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

Extreme events and extreme multistability in a nearly conservative system

This study investigates the emergence of extreme events in a complex variable dynamical system. In the absence of an external forcing, the model exhibits nearly Hamiltonian dynamics. When we set the system to a nearly conservative state and perturb it with external forcing, the formation of the onset of the extreme events was detected. By applying nullcline analysis and the system's vector field, we explored the underlying mechanism that leads to extreme events. Furthermore, we have conducted a thorough investigation to show the dynamic origins of extreme amplitude events and their transitions. The hardware electronic experiment is used to validate the numerical results of the onset of extreme events, and the results obtained are in good agreement with one another.

Cluster formation due to repulsive spanning trees in attractively coupled networks

Ensembles of coupled nonlinear oscillators are a popular paradigm and an ideal benchmark for analyzing complex collective behaviors. The onset of cluster synchronization is found to be at the core of various technological and biological processes. The current literature has investigated cluster synchronization by focusing mostly on the case of attractive coupling among the oscillators. However, the case of two coexisting competing interactions is of practical interest due to their relevance in diverse natural settings, including neuronal networks consisting of excitatory and inhibitory neurons, the coevolving social model with voters of opposite opinions, and ecological plant communities with both facilitation and competition, to name a few. In the present article, we investigate the impact of repulsive spanning trees on cluster formation within a connected network of attractively coupled limit-cycle oscillators. We successfully predict which nodes belong to each cluster and the emergent frustration of the connected networks independent of the particular local dynamics at the network nodes. We also determine local asymptotic stability of the cluster states using an approach based on the formulation of a master stability function. We additionally validate the emergence of solitary states and antisynchronization for some specific choices of spanning trees and networks.

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.

Chaotic saddles and interior crises in a dissipative nontwist system

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems; they are responsible for chaotic transients, fractal basin boundaries, and chaotic scattering, and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increases the transient times and we analyze the phenomenon of crisis induced intermittency.

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 transient dynamics

We study the extreme transient dynamics of four self-excited pendula coupled via the movable beam. A slight difference in the pendula lengths induces the appearance of traveling phase behavior, within which the oscillators synchronize, but the phases between the nodes change in time. We discuss various scenarios of traveling states (involving different pendula) and their properties, comparing them with classical synchronization patterns of phase-locking. The research investigates the problem of transient dynamics preceding the stabilization of the network on a final synchronous attractor, showing that the width of transient windows can become extremely long. The relation between the behavior of the system within the transient regime and its initial conditions is examined and described. Our results include both identical and non-identical pendula masses, showing that the distribution of the latter ones is related to the transients. The research performed in this paper underlines possible transient problems occurring during the analysis of the systems when the slow evolution of the dynamics can be misinterpreted as the final behavior.

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.

Forecasting and diagnostics of critical states in platform-vibrator with shock

A platform-vibrator with shock is a low-frequency machine used in the construction industry for compaction and molding of large concrete products. Its mathematical model is a two-degree-of-freedom two-body vibro-impact system with a soft impact. Some changes in its parameters can increase the machine performance and improve the product quality, but these same changes may lead to the emergence of critical states, such as coexisting regimes in hysteresis zone, chaotic motion, intermittency and crisis-induced intermittency, crises, and transient chaos. Some of them can be undesirable and dangerous. This article shows their diagnostics and recognition, the possibility of their prediction, as well as the criterion determining the set of parameter ranges where critical states can occur. Diagnostics is carried out both by traditional tools and by the less common ones, such as the construction of fractal structures and wavelet characteristics.

Correlation sum scalings from mixed-mode oscillations in weakly coupled molecular lasers

A model for two symmetrically coupled lasers is investigated, in which mixed-mode oscillations arise in the absence of coupling. For small enough coupling, we show that in the time series, certain dynamical transitions from different resonances in the chaotic regime may be explained by the overlap of suitable resonances. These are families of N : N + 1 resonances, which result in isolas as well as isolas consisting of intermediate-phase resonances N : N. It appears that the overlap of resonances can explain the onset of two different scaling regions in the dimension correlation sum, which display an explicit dependence on the optical coupling strength. For very small coupling ranges, there are larger scaling regions that look analogous to that for the uncoupled laser system. For larger coupling, but still well below the synchronization threshold, steeper and larger scaling regions arise, in particular, in the smaller partitions.

Transition to hyperchaos: Sudden expansion of attractor and intermittent large-amplitude events in dynamical systems

Hyperchaos is distinguished from chaos by the presence of at least two positive Lyapunov exponents instead of just one in dynamical systems. A general scenario is presented here that shows emergence of hyperchaos with a sudden large expansion of the attractor of continuous dynamical systems at a critical parameter when the temporal dynamics shows intermittent large-amplitude spiking or bursting events. The distribution of local maxima of the temporal dynamics is non-Gaussian with a tail, confirming a rare occurrence of the large-amplitude events. We exemplify our results on the sudden emergence of hyperchaos in three paradigmatic models, namely, a coupled Hindmarsh-Rose model, three coupled Duffing oscillators, and a hyperchaotic model.

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.

Chaos-hyperchaos transition in three identical quorum-sensing mean-field coupled ring oscillators

We investigate the dynamics of three identical three-dimensional ring synthetic genetic oscillators (repressilators) located in different cells and indirectly globally coupled by quorum sensing whereby it is meant that a mechanism in which special signal molecules are produced that, after the fast diffusion mixing and partial dilution in the environment, activate the expression of a target gene, which is different from the gene responsible for their production. Even at low coupling strengths, quorum sensing stimulates the formation of a stable limit cycle, known in the literature as a rotating wave (all variables have identical waveforms shifted by one third of the period), which, at higher coupling strengths, converts to complex tori. Further torus evolution is traced up to its destruction to chaos and the appearance of hyperchaos. We hypothesize that hyperchaos is the result of merging the saddle-focus periodic orbit (or limit cycle) corresponding to the rotating wave regime with chaos and present considerations in favor of this conclusion.

Traveling of extreme events in network of counter-rotating nonlinear oscillators

We study the propagation of rare or extreme events in a network of coupled nonlinear oscillators, where counter-rotating oscillators play the role of the malfunctioning agents. The extreme events originate from the coupled counter-oscillating pair of oscillators through a mechanism of saddle-node bifurcation. A detailed study of the propagation and the destruction of the extreme events and how these events depend on the strength of the coupling is presented. Extreme events travel only when nearby oscillators are in synchronization. The emergence of extreme events and their propagation are observed in a number of excitable systems for different network sizes and for different topologies.

Chaos in fractional system with extreme events

Understanding extreme events attracts scientists due to substantial impacts. In this work, we study the emergence of extreme events in a fractional system derived from a Liénard-type oscillator. The effect of fractional-order derivative on the extreme events has been investigated for both commensurate and incommensurate fractional orders. Especially, such a system displays multistability and coexistence of multiple extreme events.

Characteristics of 2020 stock market crash: The COVID-19 induced extreme event

A sudden fall of stock prices happens during a pandemic due to the panic sell-off by the investors. Such a sell-off may continue for more than a day, leading to a significant crash in the stock price or, more specifically, an extreme event (EE). In this paper, Hilbert-Huang transformation and a structural break analysis (SBA) have been applied to identify and characterize an EE in the stock market due to the COVID-19 pandemic. The Hilbert spectrum shows a maximum energy concentration at the time of an EE, and hence, it is useful to identify such an event. The EE's significant energy concentration is more than four times the standard deviation above the mean energy of the normal fluctuation of stock prices. A statistical significance test for the intrinsic mode functions is applied, and the test found that the signal is not noisy. The degree of nonstationarity test shows that the indices and stock prices are nonstationary. We identify the time of influence of the EE on the stock price by using SBA. Furthermore, we have identified the time scale ( τ) of the shock and recovery of the stock price during the EE using the intrinsic mode function obtained from the empirical mode decomposition technique. The quality stocks with V-shape recovery during the COVID-19 pandemic have definite τ of shock and recovery, whereas the stressed stocks with L-shape recovery have no definite τ. The identification of τ of shock and recovery during an EE will help investors to differentiate between quality and stressed stocks. These studies will help investors to make appropriate investment decisions.

Type III intermittency without characteristic relation

Chaotic intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories from the chaotic region into the laminar one. The RPD classically was taken as a constant. This hypothesis is behind the classically reported characteristic relations, a tool describing how the mean value of the laminar length goes to infinity as the control parameter goes to zero. Recently, a generalized non-uniform RPD has been observed in a wide class of 1D maps; hence, the intermittency theory has been generalized. Consequently, the characteristic relations were also generalized. However, the RPD and the characteristic relations observed in some experimental Poincaré maps still cannot be well explained in the actual intermittency framework. We extend the previous analytical results to deal with the mentioned class of maps. We found that in the mentioned maps, there is not a well-defined RPD in the sense that its shape drastically changes depending on a small variation of the parameter of the map. Consequently, the characteristic relation classically associated to every type of intermittency is not well defined and, in general, cannot be determined experimentally. We illustrate the results with a 1D map and we develop the analytical expressions for every RPD and its characteristic relations. Moreover, we found a characteristic relation going to a constant value, instead of increasing to infinity. We found a good agreement with the numerical simulation.