Nonlinear-time-series analysis of chaotic laser dynamics

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.

Fluid flow signals processing based on fractional Fourier transform in a stirred tank reactor

The analysis of fluid flow signals and the characterization of fluid flow behavior are of great importance for two-phase flow studies. In this work, the fractional Fourier transform (FRFT), which was based on the optimum order calculated by stepping search method, was proposed to extract the characteristics of fluid flow signals. Meanwhile, the largest Lyapunov exponent (LLE), which is an indication of the chaotic degree of mixing process, was adopted to quantify fluid flow behavior. The maximum amplitude (MA) and LLE value were taken together to inquire into the relationship between the characteristics of fluid flow signals and the characterization of fluid flow behavior. In addition, differences between the two adjacent values (AD) and the maximum differences (MD) are employed to further analyze the differences in behavioral characterization with MA and LLE. The results show that the MA value performs the same increasing trend as the LLE value when the gas flow rate and agitation speed increase. AD and MD values of the MA are one to two orders of magnitude greater than those of the LLE. The eigenvalues (MA) solved by the FRFT method is facilitates capturing small changes owing to changes in external conditions. These findings can provide new ideas for the extraction and characterization of fluid flow behavior.

Time series analysis of spontaneous upper-extremity movements of premature infants with brain injuries

BACKGROUND AND PURPOSE

Comparisons of spontaneous movements of premature infants with brain injuries and those without brain injuries can provide insights into normal and abnormal processes in the ontogeny of motor development. In this study, the characteristics of spontaneous upper-extremity movements of premature infants with brain injuries and those without brain injuries were examined with time series analysis.

SUBJECTS

Participants were 7 premature infants with brain injuries and 7 matched, low-risk, premature infants at the age of 1 month after term.

METHODS

A triaxial accelerometer was used to measure upper-extremity limb acceleration in 3-dimensional space. Acceleration signals were recorded from the right wrist when the infant was in an active, alert state and lying in the supine position. The recording time was 200 seconds. The acceleration signal was sampled at a rate of 200 Hz. The acceleration time series data were analyzed by nonlinear analysis as well as linear analysis.

RESULTS

The nonlinear time series analysis indicated that spontaneous movements of premature infants have nonlinear, chaotic, dynamic characteristics. The movements of the infants with brain injuries were characterized by larger dimensionality, and they were more unstable and unpredictable than those of infants without brain injuries.

DISCUSSION AND CONCLUSION

As determined by nonlinear analysis, the spontaneous movements of the premature infants with brain injuries had the characteristics of increased disorganization compared with those of the infants without brain injuries. Infants with brain injuries may manifest problems with self-organization as a function of the coordination of subsystems. Physical therapists should be able to support interactions among the subsystems and promote self-organization of motor learning through the individualized provision of various sensorimotor experiences for infants.

Regularity of center-of-pressure trajectories depends on the amount of attention invested in postural control

The influence of attention on the dynamical structure of postural sway was examined in 30 healthy young adults by manipulating the focus of attention. In line with the proposed direct relation between the amount of attention invested in postural control and regularity of center-of-pressure (COP) time series, we hypothesized that: (1) increasing cognitive involvement in postural control (i.e., creating an internal focus by increasing task difficulty through visual deprivation) increases COP regularity, and (2) withdrawing attention from postural control (i.e., creating an external focus by performing a cognitive dual task) decreases COP regularity. We quantified COP dynamics in terms of sample entropy (regularity), standard deviation (variability), sway-path length of the normalized posturogram (curviness), largest Lyapunov exponent (local stability), correlation dimension (dimensionality) and scaling exponent (scaling behavior). Consistent with hypothesis 1, standing with eyes closed significantly increased COP regularity. Furthermore, variability increased and local stability decreased, implying ineffective postural control. Conversely, and in line with hypothesis 2, performing a cognitive dual task while standing with eyes closed led to greater irregularity and smaller variability, suggesting an increase in the “efficiency, or “automaticity” of postural control”. In conclusion, these findings not only indicate that regularity of COP trajectories is positively related to the amount of attention invested in postural control, but also substantiate that in certain situations an increased internal focus may in fact be detrimental to postural control.

Chaos control using notch filter feedback

A method for stabilizing periodic orbits and steady states of chaotic systems is presented using specifically filtered feedback signals. The efficiency of this control technique is illustrated with simulations (Rössler system, laser model) and a successful experimental application for stabilizing intensity fluctuations of an intracavity frequency-doubled Nd:YAG laser.

Correlation dimension signature of wideband chaos synchronization of semiconductor lasers

Chaos data analysis has been performed on the chaotic output power time series data from a synchronized transmitter-receiver pair of semiconductor lasers. The system uses an asymmetric, bidirectional coupling configuration between the master (transmitter), which is a laser diode with optical feedback, and a stand-alone slave semiconductor laser. The correlation dimension of the chaotic time series has a minimum value of 4, which was obtained from high-bandwidth measurements. The correlation dimensions for both the master and the synchronized slave are identical when the cross-correlation coefficient of the synchronized chaos is above 0.9. These results establish correlation dimension analysis as an effective tool for the determination of the quality of wideband chaos synchronization.

Stabilizing unstable steady states using multiple delay feedback control

Feedback control with different and independent delay times is introduced and shown to be an efficient method for stabilizing fixed points (equilibria) of dynamical systems. In comparison to other delay based chaos control methods multiple delay feedback control is superior for controlling steady states and works also for relatively large delay times (sometimes unavoidable in experiments due to system dead times). To demonstrate this approach for stabilizing unstable fixed points we present numerical simulations of Chua's circuit and a successful experimental application for stabilizing a chaotic frequency doubled Nd-doped yttrium aluminum garnet laser.

False neighbors and false strands: a reliable minimum embedding dimension algorithm

The time-delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension. We demonstrate a reliable method to estimate the minimum necessary embedding dimension that improves upon previous methods by correcting for systematic effects due to temporal oversampling, autocorrelation, and changing time lag. The method gives a sharp and reliable indication of the proper dimension. With little computational cost, the method also distinguish easily between infinite-dimensional colored noise-including noisy periodicity-and low-dimensional dynamics.

Modeling continuous processes from data

Experimental and simulated time series are necessarily discretized in time. However, many real and artificial systems are more naturally modeled as continuous-time systems. This paper reviews the major techniques employed to estimate a continuous vector field from a finite discrete time series. We compare the performance of various methods on experimental and artificial time series and explore the connection between continuous (differential) and discrete (difference equation) systems. As part of this process we propose improvements to existing techniques. Our results demonstrate that the continuous-time dynamics of many noisy data sets can be simulated more accurately by modeling the one-step prediction map than by modeling the vector field. We also show that radial basis models provide superior results to global polynomial models.

Embedding of multidimensional time-dependent observations

A method is proposed to reconstruct dynamic attractors by embedding of multivariate observations of dynamic nonlinear processes. The Takens embedding theory is combined with independent component analysis to transform the embedding into a vector space of linearly independent vectors (phase variables). The method is successfully tested against prediction of the unembedded state vector in two case studies of simulated chaotic processes.

Characterization and stabilization of the unstable fixed points of a frequency doubled Nd:YAG laser

We demonstrate the successful stabilization of type II chaos of a multimode, intracavity frequency doubled, diode-pumped Nd:YAG (neodymium-doped yttrium aluminum garnet) laser in experiment using an adaptive proportional feedback control. The two orthogonal polarized infrared intensities are fed back to the injection current of the pump diode. The stability properties of the stabilized unstable fixed points are investigated and exploited to explain the performance of our control scheme and to determine suitable measurement signals for the feedback control.

Numerical study of reverse period doubling route from chaos to stability in a two-mode intracavity doubled Nd-YAG laser

We have numerically studied the behavior of a two-mode Nd-YAG laser with an intracavity KTP crystal. It is found that when the parameter, which is a measure of the relative orientations of the KTP crystal with respect to the Nd-YAG crystal, is varied continuously, the output intensity fluctuations change from chaotic to stable behavior through a sequence of reverse period doubling bifurcations. The graph of the intensity in the X-polarized mode against that in the Y-polarized mode shows a complex pattern in the chaotic regime. The Lyapunov exponent is calculated for the chaotic and periodic regions. (c) 1999 American Institute of Physics.