Noise-induced chaos in an optically injected semiconductor laser model

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.

Chaotic dimension enhancement by optical injection into a semiconductor laser under feedback

Optical injection into a chaotic laser under feedback is investigated for dimension enhancement. Although injecting a solitary laser is known to be low-dimensional, injecting the laser under feedback is found to enhance the correlation dimension D₂ in experiments. Using an exceptionally large data size with a very large reconstruction embedding dimension, efficient computation is enabled by averaging over many short segments to carefully estimate D₂. The dimension enhancement can be achieved together with time-delay signature suppression. The enhancement of D₂ as a fundamental geometric quantifier of attractors is useful in applications of chaos.

High-Resolution Simulation of Externally Injected Lasers Revealing a Large Regime of Noise-Induced Chaos

We present comprehensive numerically simulated scans of the spectral evolution of the output from a single-mode semiconductor laser diode undergoing external light injection. The spectral scans are helpful to understand the different regimes of operation as well as the system evolution between each state: i.e., locked state, four-wave mixing, pulsations, chaos. We find that, when under strong injection, when the injected power equals about half of the laser power, two distinct regions of chaotic behaviour are observed. One of the chaotic regions arises due to the usual period-doubling route to chaos; the other chaotic region is a blurring of what would be higher-order period pulsations whose periodicity is broken by spontaneous emission and the laser spectrum is chaotic. Eliminating spontaneous emission in our simulations confirms the latter chaotic region becomes a region with higher-order pulsations.

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.

Optical feedback stabilization of photonic microwave generation using period-one nonlinear dynamics of semiconductor lasers

Effects of optical feedback on period-one nonlinear dynamics of an optically injected semiconductor laser are numerically investigated. The optical feedback can suppress the period-one dynamics and excite other more complex dynamics if the feedback level is high except for extremely short feedback delay times. Within the range of the period-one dynamics, however, the optical feedback can stabilize the period-one dynamics in such a manner that significant reduction of microwave linewidth and phase noise is achieved, up to more than two orders of magnitude. A high feedback level and/or a long feedback delay time are generally preferred for such microwave stabilization. However, considerably enhanced microwave linewidth and phase noise happen periodically at certain feedback delay times, which is strongly related to the behavior of locking between the period-one microwave oscillation and the feedback loop modes. The extent of these enhancements reduces if the feedback level is high. While the microwave frequency only slightly changes with the feedback level, it red-shifts with the feedback delay time before an abrupt blue-shift occurs periodically. With the presence of the laser intrinsic noise, frequency jitters occur around the feedback delay times leading to the abrupt blue-shifts, ranging from the order of 0.1 GHz to the order of 1 GHz.

Chaos and noise

Simple dynamical systems--with a small number of degrees of freedom--can behave in a complex manner due to the presence of chaos. Such systems are most often (idealized) limiting cases of more realistic situations. Isolating a small number of dynamical degrees of freedom in a realistically coupled system generically yields reduced equations with terms that can have a stochastic interpretation. In situations where both noise and chaos can potentially exist, it is not immediately obvious how Lyapunov exponents, key to characterizing chaos, should be properly defined. In this paper, we show how to do this in a class of well-defined noise-driven dynamical systems, derived from an underlying Hamiltonian model.

Escape statistics for parameter sweeps through bifurcations

We consider the dynamics of systems undergoing parameter sweeps through bifurcation points in the presence of noise. Of interest here are local codimension-one bifurcations that result in large excursions away from an operating point that is transitioning from stable to unstable during the sweep, since information about these "escape events" can be used for system identification, sensing, and other applications. The analysis is based on stochastic normal forms for the dynamic saddle-node and subcritical pitchfork bifurcations with a time-varying bifurcation parameter and additive noise. The results include formulation and numerical solution for the distribution of escape events in the general case and analytical approximations for delayed bifurcations for which escape occurs well beyond the corresponding quasistatic bifurcation points. These bifurcations result in amplitude jumps encountered during parameter sweeps and are particularly relevant to nano- and microelectromechanical systems, for which noise can play a significant role.

Multiscale analysis of biological data by scale-dependent lyapunov exponent

Physiological signals often are highly non-stationary (i.e., mean and variance change with time) and multiscaled (i.e., dependent on the spatial or temporal interval lengths). They may exhibit different behaviors, such as non-linearity, sensitive dependence on small disturbances, long memory, and extreme variations. Such data have been accumulating in all areas of health sciences and rapid analysis can serve quality testing, physician assessment, and patient diagnosis. To support patient care, it is very desirable to characterize the different signal behaviors on a wide range of scales simultaneously. The Scale-Dependent Lyapunov Exponent (SDLE) is capable of such a fundamental task. In particular, SDLE can readily characterize all known types of signal data, including deterministic chaos, noisy chaos, random 1/f(α) processes, stochastic limit cycles, among others. SDLE also has some unique capabilities that are not shared by other methods, such as detecting fractal structures from non-stationary data and detecting intermittent chaos. In this article, we describe SDLE in such a way that it can be readily understood and implemented by non-mathematically oriented researchers, develop a SDLE-based consistent, unifying theory for the multiscale analysis, and demonstrate the power of SDLE on analysis of heart-rate variability (HRV) data to detect congestive heart failure and analysis of electroencephalography (EEG) data to detect seizures.

Complexity measures of brain wave dynamics

To understand the nature of brain dynamics as well as to develop novel methods for the diagnosis of brain pathologies, recently, a number of complexity measures from information theory, chaos theory, and random fractal theory have been applied to analyze the EEG data. These measures are crucial in quantifying the key notions of neurodynamics, including determinism, stochasticity, causation, and correlations. Finding and understanding the relations among these complexity measures is thus an important issue. However, this is a difficult task, since the foundations of information theory, chaos theory, and random fractal theory are very different. To gain significant insights into this issue, we carry out a comprehensive comparison study of major complexity measures for EEG signals. We find that the variations of commonly used complexity measures with time are either similar or reciprocal. While many of these relations are difficult to explain intuitively, all of them can be readily understood by relating these measures to the values of a multiscale complexity measure, the scale-dependent Lyapunov exponent, at specific scales. We further discuss how better indicators for epileptic seizures can be constructed.

Quasipotential approach to critical scaling in noise-induced chaos

When a dynamical system exhibits transient chaos and a nonchaotic attractor, as in a periodic window, noise can induce a chaotic attractor. In particular, when the noise amplitude exceeds a critical value, the largest Lyapunov exponent of the attractor of the system starts to increase from zero. While a scaling law for the variation of the Lyapunov exponent with noise was uncovered previously, it is mostly based on numerical evidence and a heuristic analysis. This paper presents a more general approach to the scaling law, one based on the concept of quasipotentials. Besides providing deeper insights into the problem of noise-induced chaos, the quasipotential approach enables previously unresolved issues to be addressed. The fractal properties of noise-induced chaotic attractors and applications to biological systems are also discussed.

Characterizing heart rate variability by scale-dependent Lyapunov exponent

Previous studies on heart rate variability (HRV) using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Especially the issue of whether normal HRV is chaotic or stochastic remains highly controversial. Here, we employ a new multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize HRV. SDLE has been shown to readily characterize major models of complex time series including deterministic chaos, noisy chaos, stochastic oscillations, random 1/f processes, random Levy processes, and complex time series with multiple scaling behaviors. Here we use SDLE to characterize the relative importance of nonlinear, chaotic, and stochastic dynamics in HRV of healthy, congestive heart failure, and atrial fibrillation subjects. We show that while HRV data of all these three types are mostly stochastic, the stochasticity is different among the three groups.

Using chaos for remote sensing of laser radiation

An idea is proposed for detecting a weak laser signal from a remote source in the presence of strong background noise. The scheme exploits dynamical nonlinearities arising from heterodyning signal and reference fields inside an active reference laser cavity. This paper shows that for certain reference laser configurations, the resulting bifurcations in the reference laser may be used as warning of irradiation by a laser source.

Chaotic component obscured by strong periodicity in voice production system

The effect of glottal aerodynamics in producing the nonlinear characteristics of voice is investigated by comparing the outputs of the asymmetric composite model and the two-mass model. The two-mass model assumes the glottal airflow to be laminar, nonviscous, and incompressible. In this model, when the asymmetric factor is decreased from 0.65 to 0.35, only 1:1 and 1:2 modes are detectable. However, with the same parameters, four vibratory modes (1:1, 1:2, 2:4, 2:6) are found in the asymmetric composite model using the Navier-Stokes equations to describe the complex aerodynamics in the glottis. Moreover, the amplitude of the waveform is modulated by a small-amplitude noiselike series. The nonlinear detection method reveals that this noiselike modulation is not random, but rather it is deterministic chaos. This result agrees with the phenomenon often seen in voice, in which the voice signal is strongly periodic but modulated by a small-amplitude chaotic component. The only difference between the two-mass model and the composite model is in their descriptions of glottal airflow. Therefore, the complex aerodynamic characteristics of glottal airflow could be important in generating the nonlinear dynamic behavior of voice production, including bifurcation and a small-amplitude chaotic component obscured by strong periodicity.

Distinguishing chaos from noise by scale-dependent Lyapunov exponent

Time series from complex systems with interacting nonlinear and stochastic subsystems and hierarchical regulations are often multiscaled. In devising measures characterizing such complex time series, it is most desirable to incorporate explicitly the concept of scale in the measures. While excellent scale-dependent measures such as epsilon entropy and the finite size Lyapunov exponent (FSLE) have been proposed, simple algorithms have not been developed to reliably compute them from short noisy time series. To promote widespread application of these concepts, we propose an efficient algorithm to compute a variant of the FSLE, the scale-dependent Lyapunov exponent (SDLE). We show that with our algorithm, the SDLE can be accurately computed from short noisy time series and readily classify various types of motions, including truly low-dimensional chaos, noisy chaos, noise-induced chaos, random 1/f alpha and alpha-stable Levy processes, stochastic oscillations, and complex motions with chaotic behavior on small scales but diffusive behavior on large scales. To our knowledge, no other measures are able to accurately characterize all these different types of motions. Based on the distinctive behaviors of the SDLE for different types of motions, we propose a scheme to distinguish chaos from noise.

Reliability of the 0-1 test for chaos

In time series analysis, it has been considered of key importance to determine whether a complex time series measured from the system is regular, deterministically chaotic, or random. Recently, Gottwald and Melbourne have proposed an interesting test for chaos in deterministic systems. Their analyses suggest that the test may be universally applicable to any deterministic dynamical system. In order to fruitfully apply their test to complex experimental data, it is important to understand the mechanism for the test to work, and how it behaves when it is employed to analyze various types of data, including those not from clean deterministic systems. We find that the essence of their test can be described as to first constructing a random walklike process from the data, then examining how the variance of the random walk scales with time. By applying the test to three sets of data, corresponding to (i) 1/falpha noise with long-range correlations, (ii) edge of chaos, and (iii) weak chaos, we show that the test mis-classifies (i) both deterministic and weakly stochastic edge of chaos and weak chaos as regular motions, and (ii) strongly stochastic edge of chaos and weak chaos, as well as 1/falpha noise as deterministic chaos. Our results suggest that, while the test may be effective to discriminate regular motion from fully developed deterministic chaos, it is not useful for exploratory purposes, especially for the analysis of experimental data with little a priori knowledge. A few speculative comments on the future of multiscale nonlinear time series analysis are made.

Quasiperiodic route to chaotic dynamics of internet transport protocols

We show that the dynamics of transmission control protocol (TCP) may often be chaotic via a quasiperiodic route consisting of more than two independent frequencies, by employing a commonly used ns-2 network simulator. To capture the essence of the additive increase and multiplicative decrease mechanism of TCP congestion control, and to qualitatively describe why and when chaos may occur in TCP dynamics, we develop a 1D discrete map. The relevance of these chaotic transport dynamics to real Internet connections is discussed.

Dynamics of two mutually coupled semiconductor lasers: instantaneous coupling limit

We consider two semiconductor lasers coupled face to face under the assumption that the delay time of the injection is small. The model under consideration consists of two coupled rate equations, which approximate the coupled Lang-Kobayashi system as the delay becomes small. We perform a detailed study of the synchronized and antisynchronized solutions for the case of identical systems and compare results from two models: with the delay and with instantaneous coupling. The bifurcation analysis of systems with detuning reveals that self-pulsations appear via bifurcations of stationary (i.e., continuous wave) solutions. We discover the connection between stationary solutions in systems with detuning and synchronous (also antisynchronous) solutions of coupled identical systems. We also identify a codimension 2 bifurcation point as an organizing center for the emergence of chaotic behavior.

Effect of noise on the neutral direction of chaotic attractor

A chaotic attractor from a deterministic flow must necessarily possess a neutral direction, as characterized by a null Lyapunov exponent. We show that for a wide class of chaotic attractors, particularly those having multiple scrolls in the phase space, the existence of the neutral direction can be extremely fragile in the sense that it is typically destroyed by noise of arbitrarily small amplitude. A universal scaling law quantifying the increase of the Lyapunov exponent with noise is obtained. A way to observe the scaling law in experiments is suggested.

Experimental characterization of transition to chaos in the presence of noise

Transition to chaos in the presence of noise is an important problem in nonlinear and statistical physics. Recently, a scaling law has been theoretically predicted which relates the Lyapunov exponent to the noise variation near the transition. Here we present experimental observation of noise-induced chaos in an electronic circuit and obtain the fundamental scaling law characterizing the transition. The experimentally obtained scaling exponent agrees very well with that predicted by theory.

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

Noise-induced Hopf-bifurcation-type sequence and transition to chaos in the lorenz equations

We study the effects of noise on the Lorenz equations in the parameter regime admitting two stable fixed point solutions and a strange attractor. We show that noise annihilates the two stable fixed point attractors and evicts a Hopf-bifurcation-like sequence and transition to chaos. The noise-induced oscillatory motions have very well defined period and amplitude, and this phenomenon is similar to stochastic resonance, but without a weak periodic forcing. When the noise level exceeds certain threshold value but is not too strong, the noise-induced signals enable an objective computation of the largest positive Lyapunov exponent, which characterize the signals to be truly chaotic.

Chaotic vibration induced by turbulent noise in a two-mass model of vocal folds

The contribution of turbulent noise was modeled in symmetric vocal folds. A two-mass model was used to simulate irregular vocal fold vibrations. The threshold values of system parameters to produce irregular vibrations were decreased as a result of turbulent airflow. Periodic vibrations were then driven into the regions of irregular vibrations. Using nonlinear dynamics including Poincaré map and Lyapunov exponents, irregular vibrations were demonstrated as chaos. For the deterministic vocal-fold model with noise free and steady airflow, a fine period-doubling bifurcation cascade was shown in a bifurcation diagram. However, turbulent noise added to the vocal-fold model would induce chaotic vibrations, broaden the regions of irregular vocal fold vibrations, and inhibit the fine period-doubling bifurcations in the bifurcation diagrams. The perturbations from neurological and biomechanical effects were simulated as a random variation of the vocal fold stiffness. Turbulent noise as an external random source, as well as random stiffness perturbation as an internal random source, played important roles in the presence of irregular vocal fold vibrations.