Numerical study of extreme events in a laser diode with phase-conjugate optical feedback

Extreme events in gene regulatory networks with time-delays

This work explores distinct complex dynamics of simplified two nodes of coupled gene regulatory networks with multiple delays in two self-inhibitory and mutually activated genes. We have identified the emergence of extreme events within a specific range of system parameter values. A detailed analysis of the time delay-induced emergence of extreme events is illustrated using bifurcation analysis, two-parameter phase diagrams, return maps, temporal plots, and probability density functions. The reasons behind the advent of extreme events are discussed in detail, with possible analogies to simplified two nodes of gene regulatory networks. The occasional large-amplitude bursting originated in the system via interior crisis-induced intermittency, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic intermittency routes. Additionally, we have used various recurrence quantification statistical measures, such as mean recurrence time, determinism, and recurrence time entropy, to describe the transition from periodic or chaotic to unforeseen large deviations. Our approach shows that the sudden surge of variance and mean recurrence time at the transition points can be used as a new metric to detect the critical transitions of distinct extreme bursting events. The comprehensive overview of the interaction between gene regulatory networks, with insights into the formation of unusual dynamics, is beneficial to grasping different neuronal diseases.

Extreme events in frequency-swept semiconductor lasers

We analyze a delay differential equation model for a frequency-swept semiconductor laser and demonstrate existence of extreme events in its dynamics, with probabilities heavily dependent on the sweep rate. The extreme events appear even in absence of any noise in the system and do not exhibit significant dependence on its presence. We investigate the problem numerically and show that intensity dynamics of these events are highly localized in the filter detuning space. Overlaying it with the structure of steady-state and periodic solutions of the static system, we show that the dynamics is governed by attraction to these periodic solutions and such extreme events occur as a result of passing through the region of stable high-intensity bridge of periodic solutions.

Enhanced extreme events in three cascade-coupled semiconductor lasers

Extreme events (EEs) are rare and unpredictable, as have been observed in nature. Up to now, manipulating EEs has remained a challenge. Here, we experimentally observe the enhancement of EEs in a three cascade-coupled semiconductor laser system. Specifically, a continuous-wave optical injection semiconductor laser acts as the chaotic source with rare EEs, which is subsequently injected into a second laser for increasing the number of EEs. Interestingly, we find that the number and region size of EEs can be further enhanced by sequentially injecting into a third laser, i.e., a cascade-injection structure. Our experimental observations are in good agreement with the numerical results, which indicate that EEs can be significantly enhanced in wide injection parameter space due to the cascade-injection effect. Furthermore, our simulations show that the evoluation of the regions with enhanced EEs may be associated with the noise considered.

Different routes to large-intensity pulses in Zeeman laser model

In this study, we report a rich variety of large-intensity pulses exhibited by a Zeeman laser model. The instabilities in the system occur via three different dynamical processes, such as quasiperiodic intermittency, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion to chaos followed by an interior crisis. This Zeeman laser model is more capable of exploring the major possible types of instabilities when changing a specific system's parameter in a particular range. We exemplified distinct dynamical transitions of the Zeeman laser model. The statistical measures reveal the appearance of the low probability of large-intensity pulses above the qualifier threshold value. Moreover, they seem to follow an exponential decay that shows a Poisson-like distribution. The impact of noise and time delay effects have been analyzed near the transition point of the system.

Rogue wave generation using a chaotic semiconductor laser with energy redistribution

We demonstrate for the first time that optical rogue waves (RWs) can be generated using a chaotic semiconductor laser with energy redistribution. Chaotic dynamics are numerically generated using the rate equation model of an optically injected laser. The chaotic emission is then sent to an energy redistribution module (ERM) that consists of a temporal phase modulation and a dispersive propagation. The process enables a temporal energy redistribution of the chaotic emission waveforms, where coherent summation of consecutive laser pulses leads to random generation of giant intensity pulses. Efficient generation of optical RWs are numerically demonstrated by varying the ERM operating parameters in the entire injection parameter space. The effects of the laser spontaneous emission noise on the generation of RWs are further investigated. The RW generation approach offers a relatively high flexibility and tolerance in the choice of ERM parameters according to the simulation results.

Controlling the likelihood of extreme events in an optically pumped spin-VCSEL via chaotic optical injection

We report on the manipulation of extreme events (EEs) in a slave spin-polarized vertical-cavity surface-emitting laser (spin-VCSEL) subject to chaotic optical injection from a master spin-VCSEL. The master laser is free-running but yielding a chaotic regime with obvious EEs, while the slave laser originally (i.e., without external injection) operates in either continuous-wave (CW), period-one (P1), period-two (P2), or a chaotic state. We systematically investigate the influence of injection parameters, i.e., injection strength and frequency detuning, on the characteristics of EEs. We find that injection parameters can regularly trigger, enhance, or suppress the relative number of EEs in the slave spin-VCSEL, where the large ranges of enhanced vectorial EEs and average intensity of both vectorial and scalar EEs can be achieved with suitable parameter conditions. Moreover, with the help of two-dimensional correlation maps, we confirm that the probability of occurrence of EEs in the slave spin-VCSEL is associated with the injection locking regions, outside which enhanced relative number of EEs regions can be obtained and expanded with augmenting the complexity of the initial dynamic state of the slave spin-VCSEL.

Emergence of extreme events in a quasiperiodic oscillator

Extreme events are unusual and rare large-amplitude fluctuations can occur unexpectedly in nonlinear dynamical systems. Events above the extreme event threshold of the probability distribution of a nonlinear process characterize extreme events. Different mechanisms for the generation of extreme events and their prediction measures have been reported in the literature. Based on the properties of extreme events, such as those that are rare in the frequency of occurrence and extreme in amplitude, various studies have shown that extreme events are both linear and nonlinear in nature. Interestingly, in this Letter, we report on a special class of extreme events which are nonchaotic and nonperiodic. These nonchaotic extreme events appear in between the quasiperiodic and chaotic dynamics of the system. We report the existence of such extreme events with various statistical measures and characterization techniques.

Transition to hyperchaos and rare large-intensity pulses in Zeeman laser

A discontinuous transition to hyperchaos is observed at discrete critical parameters in the Zeeman laser model for three well known nonlinear sources of instabilities, namely, quasiperiodic breakdown to chaos followed by interior crisis, quasiperiodic intermittency, and Pomeau-Manneville intermittency. Hyperchaos appears with a sudden expansion of the attractor of the system at a critical parameter for each case and it coincides with triggering of occasional and recurrent large-intensity pulses. The transition to hyperchaos from a periodic orbit via Pomeau-Manneville intermittency shows hysteresis at the critical point, while no hysteresis is recorded during the other two processes. The recurrent large-intensity pulses show characteristic features of extremes with their height larger than a threshold and the probability of a rare occurrence. The phenomenon is robust to weak noise although the critical parameter of transition to hyperchaos shifts with noise strength. This phenomenon appears as common in many low dimensional systems as reported earlier by Chowdhury et al. [Phys. Rep. 966, 1-52 (2022)], there the emergent large-intensity events or extreme events dynamics have been recognized simply as chaotic in nature although the temporal dynamics shows occasional large deviations from the original chaotic state in many examples. We need a new metric, in the future, that would be able to classify such significantly different dynamics and distinguish from chaos.

Extreme events in a complex network: Interplay between degree distribution and repulsive interaction

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.

Extreme events in two laterally-coupled semiconductor lasers

Rogue waves (RWs) are extreme and rare waves that emerge unexpectedly in many natural systems and their formation mechanism and prediction have been extensively studied. Here, we numerically demonstrate the appearance of extreme events (EEs) for the first time, to the best of our knowledge, in the chaotic regimes of a two-element coupled semiconductor laser array. Based on coupled-mode theory, we characterize the occurrence of EEs by calculating the probability distribution, which confirms the RW-type feature of the intensity pulses, i.e., non-Gaussian distribution. Combining with the results of the 0-1 test for chaos, we confirm that EEs originate from deterministic nonlinearities in coupled semiconductor laser systems. We show that EEs can be predicted with a long anticipation time. Furthermore, simulation results manifest that the occurrence probability of EEs can be flexibly tuned by tailoring the coupling parameter space. With the help of two-dimension maps, the effects of key parameters, i.e., the waveguide structure and the pump level, on the formation of EEs are discussed systematically. This work provides a new platform for the research of EEs in a highly integrated structure and opens up a novel investigation field for coupled semiconductor laser arrays.

Extreme events in optically pumped spin-VCSELs

Extreme events (EEs) are predicted for the first time, to the best of our knowledge, in the chaotic dynamics of a free-running spin-polarized vertical-cavity surface-emitting laser (spin-VCSEL). Here, we not only show two types of EEs, i.e., vectorial and scalar EEs separately corresponding to the emission of a high-power pulse in both linear polarizations (LPs) simultaneously and in single LP, but we also observe a new EE type that only occurs in total intensity. We also confirm that the observed EEs follow similar statistical distributions to conventional rogue waves. Moreover, the effects of pump power and pump ellipticity on the generation of EEs are analyzed. Finally, we compare free-running and optical feedback spin-VCSELs, which provides more insights into the study of EEs. More importantly, this work offers a novel platform for the study of EEs with a simple structure and opens up new research fields into spin-VCSELs.

Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known; however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

Classification strategies in machine learning techniques predicting regime changes and durations in the Lorenz system

In this paper, we use machine learning strategies aiming to predict chaotic time series obtained from the Lorenz system. Such strategies prove to be successful in predicting the evolution of dynamical variables over a short period of time. Transitions between the regimes and their duration can be predicted with great accuracy by means of counting and classification strategies, for which we train multi-layer perceptron ensembles. Even for the longest regimes the occurrences and duration can be predicted. We also show the use of an echo state network to generate data of the time series with an accuracy of up to a few hundreds time steps. The ability of the classification technique to predict the regime duration of more than 11 oscillations corresponds to around 10 Lyapunov times.

Rare events in mixed-mode oscillations from weakly coupled lasers

We study a dynamical system consisting of two mutually coupled molecular lasers, each of which shows mixed-mode oscillations and chaos when uncoupled. The type of coupling, incoherent laser interaction via saturable absorbers is an example of inhibitory nonlinear coupling, which is also found in Hodgkin-Huxley models that describe action potentials in neurons. We have carried out extensive numerical bifurcation analysis and numerical simulations to show that for small-enough coupling, well below the chaotic synchronization threshold, the presence of distinctive resonances in a symmetric mirror configuration of the system generates a type of rare events characterized by very small amplitudes. When this symmetry is broken by introducing a relatively small difference between the lasers pump parameters near an in-phase Hopf bifurcation, we observe extreme rare events (rogue waves) in one of the lasers. In this case the outliers deviate from power-law distributions and are reminiscent of those known as dragon kings. We consider the conditions for both types of rare events to occur, their origin, as well as relevant statistical features.

Improving clinical refractive results of cataract surgery by machine learning

AIM

To evaluate the potential of the Support Vector Machine Regression model (SVM-RM) and Multilayer Neural Network Ensemble model (MLNN-EM) to improve the intraocular lens (IOL) power calculation for clinical workflow.

BACKGROUND

Current IOL power calculation methods are limited in their accuracy with the possibility of decreased accuracy especially in eyes with an unusual ocular dimension. In case of an improperly calculated power of the IOL in cataract or refractive lens replacement surgery there is a risk of re-operation or further refractive correction. This may create potential complications and discomfort for the patient.

METHODS

A dataset containing information about 2,194 eyes was obtained using data mining process from the Electronic Health Record (EHR) system database of the Gemini Eye Clinic. The dataset was optimized and split into the selection set (used in the design for models and training), and the verification set (used in the evaluation). The set of mean prediction errors (PEs) and the distribution of predicted refractive errors were evaluated for both models and clinical results (CR).

RESULTS

Both models performed significantly better for the majority of the evaluated parameters compared with the CR. There was no significant difference between both evaluated models. In the ±0.50 D PE category both SVM-RM and MLNN-EM were slightly better than the Barrett Universal II formula, which is often presented as the most accurate calculation formula.

CONCLUSION

In comparison to the current clinical method, both SVM-RM and MLNN-EM have achieved significantly better results in IOL calculations and therefore have a strong potential to improve clinical cataract refractive outcomes.

Predictability of the onset of spiking and bursting in complex chemical reactions

For three complex chemical reactions displaying intricate dynamics, we assess the effectiveness of a recently proposed quantitative method to forecast bursting and large spikes, i.e. extreme events. Specifically, we consider predicting extreme events in (i) a copper dissolution model where Bassett and Hudson experimentally observed homoclinic (Shilnikov) chaos, (ii) a model derived from the mass action law of chemical kinetics, and (iii) an autocatalator model. For these systems, we describe how the alignment of Lyapunov vectors can be used to predict the imminence of large-amplitude events and the onset of complex dynamics in chaotic time-series of observables.

Dragon-king-like extreme events in coupled bursting neurons

We present evidence of extreme events in two Hindmarsh-Rose (HR) bursting neurons mutually interacting via two different coupling configurations: chemical synaptic- and gap junctional-type diffusive coupling. A dragon-king-like probability distribution of the extreme events is seen for combinations of synaptic coupling where small- to medium-size events obey a power law and the larger events that cross an extreme limit are outliers. The extreme events originate due to instability in antiphase synchronization of the coupled systems via two different routes, intermittency and quasiperiodicity routes to complex dynamics for purely excitatory and inhibitory chemical synaptic coupling, respectively. For a mixed type of inhibitory and excitatory chemical synaptic interactions, the intermittency route to extreme events is only seen. Extreme events with our suggested distribution is also seen for gap junctional-type diffusive, but repulsive, coupling where the intermittency route to complexity is found. A simple electronic experiment using two diffusively coupled analog circuits of the HR neuron model, but interacting in a repulsive way, confirms occurrence of the dragon-king-like extreme events.

Extreme events in the forced Liénard system

We observe extremely large amplitude intermittent spikings in a dynamical variable of a periodically forced Liénard-type oscillator and characterize them as extreme events, which are rare, but recurrent and larger in amplitude than a threshold. The extreme events occur via two processes, an interior crisis and intermittency. The probability of occurrence of the events shows a long-tail distribution in both the cases. We provide evidence of the extreme events in an experiment using an electronic analog circuit of the Liénard oscillator that shows good agreement with our numerical results.

Mapping of external cavity modes for a laser diode subject to phase-conjugate feedback

We numerically investigate the dynamics of a semiconductor laser subject to phase-conjugate optical feedback. We explore the effects of the laser model and feedback parameters for the generation of time-periodic oscillations of the output power at harmonics of the external cavity frequency, i.e., dynamical solutions that have been named external cavity modes. We point out that both the experimentally tunable and other parameters have an influence on the frequency of such dynamics. Since the delay has to exist, it is not the relevant parameter as we show that the feedback rate fixes the frequency of the periodic self-pulsations. The interaction length of the crystal and the ratio between carrier and photon lifetimes tend to filter out high frequencies as they increase. Finally, the linewidth enhancement factor unlocks high frequencies as it increases. We conclude by providing a situation which leads to periodic solutions with higher frequencies using a set of realistic values of parameters.

Extreme events induced by collisions in a forced semiconductor laser

We report on the experimental study of an optically driven multimode semiconductor laser with a 1 m cavity length. We observed a spatiotemporal regime where real-time measurements reveal very high-intensity peaks in the laser field. Such a regime, which coexists with the locked state and with stable phase solitons, is characterized by the emergence of extreme events that produce heavy tail statistics in the probability density function. We interpret the extreme events as collisions of spatiotemporal structures with opposite chirality. Numerical simulations of the semiconductor laser model, showing very similar dynamical behavior, substantiate our evidences and corroborate the description of interactions such as collisions between phase solitons and transient structures with different phase rotations.

Vectorial extreme events in VCSEL polarization dynamics

We report on the occurrence of extreme events (EEs) in the polarization dynamics of vertical cavity surface emitting lasers with optical feedback. We have identified two types of EEs based on numerical simulations: vectorial and scalar events corresponding, respectively, to the emission of a high-power pulse in both linear polarizations simultaneously and in single linear polarization. We show that these two types of events follow the typical statistics of rogue waves. Finally, we observe that an emission in both polarizations leads to a larger generation rate of EEs with a saturation over a wide range of feedback strength by comparison to a single-polarization mode emission.

Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?

We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.

Correlations in electrically coupled chaotic lasers

We show how two electrically coupled semiconductor lasers having optical feedback can present simultaneous antiphase correlated fast power fluctuations, and strong in-phase synchronized spikes of chaotic power drops. This quite counterintuitive phenomenon is demonstrated experimentally and confirmed by numerical solutions of a deterministic dynamical system of rate equations. The occurrence of negative and positive cross correlation between parts of a complex system according to time scales, as proved in our simple arrangement, is relevant for the understanding and characterization of collective properties in complex networks.

Extreme events induced by spatiotemporal chaos in experimental optical patterns

Extreme events such as rogue waves are often associated with the merging of coherent structures. We report on experimental results in the physics of extreme events emerging in a liquid-crystal light valve subjected to optical feedback, and we establish the relation of this phenomenon with the appearance of spatiotemporal chaos. This system, under particular conditions, exhibits stationary roll patterns that can be destabilized into quasi-periodic and chaotic textures when control parameters are properly modified. We have identified the parameter regions where extreme fluctuations of the amplitude can emerge and established their origin through its direct relation with the experimental largest Lyapunov exponents, the proportion of extreme events, and the normed kurtosis.

Low-frequency fluctuations in an external-cavity laser leading to extreme events

We experimentally investigate the dynamical regimes of a laser diode subject to external optical feedback in light of extreme-event (EE) analysis. We observe EEs in the low-frequency fluctuations (LFFs) regime. This number decreases to negligible values when the laser transitions towards fully developed coherence collapse as the injection current is increased. Moreover, we show that EEs observed in the LFF regime are linked to high-frequency pulsing events observed after a power dropout. Finally, we prove experimentally that the observation of EEs in the LFF regimes is robust to changes in operational parameters.