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In real-world dynamical systems, technical limitations may prevent complete access to their dynamical variables. Such a lack of information may cause significant problems, especially when monitoring or controlling the dynamics of the system is required or when decisions need to be taken based on the dynamical state of the system. Cross-predicting the missing data is, therefore, of considerable interest. Here, we use a machine learning algorithm based on reservoir computing to perform cross-prediction of unknown variables of a chaotic dynamical laser system. In particular, we chose a realistic model of an optically injected single-mode semiconductor laser. While the intensity of the laser can often be acquired easily, measuring the phase of the electric field and the carriers in real time, although possible, requires a more demanding experimental scheme. We demonstrate that the dynamics of two of the three dynamical variables describing the state of the laser can be reconstructed accurately from the knowledge of only one variable, if our algorithm has been trained beforehand with all three variables for a limited period of time. We analyze the accuracy of the method depending on the parameters of the laser system and the reservoir. Finally, we test the robustness of the cross-prediction method when adding noise to the time series. The suggested reservoir computing state observer might be used in many applications, including reconstructing time series, recovering lost time series data and testing data encryption security in cryptography based on chaotic synchronization of lasers. [ABSTRACT FROM AUTHOR]

In this paper we present a unified framework for extreme learning machines and reservoir computing (echo state networks), which can be physically implemented using a single nonlinear neuron subject to delayed feedback. The reservoir is built within the delay-line, employing a number of "virtual" neurons. These virtual neurons receive random projections from the input layer containing the information to be processed. One key advantage of this approach is that it can be implemented efficiently in hardware. We show that the reservoir computing implementation, in this case optoelectronic, is also capable to realize extreme learning machines, demonstrating the unified framework for both schemes in software as well as in hardware. [ABSTRACT FROM AUTHOR]

We perform phase-space tomography of semiconductor laser dynamics by simultaneous experimental determination of optical intensity, frequency, and population inversion with high temporal resolution. We apply this technique to a laser with delayed feedback, serving as prominent example for high-dimensional chaotic dynamics and as model system for fundamental investigations of complex systems. Our approach allows us to explore so far unidentified trajectories in phase space and identify the underlying physical mechanism. [ABSTRACT FROM AUTHOR]

In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of semiconductor lasers subject to time-delayed optical feedback from the perspective of dynamical self-injection locking. Based on the Lang-Kobayashi model, we perform an analysis of the well-known Low Frequency Fluctuations (LFFs) in the frequencyintensity plane. Moreover, we investigate a recently found dynamical regime of fragmented LFFs by means of a locking-range analysis, spectral comparison and precursor pulse identification. We show that LFF dynamics can be explained by dynamical optical injection locking due to the delayed optical feedback. Moreover, the fragmented LFFs occur due to a re-injection locking induced by a particular optical pulse structure in the chaotic feedback dynamics. This is corroborated by experiments with a semiconductor laser experiencing delayed feedback from an optical fiber loop. The dynamical nature of the feedback injection results in an eventual loss, but also possible regaining, of the locking, explaining the recently observed phenomenon of fragmented LFFs. [ABSTRACT FROM AUTHOR]

We show experimentally that two semiconductor lasers mutually coupled via a passive relay fiber loop exhibit chaos synchronization at zero lag, and study how this synchronized regime is lost as the lasers' pump currents are increased. We characterize the synchronization properties of the system with high temporal resolution in two different chaotic regimes, namely, low-frequency fluctuations and coherence collapse, identifying significant differences between them. In particular, a marked decrease in synchronization quality develops as the lasers enter the coherence collapse regime. Our high-resolution measurements allow us to establish that synchronization loss is associated with bubbling events, the frequency of which increases with increasing pump current. [ABSTRACT FROM AUTHOR]