Consistency properties of a chaotic semiconductor laser driven by optical feedback

High-resolution dynamic consistency analysis of photonic time-delay reservoir computer

We numerically investigate a time-delayed reservoir computer architecture based on a single-mode laser diode with optical injection and optical feedback. Through a high-resolution parametric analysis, we reveal unforeseen regions of high dynamic consistency. We demonstrate furthermore that the best computing performance is not achieved at the edge of consistency, as previously suggested in a coarser parametric analysis. This region of high consistency and optimal reservoir performances is highly sensitive to the data input modulation format.

Deep optical reservoir computing and chaotic synchronization predictions based on the cascade coupled optically pumped spin-VCSELs

In this work, we utilize two cascade coupling modes (unidirectional coupling and bidirectional coupling) to construct a four-layer deep reservoir computing (RC) system based on the cascade coupled optically-pumped spin-VCSEL. In such a system, there are double sub-reservoirs in each layer, which are formed by the chaotic x-PC and y-PC emitted by the reservoir spin-VCSEL in each layer. Under these two coupling modes, the chaotic x-PC and y-PC emitted by the driving optically-pumped spin-VCSEL (D-Spin-VCSEL), as two learning targets, are predicted by utilizing the four-layer reservoirs. In different parameter spaces, it is further explored that the outputs of the double sub-reservoirs in each layer are respectively synchronized with the chaotic x-PC and y-PC emitted by the D-Spin-VCSEL. The memory capacities (MCs) for the double sub-reservoirs in each layer are even further investigated. The results show that under two coupling modes, the predictions of the double sub-reservoirs with higher-layer for these two targets have smaller errors, denoting that the higher-layer double sub-reservoirs possess better predictive learning ability. Under the same system parameters, the outputs of the higher-layer dual parallel reservoirs are better synchronized with two chaotic PCs emitted by the D-Spin-VCSEL, respectively. The larger MCs can also be obtained by the higher-layer double reservoirs. In particular, compared with the four-layer reservoir computing system under unidirectional coupling, the four-layer reservoir computing system under bidirectional coupling shows better predictive ability in the same parameter space. The chaotic synchronizations predicted by each layer double sub-reservoirs under bidirectional coupling can be obtained higher qualities than those under unidirectional coupling. By the optimization of the system parameters, the outputs of the fourth-layer double sub-reservoirs are almost completely synchronized with the chaotic x-PC and y-PC emitted by the D-Spin-VCSEL, respectively, due to their correlation coefficient used to measure synchronization quality can be obtained as 0.99. These results have potential applications in chaotic computation, chaotic secure communication and accurate prediction of time series.

Consistency Hierarchy of Reservoir Computers

We study the propagation and distribution of information-carrying signals injected in dynamical systems serving as reservoir computers. Through different combinations of repeated input signals, a multivariate correlation analysis reveals measures known as the consistency spectrum and consistency capacity. These are high-dimensional portraits of the nonlinear functional dependence between input and reservoir state. For multiple inputs, a hierarchy of capacities characterizes the interference of signals from each source. For an individual input, the time-resolved capacities form a profile of the reservoir's nonlinear fading memory. We illustrate this methodology for a range of echo state networks.

Predicting the dynamical behaviors for chaotic semiconductor lasers by reservoir computing

We demonstrate the successful prediction of the continuous intensity time series and reproduction of the underlying dynamical behaviors for a chaotic semiconductor laser by reservoir computing. The laser subject to continuous-wave optical injection is considered using the rate-equation model. A reservoir network is constructed and trained using over 2 × 10⁴ data points sampled every 1.19 ps from the simulated chaotic intensity time series. Upon careful optimization of the reservoir parameters, the future evolution of the continuous intensity time series can be accurately predicted for a time duration of longer than 0.6 ns, which is six times the reciprocal of the relaxation resonance frequency of the laser. Moreover, we demonstrate for the first time, to the best of our knowledge, that the predicted intensity time series allows for accurate reproduction of the chaotic dynamical behaviors, including the microwave power spectrum, probability density function, and the chaotic attractor. In general, the demonstrated approach offers a relatively high flexibility in the choice of reservoir parameters according to the simulation results, and it provides new insights into the learning and prediction of semiconductor laser dynamics based on measured intensity time series.

Consciousness is supported by near-critical slow cortical electrodynamics

Mounting evidence suggests that during conscious states, the electrodynamics of the cortex are poised near a critical point or phase transition and that this near-critical behavior supports the vast flow of information through cortical networks during conscious states. Here, we empirically identify a mathematically specific critical point near which waking cortical oscillatory dynamics operate, which is known as the edge-of-chaos critical point, or the boundary between stability and chaos. We do so by applying the recently developed modified 0-1 chaos test to electrocorticography (ECoG) and magnetoencephalography (MEG) recordings from the cortices of humans and macaques across normal waking, generalized seizure, anesthesia, and psychedelic states. Our evidence suggests that cortical information processing is disrupted during unconscious states because of a transition of low-frequency cortical electric oscillations away from this critical point; conversely, we show that psychedelics may increase the information richness of cortical activity by tuning low-frequency cortical oscillations closer to this critical point. Finally, we analyze clinical electroencephalography (EEG) recordings from patients with disorders of consciousness (DOC) and show that assessing the proximity of slow cortical oscillatory electrodynamics to the edge-of-chaos critical point may be useful as an index of consciousness in the clinical setting.

0.75 Gbit/s high-speed classical key distribution with mode-shift keying chaos synchronization of Fabry-Perot lasers

High-speed physical key distribution is diligently pursued for secure communication. In this paper, we propose and experimentally demonstrate a scheme of high-speed key distribution using mode-shift keying chaos synchronization between two multi-longitudinal-mode Fabry-Perot lasers commonly driven by a super-luminescent diode. Legitimate users dynamically select one of the longitudinal modes according to private control codes to achieve mode-shift keying chaos synchronization. The two remote chaotic light waveforms are quantized to generate two raw random bit streams, and then those bits corresponding to chaos synchronization are sifted as shared keys by comparing the control codes. In this method, the transition time, i.e., the chaos synchronization recovery time is determined by the rising time of the control codes rather than the laser transition response time, so the key distribution rate is improved greatly. Our experiment achieved a 0.75-Gbit/s key distribution rate with a bit error rate of 3.8 × 10^-3 over 160-km fiber transmission with dispersion compensation. The entropy rate of the laser chaos is evaluated as 16 Gbit/s, which determines the ultimate final key rate together with the key generation ratio. It is therefore believed that the method pays a way for Gbit/s physical key distribution.

Reservoir computing with swarms

We study swarms as dynamical systems for reservoir computing (RC). By example of a modified Reynolds boids model, the specific symmetries and dynamical properties of a swarm are explored with respect to a nonlinear time-series prediction task. Specifically, we seek to extract meaningful information about a predator-like driving signal from the swarm's response to that signal. We find that the naïve implementation of a swarm for computation is very inefficient, as permutation symmetry of the individual agents reduces the computational capacity. To circumvent this, we distinguish between the computational substrate of the swarm and a separate observation layer, in which the swarm's response is measured for use in the task. We demonstrate the implementation of a radial basis-localized observation layer for this task. The behavior of the swarm is characterized by order parameters and measures of consistency and related to the performance of the swarm as a reservoir. The relationship between RC performance and swarm behavior demonstrates that optimal computational properties are obtained near a phase transition regime.

Identifiability of structural networks of nonlinear electronic oscillators

The interplay between structure and function is critical in the understanding of complex systems, their dynamics and their behavior. We investigated the interplay between structural and functional networks by means of the differential identifiability framework, which here quantifies the ability of identifying a particular network structure based on (1) the observation of its functional network and (2) the comparison with a prior observation under different initial conditions. We carried out an experiment consisting of the construction of [Formula: see text] different structural networks composed of [Formula: see text] nonlinear electronic circuits and studied the regions where network structures are identifiable. Specifically, we analyzed how differential identifiability is related to the coupling strength between dynamical units (modifying the level of synchronization) and what are the consequences of increasing the amount of noise existing in the functional networks. We observed that differential identifiability reaches its highest value for low to intermediate coupling strengths. Furthermore, it is possible to increase the identifiability parameter by including a principal component analysis in the comparison of functional networks, being especially beneficial for scenarios where noise reaches intermediate levels. Finally, we showed that the regime of the parameter space where differential identifiability is the highest is highly overlapped with the region where structural and functional networks correlate the most.

Laminar chaos in nonlinear electronic circuits with delay clock modulation

We study laminar chaos in an electronic experiment. A two-diode nonlinear circuit with delayed feedback shows chaotic dynamics similar to the Mackey-Glass or Ikeda delay systems. Clock modulation of a single delay line leads to a conservative variable delay, which with a second delay line is augmented to dissipative delays, leading to laminar chaotic regimes. We discuss the properties of this particular delay modulation and demonstrate experimental aspects of laminar chaos in terms of power spectra and return maps.

The reservoir's perspective on generalized synchronization

We employ reservoir computing for a reconstruction task in coupled chaotic systems, across a range of dynamical relationships including generalized synchronization. For a drive-response setup, a temporal representation of the synchronized state is discussed as an alternative to the known instantaneous form. The reservoir has access to both representations through its fading memory property, each with advantages in different dynamical regimes. We also extract signatures of the maximal conditional Lyapunov exponent in the performance of variations of the reservoir topology. Moreover, the reservoir model reproduces different levels of consistency where there is no synchronization. In a bidirectional coupling setup, high reconstruction accuracy is achieved despite poor observability and independent of generalized synchronization.

Consistency in echo-state networks

Consistency is an extension to generalized synchronization which quantifies the degree of functional dependency of a driven nonlinear system to its input. We apply this concept to echo-state networks, which are an artificial-neural network version of reservoir computing. Through a replica test, we measure the consistency levels of the high-dimensional response, yielding a comprehensive portrait of the echo-state property.

Consistency properties of chaotic systems driven by time-delayed feedback

Consistency refers to the property of an externally driven dynamical system to respond in similar ways to similar inputs. In a delay system, the delayed feedback can be considered as an external drive to the undelayed subsystem. We analyze the degree of consistency in a generic chaotic system with delayed feedback by means of the auxiliary system approach. In this scheme an identical copy of the nonlinear node is driven by exactly the same signal as the original, allowing us to verify complete consistency via complete synchronization. In the past, the phenomenon of synchronization in delay-coupled chaotic systems has been widely studied using correlation functions. Here, we analytically derive relationships between characteristic signatures of the correlation functions in such systems and unequivocally relate them to the degree of consistency. The analytical framework is illustrated and supported by numerical calculations of the logistic map with delayed feedback for different replica configurations. We further apply the formalism to time series from an experiment based on a semiconductor laser with a double fiber-optical feedback loop. The experiment constitutes a high-quality replica scheme for studying consistency of the delay-driven laser and confirms the general theoretical results.

Quantitative identification of dynamical transitions in a semiconductor laser with optical feedback

Identifying transitions to complex dynamical regimes is a fundamental open problem with many practical applications. Semi- conductor lasers with optical feedback are excellent testbeds for studying such transitions, as they can generate a rich variety of output signals. Here we apply three analysis tools to quantify various aspects of the dynamical transitions that occur as the laser pump current increases. These tools allow to quantitatively detect the onset of two different regimes, low-frequency fluctuations and coherence collapse, and can be used for identifying the operating conditions that result in specific dynamical properties of the laser output. These tools can also be valuable for analyzing regime transitions in other complex systems.

Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction

State-space reconstruction is investigated for evaluating the randomness generated by an optically injected semiconductor laser in chaos. The reconstruction of the attractor requires only the emission intensity time series, allowing both experimental and numerical evaluations with good qualitative agreement. The randomness generation is evaluated by the divergence of neighboring states, which is quantified by the time-dependent exponents (TDEs) as well as the associated entropies. Averaged over the entire attractor, the mean TDE is observed to be positive as it increases with the evolution time through chaotic mixing. At a constant laser noise strength, the mean TDE for chaos is observed to be greater than that for periodic dynamics, as attributed to the effect of noise amplification by chaos. After discretization, the Shannon entropies continually generated by the laser for the output bits are estimated in providing a fundamental basis for random bit generation, where a combined output bit rate reaching 200 Gb/s is illustrated using practical tests. Overall, based on the reconstructed states, the TDEs and entropies offer a direct experimental verification of the randomness generated in the chaotic laser.

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.

Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal

We numerically investigate reservoir computing based on the consistency of a semiconductor laser subjected to optical feedback and injection. We introduce a chaos mask signal as an input temporal mask for reservoir computing and perform a time-series prediction task. We compare the errors of the task obtained from the chaos mask signal with those obtained from other digital and analog masks. The performance of the prediction task can be improved by using the chaos mask signal due to complex dynamical response.

Determining the sub-Lyapunov exponent of delay systems from time series

For delay systems the sign of the sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes the properties of strong and weak chaos and of consistency. Here we present a robust algorithm based on reconstruction of the local linearized equations of motion, which allows for calculating the sub-LE from time series. The algorithm is inspired by a method introduced by Pyragas for a nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In the presented extension to delay systems, the delayed feedback takes over the role of the drive, whereas the response of the low-dimensional node leads to the sub-Lyapunov exponent. Our method is based on a low-dimensional representation of the delay system. We introduce the basic algorithm for a discrete scalar map, extend the concept to scalar continuous delay systems, and give an outlook to the case of a full vector-state system, from which only a scalar observable is recorded.

The transition between strong and weak chaos in delay systems: Stochastic modeling approach

We investigate the scaling behavior of the maximal Lyapunov exponent in chaotic systems with time delay. In the large-delay limit, it is known that one can distinguish between strong and weak chaos depending on the delay scaling, analogously to strong and weak instabilities for steady states and periodic orbits. Here we show that the Lyapunov exponent of chaotic systems shows significant differences in its scaling behavior compared to constant or periodic dynamics due to fluctuations in the linearized equations of motion. We reproduce the chaotic scaling properties with a linear delay system with multiplicative noise. We further derive analytic limit cases for the stochastic model illustrating the mechanisms of the emerging scaling laws.