Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events

Extreme events in two coupled chaotic oscillators

Since 1970, the Rössler system has remained as a considerably simpler and minimal-dimensional chaos serving system. Unveiling the dynamics of a system of two coupled chaotic oscillators that lead to the emergence of extreme events in the system is an engrossing and crucial scientific research area. Our present study focuses on the emergence of extreme events in a system of diffusively and bidirectionally two coupled Rössler oscillators and unraveling the mechanism behind the genesis of extreme events. We find the appearance of extreme events in three different observables: average velocity, synchronization error, and one transverse directional variable to the synchronization manifold. The emergence of extreme events in average velocity variables happens due to the occasional in-phase synchronization. The on-off intermittency plays a crucial role in the genesis of extreme events in the synchronization error dynamics and in the transverse directional variable to the synchronization manifold. The bubble transition of the chaotic attractor due to the on-off intermittency is illustrated for the transverse directional variable. We use generalized extreme value theory to study the statistics of extremes. The extreme events data sets concerning the average velocity variable follow a generalized extreme value distribution. The inter-event intervals of the extreme events in the average velocity variable spread well exponentially. The upshot of the interplay between the coupling strength and the frequency mismatch between the oscillators in the genesis of extreme events in the coupled system is depicted numerically.

Novel patterns in discrete Ikeda map: Quint points and complex non-quantum chirality

In this paper, the complex and dynamically rich distribution of stable phases in the well-known discrete Ikeda map is studied in detail. The unfolding patterns of these stable phases are described through three complementary stability diagrams: the Lyapunov stability diagram, the isoperiod stability diagram, and the isospike stability diagram. The adding-doubling complexification cascade and fascinating non-quantum chiral pairs are discovered, marking the first report of such structures in discrete mapping. The inherent symmetry of the Ikeda map also leads to the emergence of even more complex chiral formations. Additionally, the effects of initial value perturbations on stable phase topology are explored, revealing that in near-conservative states, small changes in initial conditions significantly disturb the system, resulting in the discovery of a multitude of previously hidden shrimp islands. Our findings enhance the understanding of non-quantum chiral structures within discrete systems and offer new insights into the intricate manifestations of stability and multistability in complex mappings.

Intermittent cluster synchronization in a unidirectional ring of bursting neurons

We report a new mechanism through which extreme events with a dragon-king-like distribution emerge in a network of unidirectional ring of Hindmarsh-Rose bursting neurons interacting through chemical synapses. We establish and substantiate the fact that depending on the choice of initial conditions, the neurons are divided into different clusters. These clusters transit from a phase-locked state (antiphase) to phase synchronized regime with increasing value of the coupling strength. Before attaining phase synchronization, there exist some regions of the coupling strength where these clusters are phase synchronized intermittently. During such intermittent phase synchronization, extreme events originate in the mean field of the membrane potential. This mechanism, which we name as intermittent cluster synchronization, is proposed as the new precursor for the generation of emergent extreme events in this system. These results are also true for diffusive coupling (gap junctions). The distribution of the local maxima of the collective observable shows a long-tailed non-Gaussian while the interevent interval follows the Weibull distribution. The goodness of fit is corroborated using probability-probability plot and quantile-quantile plot. This intermittent phase synchronization becomes rarer and rarer with an increase in the number of clusters of initial conditions.

Studying extreme events: An interdisciplinary review of recent research

While extreme events have been a focus of research for several decades, often centered around the causes and impacts of meteorological and climatological events, the term has expanded into a range of other disciplines, exploring a wide variety of associated topics. Analytical tools and definitions have hereby posed a challenge that has been addressed in different ways. Drawing from a broad body of research on extreme events, this review takes into account the often complex and cascading nature of extreme events in order to provide a large-scale overview of the main themes, discussions and trends of extreme event research. It does so by combining a systematic, large-scale analysis of publications on extreme events from 2019 to 2023 with additional database information on extreme events in the first part and a more in-depth narrative review on the main issues in the second part. The results show that extreme event research is dominated by meteorological and climatological extreme events and publications in the physical and life sciences. Identified focal areas in current research activities on extreme events are discussed in regards to issues of definitions and mathematical comprehension, extreme events and their impact in nature, and the interrelation of extremes and humans, including impacts on humans, extremes in human cognition, and human behaviors as causes and responses to extreme events.

Recent achievements in nonlinear dynamics, synchronization, and networks

This Focus Issue covers recent developments in the broad areas of nonlinear dynamics, synchronization, and emergent behavior in dynamical networks. It targets current progress on issues such as time series analysis and data-driven modeling from real data such as climate, brain, and social dynamics. Predicting and detecting early warning signals of extreme climate conditions, epileptic seizures, or other catastrophic conditions are the primary tasks from real or experimental data. Exploring machine-based learning from real data for the purpose of modeling and prediction is an emerging area. Application of the evolutionary game theory in biological systems (eco-evolutionary game theory) is a developing direction for future research for the purpose of understanding the interactions between species. Recent progress of research on bifurcations, time series analysis, control, and time-delay systems is also discussed.

Impact of time varying interaction: Formation and annihilation of extreme events in dynamical systems

This study investigates the emergence of extreme events in two different coupled systems: the FitzHugh-Nagumo neuron model and the forced Liénard system, both based on time-varying interactions. The time-varying coupling function between the systems determines the duration and frequency of their interaction. Extreme events in the coupled system arise as a result of the influence of time-varying interactions within various parameter regions. We specifically focus on elucidating how the transition point between extreme events and regular events shifts in response to the duration of interaction time between the systems. By selecting the appropriate interaction time, we can effectively mitigate extreme events, which is highly advantageous for controlling undesired fluctuations in engineering applications. Furthermore, we extend our investigation to networks of oscillators, where the interactions among network elements are also time dependent. The proposed approach for coupled systems holds wide applicability to oscillator networks.

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.

Extreme rotational events in a forced-damped nonlinear pendulum

Since Galileo's time, the pendulum has evolved into one of the most exciting physical objects in mathematical modeling due to its vast range of applications for studying various oscillatory dynamics, including bifurcations and chaos, under various interests. This well-deserved focus aids in comprehending various oscillatory physical phenomena that can be reduced to the equations of the pendulum. The present article focuses on the rotational dynamics of the two-dimensional forced-damped pendulum under the influence of the ac and dc torque. Interestingly, we are able to detect a range of the pendulum's length for which the angular velocity exhibits a few intermittent extreme rotational events that deviate significantly from a certain well-defined threshold. The statistics of the return intervals between these extreme rotational events are supported by our data to be spread exponentially at a specific pendulum's length beyond which the external dc and ac torque are no longer sufficient for a full rotation around the pivot. The numerical results show a sudden increase in the size of the chaotic attractor due to interior crisis, which is the source of instability that is responsible for triggering large amplitude events in our system. We also notice the occurrence of phase slips with the appearance of extreme rotational events when the phase difference between the instantaneous phase of the system and the externally applied ac torque is observed.

Modeling directed weighted network based on event coincidence analysis and its application on spatial propagation characteristics

The problem of synchronicity quantification, based on event occurrence time, has become the research focus in different fields. Methods of synchrony measurement provide an effective way to explore spatial propagation characteristics of extreme events. Using the synchrony measurement method of event coincidence analysis, we construct a directed weighted network and innovatively explore the direction of correlations between event sequences. Based on trigger event coincidence, the synchrony of traffic extreme events of base stations is measured. Analyzing topology characteristics of the network, we study the spatial propagation characteristics of traffic extreme events in the communication system, including the propagation area, propagation influence, and spatial aggregation. This study provides a framework of network modeling to quantify the propagation characteristics of extreme events, which is helpful for further research on the prediction of extreme events. In particular, our framework is effective for events that occurred in time aggregation. In addition, from the perspective of a directed network, we analyze differences between the precursor event coincidence and the trigger event coincidence and the impact of event aggregation on the synchrony measurement methods. The precursor event coincidence and the trigger event coincidence are consistent when identifying event synchronization, while there are differences when measuring the event synchronization extent. Our study can provide a reference for the analysis of extreme climatic events such as rainstorms, droughts, and others in the climate field.

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.

Using machine learning to anticipate tipping points and extrapolate to post-tipping dynamics of non-stationary dynamical systems

The ability of machine learning (ML) models to "extrapolate" to situations outside of the range spanned by their training data is crucial for predicting the long-term behavior of non-stationary dynamical systems (e.g., prediction of terrestrial climate change), since the future trajectories of such systems may (perhaps after crossing a tipping point) explore regions of state space which were not explored in past time-series measurements used as training data. We investigate the extent to which ML methods can yield useful results by extrapolation of such training data in the task of forecasting non-stationary dynamics, as well as conditions under which such methods fail. In general, we find that ML can be surprisingly effective even in situations that might appear to be extremely challenging, but do (as one would expect) fail when "too much" extrapolation is required. For the latter case, we show that good results can potentially be obtained by combining the ML approach with an available inaccurate conventional model based on scientific knowledge.

Route to extreme events in a parametrically driven position-dependent nonlinear oscillator

We explore the dynamics of a damped and driven Mathews-Lakshmanan oscillator type model with position-dependent mass term and report two distinct bifurcation routes to the advent of sudden, intermittent large-amplitude chaotic oscillations in the system. We characterize these infrequent and recurrent large oscillations as extreme events (EE) when they are significantly greater than the pre-defined threshold height. In the first bifurcation route, the system exhibits a bifurcation from quasiperiodic (QP) attractor to chaotic attractor via strange non-chaotic (SNA) attractor as a function of damping parameter. In the second route, the chaotic attractor in the form of EE has emerged directly from the QP attractor. Hence, to the best of our knowledge, this is the first study to report the birth of EE from these two distinct bifurcation routes. We also discuss that EE are emerged due to the sudden expansion of the chaotic attractor via interior crisis in the system. Regions of different dynamical states are distinguished using the Lyapunov exponent spectrum. Further, SNA and QP dynamics are determined using the singular spectrum analysis and 0-1 test. The region of EE is characterized using the threshold height.

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.

Identifying the spatiotemporal organization of high-traffic events in a mobile communication system using event synchronization and complex networks

In mobile communication systems, congestion is related to high-traffic events (HTEs) that occur in the coverage areas of base stations. Understanding, recognizing, and predicting these HTEs and researching their occurrence rules provides theoretical and decision-making support for preventing system congestion. Communication sectors are regarded as nodes, and if HTEs occur synchronously among sectors, then the corresponding nodes are connected. The total number of synchronous HTEs determines the edge weights. The mobile-communication spatiotemporal data are mapped to a weighted network, with the occurrence locations of HTEs as the basic elements. Network analysis provides a structure for representing the interaction of HTEs. By analyzing the topological features of the event synchronization network, the associations among the occurrence times of HTEs can be mined. We find that the event synchronization network is a small-world network, the cumulative strength distribution is exponential, and the edge weight obeys a power law. Moreover, the node clustering coefficient is negatively correlated with the node degree. A congestion coefficient based on several topological parameters is proposed, and the system congestion is visualized. The congestion coefficient contains information about the synchronous occurrence of HTEs between a sector and its neighbors and information about the synchronous occurrence of HTEs among its neighbors. For the mobile communication system considered in this study, the congestion coefficient of a large number of sectors is small and the risk of system congestion is low.

Optimized ensemble deep learning framework for scalable forecasting of dynamics containing extreme events

The remarkable flexibility and adaptability of both deep learning models and ensemble methods have led to the proliferation for their application in understanding many physical phenomena. Traditionally, these two techniques have largely been treated as independent methodologies in practical applications. This study develops an optimized ensemble deep learning framework wherein these two machine learning techniques are jointly used to achieve synergistic improvements in model accuracy, stability, scalability, and reproducibility, prompting a new wave of applications in the forecasting of dynamics. Unpredictability is considered one of the key features of chaotic dynamics; therefore, forecasting such dynamics of nonlinear systems is a relevant issue in the scientific community. It becomes more challenging when the prediction of extreme events is the focus issue for us. In this circumstance, the proposed optimized ensemble deep learning (OEDL) model based on a best convex combination of feed-forward neural networks, reservoir computing, and long short-term memory can play a key role in advancing predictions of dynamics consisting of extreme events. The combined framework can generate the best out-of-sample performance than the individual deep learners and standard ensemble framework for both numerically simulated and real-world data sets. We exhibit the outstanding performance of the OEDL framework for forecasting extreme events generated from a Liénard-type system, prediction of COVID-19 cases in Brazil, dengue cases in San Juan, and sea surface temperature in the Niño 3.4 region.

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.

Understanding the origin of extreme events in El Niño southern oscillation

We investigate a low-dimensional slow-fast model to understand the dynamical origin of El Niño southern oscillation. A close inspection of the system dynamics using several bifurcation plots reveals that a sudden large expansion of the attractor occurs at a critical system parameter via a type of interior crisis. This interior crisis evolves through merging of a cascade of period-doubling and period-adding bifurcations that leads to the origin of occasional amplitude-modulated extremely large events. More categorically, a situation similar to homoclinic chaos arises near the critical point; however, atypical global instability evolves as a channellike structure in phase space of the system that modulates variability of amplitude and return time of the occasional large events and makes a difference from the homoclinic chaos. The slow-fast timescale of the low-dimensional model plays an important role on the onset of occasional extremely large events. Such extreme events are characterized by their heights when they exceed a threshold level measured by a mean-excess function. The probability density of events' height displays multimodal distribution with an upper-bounded tail. We identify the dependence structure of interevent intervals to understand the predictability of return time of such extreme events using autoregressive integrated moving average model and box-plot analysis.

Identifying edges that facilitate the generation of extreme events in networked dynamical systems

The collective dynamics of complex networks of FitzHugh-Nagumo units exhibits rare and recurrent events of high amplitude (extreme events) that are preceded by so-called proto-events during which a certain fraction of the units become excited. Although it is well known that a sufficiently large fraction of excited units is required to turn a proto-event into an extreme event, it is not yet clear how the other units are being recruited into the final generation of an extreme event. Addressing this question and mimicking typical experimental situations, we investigate the centrality of edges in time-dependent interaction networks. We derived these networks from time series of the units' dynamics employing a widely used bivariate analysis technique. Using our recently proposed edge-centrality concepts together with an edge-based network decomposition technique, we observe that the recruitment is primarily facilitated by sets of certain edges that have no equivalent in the underlying topology. Our finding might aid to improve the understanding of generation of extreme events in natural networked dynamical systems.

Routes to extreme events in dynamical systems: Dynamical and statistical characteristics

Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system, to illustrate how the instabilities work to originate extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

Extreme events in a network of heterogeneous Josephson junctions

We report intermittent large spiking events in a heterogeneous network of forced Josephson junctions under the influence of repulsive interaction. The response of the individual junctions has been inspected instead of the collective response of the ensemble, which reveals the large spiking events in a subpopulation with characteristic features of extreme events (EE). The network splits into three clusters of junctions, one in coherent libration, one in incoherent rotational motion, and another subpopulation originating EE, which resembles a chimeralike pattern. EE migrates spatially from one to another subpopulation of junctions with the repulsive strength. The origin of EE in a subpopulation and chimera pattern is a generic effect of distributed damping parameter and repulsive interaction, which we verify with another network of the Liénard system. EE originates in the subpopulation via a local riddling of in-phase synchronization. The probability density function of event heights confirms the rare occurrence of large events and the return time of EE as expressed by interevent intervals in the subgroup follows a Poisson distribution. The mechanism of the origin of such a unique clustering is explained qualitatively.