few nearest neighbors. In our experiment, the time delays and coupling are implemented on a field-programmable gate array, allowing the creation of networks with arbitrary coupling topology. This system has many advantages: the network nodes are truly identical, the network is easily reconfigurable, and the network dynamics occur at high speeds. We use this system to study cluster synchronization and chimera states in both small and large networks of different topologies.

Two distinct bifurcation routes for delayed optoelectronic oscillators

We investigate the coexistence of low- and high-frequency oscillations in a delayed optoelectronic oscillator. We identify two nearby Hopf bifurcation points exhibiting low and high frequencies and demonstrate analytically how they lead to stable solutions. We then show numerically that these two branches of solutions undergo higher order instabilities as the feedback rate is increased but remain separated in the bifurcation diagram. The two bifurcation routes can be followed independently by either progressively increasing or decreasing the bifurcation parameter.

Interaction between Liénard and Ikeda dynamics in a nonlinear electro-optical oscillator with delayed bandpass feedback

We report on experimental and theoretical analysis of the complex dynamics generated by a nonlinear time-delayed electro-optic bandpass oscillator. We investigate the interaction between the slow- and fast-scale dynamics of autonomous oscillations in the breather regime. We analyze in detail the coupling between the fast-scale behavior associated to a characteristic low-pass Ikeda behavior and the slow-scale dynamics associated to a Liénard limit-cycle. Finally, we show that when projected onto a two-dimensional phase space, the attractors corresponding to periodic and chaotic breathers display a spiral-like pattern, which strongly depends on the shape of the nonlinear function.

Electro-optic chaotic system based on the reverse-time chaos theory and a nonlinear hybrid feedback loop

A novel electro-optic chaos source is proposed on the basis of the reverse-time chaos theory and an analog-digital hybrid feedback loop. The analog output of the system can be determined by the numeric states of shift registers, which makes the system robust and easy to control. The dynamical properties as well as the complexity dependence on the feedback parameters are investigated in detail. The correlation characteristics of the system are also studied. Two improving strategies which were established in digital field and analog field are proposed to conceal the time-delay signature. The proposed scheme has the potential to be used in radar and optical secure communication systems.

Consistency in experiments on multistable driven delay systems

We investigate the consistency properties in the responses of a nonlinear delay optoelectronic intensity oscillator subject to different drives, in particular, harmonic and self-generated waveforms. This system, an implementation of the Ikeda oscillator, is operating in a closed-loop configuration, exhibiting its autonomous dynamics while the drive signals are additionally introduced. Applying the same drive multiple times, we compare the dynamical responses of the optoelectronic oscillator and quantify the degree of consistency among them via their correlation. Our results show that consistency is not restricted to conditions close to the first Hopf bifurcation but can be found in a broad range of dynamical regimes, even in the presence of multistability. Finally, we discuss the dependence of consistency on the nature of the drive signal.

Fully analogue photonic reservoir computer

Introduced a decade ago, reservoir computing is an efficient approach for signal processing. State of the art capabilities have already been demonstrated with both computer simulations and physical implementations. If photonic reservoir computing appears to be promising a solution for ultrafast nontrivial computing, all the implementations presented up to now require digital pre or post processing, which prevents them from exploiting their full potential, in particular in terms of processing speed. We address here the possibility to get rid simultaneously of both digital pre and post processing. The standalone fully analogue reservoir computer resulting from our endeavour is compared to previous experiments and only exhibits rather limited degradation of performances. Our experiment constitutes a proof of concept for standalone physical reservoir computers.

A new switching parameter varying optoelectronic delayed feedback model with computer simulation

In this paper, a new switching parameter varying optoelectronic delayed feedback model is proposed and analyzed by computer simulation. This model is switching between two parameter varying optoelectronic delayed feedback models based on chaotic pseudorandom sequences. Complexity performance results show that this model has a high complexity compared to the original model. Furthermore, this model can conceal the time delay effectively against the auto-correlation function, delayed mutual information and permutation information analysis methods, and can extent the key space, which greatly improve its security.

In-phase, out-of-phase and T/4 synchronization of square waves in delay-coupled non-identical optoelectronic oscillators

We model two non-identical delay-line optoelectronic oscillators mutually coupled through delayed cross-feedback. The system can generate multi-stable nanosecond periodic square-wave solutions which arise through a Hopf instability. We show that for suitable ratios between self and cross delay times, the two oscillators generate square waves with different amplitude but synchronized in phase, out of phase or with a dephasing of a quarter of the period. We also show that the synchronization is robust to small mismatches in the delay times.

Laser chimeras as a paradigm for multistable patterns in complex systems

A chimera state is a rich and fascinating class of self-organized solutions developed in high-dimensional networks. Necessary features of the network for the emergence of such complex but structured motions are non-local and symmetry breaking coupling. An accurate understanding of chimera states is expected to bring important insights on deterministic mechanism occurring in many structurally similar high-dimensional dynamics such as living systems, brain operation principles and even turbulence in hydrodynamics. Here we report on a powerful and highly controllable experiment based on an optoelectronic delayed feedback applied to a wavelength tuneable semiconductor laser, with which a wide variety of chimera patterns can be accurately investigated and interpreted. We uncover a cascade of higher-order chimeras as a pattern transition from N to N+1 clusters of chaoticity. Finally, we follow visually, as the gain increases, how chimera state is gradually destroyed on the way to apparent turbulence-like system behaviour.

Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop

Many physical processes, including the intensity fluctuations of a chaotic laser, the detection of single photons, and the Brownian motion of a microscopic particle in a fluid are unpredictable, at least on long timescales. This unpredictability can be due to a variety of physical mechanisms, but it is quantified by an entropy rate. This rate, which describes how quickly a system produces new and random information, is fundamentally important in statistical mechanics and practically important for random number generation. We experimentally study entropy generation and the emergence of deterministic chaotic dynamics from discrete noise in a system that applies feedback to a weak optical signal at the single-photon level. We show that the dynamics transition from shot noise to chaos as the photon rate increases and that the entropy rate can reflect either the deterministic or noisy aspects of the system depending on the sampling rate and resolution.

Synchronization of tunable asymmetric square-wave pulses in delay-coupled optoelectronic oscillators

We consider a model for two delay-coupled optoelectronic oscillators under positive delayed feedback as prototypical to study the conditions for synchronization of asymmetric square-wave oscillations, for which the duty cycle is not half of the period. We show that the scenario arising for positive feedback is much richer than with negative feedback. First, it allows for the coexistence of multiple in- and out-of-phase asymmetric periodic square waves for the same parameter values. Second, it is tunable: The period of all the square-wave periodic pulses can be tuned with the ratio of the delays, and the duty cycle of the asymmetric square waves can be changed with the offset phase while the total period remains constant. Finally, in addition to the multiple in- and out-of-phase periodic square waves, low-frequency periodic asymmetric solutions oscillating in phase may coexist for the same values of the parameters. Our analytical results are in agreement with numerical simulations and bifurcation diagrams obtained by using continuation techniques.

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.

Multirhythmicity in an optoelectronic oscillator with large delay

An optoelectronic oscillator exhibiting a large delay in its feedback loop is studied both experimentally and theoretically. We show that multiple square-wave oscillations may coexist for the same values of the parameters (multirhythmicity). Depending on the sign of the phase shift, these regimes admit either periods close to an integer fraction of the delay or periods close to an odd integer fraction of twice the delay. These periodic solutions emerge from successive Hopf bifurcation points and stabilize at a finite amplitude following a scenario similar to Eckhaus instability in spatially extended systems. We find quantitative agreements between experiments and numerical simulations. The linear stability of the square waves is substantiated analytically by determining the stable fixed points of a map.

Polynomial law for controlling the generation of n-scroll chaotic attractors in an optoelectronic delayed oscillator

Controlled transitions between a hierarchy of n-scroll attractors are investigated in a nonlinear optoelectronic oscillator. Using the system's feedback strength as a control parameter, it is shown experimentally the transition from Van der Pol-like attractors to 6-scroll, but in general, this scheme can produce an arbitrary number of scrolls. The complexity of every state is characterized by Lyapunov exponents and autocorrelation coefficients.

Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems

We analyze the response of two delay-coupled optoelectronic oscillators. Each oscillator operates under its own delayed feedback. We show that the system can display square-wave periodic solutions that can be synchronized in phase or out of phase depending on the ratio between self- and cross-delay times. Furthermore, we show that multiple periodic synchronized solutions can coexist for the same values of the fixed parameters. As a consequence, it is possible to generate square-wave oscillations with different periods by just changing the initial conditions.

Enhanced secure strategy for electro-optic chaotic systems with delayed dynamics by using fractional Fourier transformation

We propose a scheme whereby a time domain fractional Fourier transform (FRFT) is used to post process the optical chaotic carrier generated by an electro-optic oscillator. The time delay signature of the delay dynamics is successfully masked by the FRFT when some conditions are satisfied. Meanwhile the dimension space of the physical parameters is increased. Pseudo random binary sequence (PRBS) with low bit rate (hundreds of Mbps) is introduced to control the parameters of the FRFT. The chaotic optical carrier, FRFT parameters and the PRBS are covered by each other so that the eavesdropper has to search the whole key space to crack the system. The scheme allows enhancing the security of communication systems based on delay dynamics without modifying the chaotic source. In this way, the design of chaos based communication systems can be implemented in a modular manner.

Complexity in electro-optic delay dynamics: modelling, design and applications

Nonlinear delay dynamics have found during the last 30 years a particularly prolific exploration area in the field of photonic systems. Besides the popular external cavity laser diode set-ups, we focus in this article on another experimental realization involving electro-optic (EO) feedback loops, with delay. This approach has strongly evolved with the important technological progress made on broadband photonic and optoelectronic devices dedicated to high-speed optical telecommunications. The complex dynamical systems performed by nonlinear delayed EO feedback loop architectures were designed and explored within a huge range of operating parameters. Thanks to the availability of high-performance photonic devices, these EO delay dynamics led also to many successful, efficient and diverse applications, beyond the many fundamental questions raised from the observation of experimental behaviours. Their chaotic motion allowed for a physical layer encryption method to secure optical data, with a demonstrated capability to operate at the typical speed of modern optical telecommunications. Microwave limit cycles generated in similar EO delay oscillators showed significantly improved spectral purity thanks to the use of a very long fibre delay line. Last but not least, a novel brain inspired computational principle has been recently implemented physically in photonics for the first time, again on the basis of an EO delay dynamical system. In this latter emerging application, the computed result is obtained by a proper 'read-out' of the complex nonlinear transients emerging from a fixed point, the transient being issued by the injection of the information signal to be processed.

Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers

We perform a detailed numerical analysis of square-wave (SW) polarization switching in two semiconductor lasers with time-delayed, orthogonal mutual coupling. An in-depth mapping of the dynamics in the two-parameter plane coupling strength versus frequency detuning shows that stable SWs occur in narrow parameter regions that are localized close to the boundary of stability of the pure-mode solution. In this steady state, the two coupled lasers emit orthogonal polarizations. We also show that there are various types of SW forms and that stable switching does not need the inclusion of noise or nonlinear gain in the model. As these narrow regions of deterministic and stable SWs occur for quite different combinations of parameters, they could potentially explain the waveforms that have been observed experimentally. However, on the other hand, these regions are narrow enough to be in fact considered as experimentally unreachable. Therefore, our results indicate that further experimental statistical studies are needed in order to distinguish deterministic and stationary square waveforms from long transients because of noise.

Virtual chimera states for delayed-feedback systems

Time-delayed systems are found to display remarkable temporal patterns the dynamics of which split into regular and chaotic components repeating at the interval of a delay. This novel long-term behavior for delay dynamics results from strongly asymmetric nonlinear delayed feedback driving a highly damped harmonic oscillator dynamics. In the corresponding virtual space-time representation, the behavior is found to develop as a chimeralike state, a new paradigmatic object from the network theory characterized by the coexistence of synchronous and incoherent oscillations. Numerous virtual chimera states are obtained and analyzed, through experiment, theory, and simulations.

Chaotic dynamics of a frequency-modulated microwave oscillator with time-delayed feedback

We present a chaotic frequency-modulated microwave source that is governed by a simple, first-order nonlinear delay differential equation. When a sinusoidal nonlinearity is incorporated, the dynamical behaviors range from fixed-point to periodic to chaotic, depending on the feedback strength. When the sinusoidal nonlinearity is replaced by a binary nonlinearity, the system exhibits a complex periodic attractor with no fixed-point solution.

Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators

We experimentally demonstrate group synchrony in a network of four nonlinear optoelectronic oscillators with time-delayed coupling. We divide the nodes into two groups of two each, by giving each group different parameters and by enabling only intergroup coupling. When coupled in this fashion, the two groups display different dynamics, with no isochronal synchrony between them, but the nodes in a single group are isochronally synchronized, even though there is no intragroup coupling. We compare experimental behavior with theoretical and numerical results.

Ultra-high-frequency piecewise-linear chaos using delayed feedback loops

We report on an ultra-high-frequency (>1 GHz), piecewise-linear chaotic system designed from low-cost, commercially available electronic components. The system is composed of two electronic time-delayed feedback loops: A primary analog loop with a variable gain that produces multi-mode oscillations centered around 2 GHz and a secondary loop that switches the variable gain between two different values by means of a digital-like signal. We demonstrate experimentally and numerically that such an approach allows for the simultaneous generation of analog and digital chaos, where the digital chaos can be used to partition the system's attractor, forming the foundation for a symbolic dynamics with potential applications in noise-resilient communications and radar.

A pseudo-matched filter for chaos

A matched filter maximizes the signal-to-noise ratio of a signal. In the recent work of Corron et al. [Chaos 20, 023123 (2010)], a matched filter is derived for the chaotic waveforms produced by a piecewise-linear system. This system produces a readily available binary symbolic dynamics that can be used to perform correlations in the presence of large amounts of noise using the matched filter. Motivated by these results, we describe a pseudo-matched filter, which operates similarly to the original matched filter. It consists of a notch filter followed by a first-order, low-pass filter. We compare quantitatively the matched filter's performance to that of our pseudo-matched filter using correlation functions. On average, the pseudo-matched filter performs with a correlation signal-to-noise ratio that is 2.0 dB below that of the matched filter. Our pseudo-matched filter, though somewhat inferior in comparison to the matched filter, is easily realizable at high speed (>1 GHz) for potential radar applications.

Hopf bifurcation to square-wave switching in mutually coupled semiconductor lasers

Using advanced continuation techniques for dynamical systems, we elucidate the bifurcations leading to asymptotically stable square-wave pulsing and polarization mode switching in semiconductor lasers with mutual time-delayed and polarization rotating coupling. We find that the increase of coupling strength leads to a cascade of Hopf bifurcations on a mixed-mode steady state up to a transcritical bifurcation on a so-called pure-mode steady state where both lasers emit with the injected polarization state. From these successive Hopf bifurcations emerge time-periodic solutions that have a period close to the laser relaxation oscillation for weak coupling but a period close to twice the time delay for large coupling strength. The wave form of the time-periodic solutions also evolves from harmonic pulsing up to square-wave pulsing as has been observed recently in experiments.

Optoelectronic reservoir computing

Reservoir computing is a recently introduced, highly efficient bio-inspired approach for processing time dependent data. The basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer. Within these constraints many implementations are possible. Here we report an optoelectronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line. Our implementation is sufficiently fast for real time information processing. We illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition, and obtain results comparable to state of the art digital implementations.

Subwavelength position sensing using nonlinear feedback and wave chaos

We demonstrate a position-sensing technique that relies on the inherent sensitivity of chaos, where we illuminate a subwavelength object with a complex structured radio-frequency field generated using wave chaos and nonlinear feedback. We operate the system in a quasiperiodic state and analyze changes in the frequency content of the scalar voltage signal in the feedback loop. This allows us to extract the object's position with a one-dimensional resolution of ~λ/10,000 and a two-dimensional resolution of ~λ/300, where λ is the shortest wavelength of the illuminating source.

Wideband chaos generation using a delayed oscillator and a two-dimensional nonlinearity induced by a quadrature phase-shift-keying electro-optic modulator

We present a new optoelectronic architecture intended for chaotic optical intensity generation. The principle relies on an electro-optic nonlinear delay dynamics, where the nonlinearity originates from an integrated four-wave optical interferometer, involving two independent electro-optic modulation inputs. Consequently, the setup involves both two-dimensional nonlinearity and dual-delay feedback dynamics, which results in enhanced chaos complexity of particular interest in chaos encryption schemes. The generated chaos observed with large feedback gains has a bandwidth ranging from 30 kHz to 13 GHz and is confirmed by numerical simulations of the proposed dynamical model and bifurcation diagram calculation.

Robustness of optimal synchronization in real networks

Experimental studies can provide powerful insights into the physics of complex networks. Here, we report experimental results on the influence of connection topology on synchronization in fiber-optic networks of chaotic optoelectronic oscillators. We find that the recently predicted nonmonotonic, cusplike synchronization landscape manifests itself in the rate of convergence to the synchronous state. We also observe that networks with the same number of nodes, same number of links, and identical eigenvalues of the coupling matrix can exhibit fundamentally different approaches to synchronization. This previously unnoticed difference is determined by the degeneracy of associated eigenvectors in the presence of noise and mismatches encountered in real-world conditions.

Scaling behavior of oscillations arising in delay-coupled optoelectronic oscillators

We study the effect of asymmetric coupling strengths on the onset of oscillations in delay-coupled nonlinear systems. Our experiment consists of two wide-band optoelectronic devices that are cross-coupled. We find that oscillations appear in the system when the product of the two coupling strengths exceeds a critical value. Beyond the oscillation threshold, the oscillation amplitudes grow smoothly and we find a scaling law that describes the dependence of the amplitude on the coupling strengths. The observations are in good agreement with predictions from linear stability analysis and normal form theory.