Extreme Events: Mechanisms and Prediction

Mohammad FarazmandDepartment of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA 02139, USAEmail: mfaraz@mit.edu

Themistoklis P. SapsisDepartment of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA 02139, USAEmail: sapsis@mit.edu

Extreme events, such as rogue waves, earthquakes and stockmarket crashes, occur spontaneously in many dynamical sys-tems. Because of their usually adverse consequences, quan-tification, prediction and mitigation of extreme events arehighly desirable. Here, we review several aspects of extremeevents in phenomena described by high-dimensional, chaoticdynamical systems. We specially focus on two pressing as-pects of the problem: (i) Mechanisms underlying the forma-tion of extreme events and (ii) Real-time prediction of ex-treme events. For each aspect, we explore methods relyingon models, data or both. We discuss the strengths and limi-tations of each approach as well as possible future researchdirections.

1 IntroductionExtreme events are observed in a variety of natural

and engineering systems. Examples include oceanic roguewaves [1, 2], extreme weather patterns [3, 4, 5, 6], earth-quakes [7] and shocks in power grids [8,9]. These events areassociated with abrupt changes in the state of the system andoften cause unfortunate humanitarian, environmental and fi-nancial impacts. As such, the prediction and mitigation ofextreme events are highly desired.

There are several outstanding challenges in dealing withextreme events. These events often arise spontaneously withlittle to no apparent early warning signs. This renders theirearly prediction from direct observations a particularly dif-ficult task [10, 11, 12]. In certain problems, such as earth-quakes, reliable mathematical models capable of predictingthe extreme events are not available yet [13].

In other areas, such as weather prediction where moreadvanced models are in hand, accurate predictions requiredetailed knowledge of the present state of the system whichis usually unavailable. The partial knowledge of the currentstate together with the chaotic nature of the system leads to

uncertainty in the future predictions. These uncertainties areparticularly significant during the extreme episodes [14, 15,16].

In addition, models of complex systems are usuallytuned using data assimilation techniques. This involves se-lecting the model parameters so that its predictions matchthe existing empirical data. The effectiveness of data assim-ilation, however, is limited when it comes to rare extremeevents due to the scarcity of observation data correspondingto these events [17, 18, 19, 20].

These challenges to modeling and prediction of ex-treme events remain largely outstanding. The purpose of thepresent article is to review some of these challenges and topresent the recent developments toward their resolution.

Mechanisms

Statistics

Indica

tor

Control Activation

Control Strategy

MitigationReal-time Prediction

Fig. 1: The study of extreme events consists mainly of fourcomponents.

The analysis of extreme events can be divided into fourcomponents as illustrated in figure 1: Mechanisms, Predic-tion, Mitigation and Statistics. Below, we briefly discusseach of these components.

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(i) Mechanisms: The mechanisms that trigger the ex-treme events are the primary focus of this article. Consideran evolving system that is known from the time series of itsobservables to produce extreme events. We are interested inunderstanding the conditions that underly the extreme eventsand trigger their formation. Even when the governing equa-tions of the system are known, it is often a difficult task todeduce the mechanism underlying the extreme events. Thisis due to the well-known fact that even seemingly simplegoverning equations can generate very complex chaotic dy-namics. The task of deducing the behavior of solutions fromthe governing equations becomes specially daunting whenthe system consists of many interacting degrees of freedomwhich give rise to a high-dimensional and complex attractor.

In section 3, we review a number of methods that un-ravel the extreme event mechanisms. These methods havebeen developed to analyze specific classes of dynamical sys-tems. For instance, the multiscale method discussed in sec-tion 3.1 only applies to systems whose degrees of freedomcan be separated into the so-called slow and fast variables.Even when such a slow-fast decomposition is available, com-puting the corresponding slow manifold and its stable andunstable manifolds become quickly prohibitive as the dimen-sion of the system increases.

As a result, a more general mathematical framework isneeded that is applicable to a broader range of dynamicalsystems and at the same time can leverage the ever growingcomputational resources. We explore such a general frame-work in section 4.

(ii) Real-time prediction: Most undesirable aspects ofextreme events can often be avoided if the events are pre-dicted in advance. For instance, if we can predict severeearthquakes a few hours in advance, many lives will be savedby evacuation of endangered zones. As a result, their real-time prediction is perhaps the most exigent aspect of extremephenomena.

Real-time prediction requires measurable observablesthat contain early warning signs of upcoming extreme events.We refer to such observable as indicators of extreme events.Reliable indicators of extreme events must have low ratesof false-positive and false-negative predictions. A false posi-tive refers to the case where the indicator incorrectly predictsan upcoming extreme event. Conversely, a false negativerefers to the case where the indicator fails to predict an ac-tual extreme event. Knowing the mechanisms that trigger theextreme events does not necessarily enable their prediction.However, as we show in section 5, even partial knowledgeof these mechanisms may lead to the discovery of reliableindicators of extreme events.

Another important aspect of extreme event prediction isthe confidence in the predictions. The sensitivity to initialconditions leads to an inherent uncertainty in chaotic systemseven when the system model is deterministic [21, 22]. Suchuncertainties permeate the prediction of extreme events. Asa result, the predictions have to be made in a probabilisticsense where the uncertainties in the predictions are properlyquantified (see section 5.1).

(iii) Mitigation: Can we control a system so as to sup-

press the formation of extreme events? This if of course be-yond reach in many natural systems such as ocean wavesand extreme weather patterns. However, in certain engi-neered systems, such as power grids, one can in principledesign control strategies to avoid the formation of extremeevents [23, 24, 25]. To this end, knowing the mechanismsthat trigger the extreme events is crucial as it informs the de-sign of the control strategy. The real-time prediction of theextreme events, on the other hand, informs the optimal timefor the activation of the control strategy (see figure 1).

The mitigation of extreme events within a dynamicalsystems framework has only recently been examined [26,27,28,29,30,31]. The research in this direction has been limitedto mitigation in simplified models by introducing arbitraryperturbations that nudge the system away from the extremeevents. However, a systematic study involving controllabil-ity and observability of extreme events in the sense of controltheory is missing.

(iv) Statistics: The statistical study of extreme eventsattempts to answer questions regarding the frequency andprobability of occurrence of extreme events from a largesample. Such statistical questions are perhaps the most in-tensely studied aspect of extreme events due to their ap-plications in finance, insurance industry and risk manage-ment [32,33,34,35]. In this article, we will limit our discussof the statistics to this section and refer the interested readerto the cited literature on the topic.

Two major frameworks for quantifying the extremestatistics of stochastic processes are the extreme value the-ory and the large deviation theory. The extreme value theorystudies the probability distribution of the random variableMn = maxX1,X2, · · · ,Xn where X1,X2, · · · is a sequence ofrandom variables [36]. The main objective in extreme valuetheory is to determine the possible limiting distributions ofMn as n tends to infinity. In particular, the Fisher–Tippett–Gnedenko theorem (also known as the extremal types theo-rem) states that, if Xii≥1 is a sequence of independent andidentically distributed (i.i.d) random variables then the lim-iting distribution of Mn can only converge to three possibledistributions and provides explicit formula for these distribu-tions [37,38,39]. This is a significant result since the extremestatistics of the random variable can be deduced even whenno extreme events have actually been observed. In manypractical cases, however, the random variables are not inde-pendent. Therefore, the more recent work in extreme valuetheory has been focused on relaxing the independence as-sumption [40, 41, 42, 43, 44, 45, 46, 47, 48]. For an extensivereview of extreme value theory in the context of dynamicalsystems, we refer to a recent book by Lucarini et al. [49].

Another prominent framework for the statistical anal-ysis of extreme events is the large deviation theory whichis concerned with the tail distribution of random variables.The tails of the probability distributions contain the ex-treme values a random variable can take, hence the namelarge deviations. The large deviations were first analyzed byCramer [50] who studied the decay of the tail distribution ofthe empirical means Zn = ∑

ni=1 Xi/n for n 1 where Xii≥1

is a sequence of i.i.d random variables. Later, Donsker and

2

Varadhan [51, 52, 53, 54] generalized the large deviation re-sults to apply them to Markov processes. The current scopeof the large deviation theory is quite broad and is applied toquantifying heavy tailed statistics in a variety of determinis-tic and stochastic dynamical systems. We refer the interestedreader to the articles by Varadhan [55] and Touchette [56,57]for a historical review of large deviation theory and its appli-cations.

The four aspects of extreme events mentioned above areintertwined. However, the discovery of mechanisms that giverise to extreme events resides in the heart of the problem (seefigure 1). For instance, even partial knowledge of the mecha-nisms that trigger the extreme events may lead to the discov-ery of indicators that facilitate their data-driven prediction(see section 5.2). In addition, once we know what mech-anisms trigger the extreme events, we can make informedchoices about the control strategies towards avoiding them.To this end, the real-time prediction of upcoming extremeevents informs the time the control strategy should be acti-vated. Knowledge of the mechanisms of the extreme eventscan also help improve the statistical estimates regarding theirlikelihood and frequency.

As a result, the main focus of this article will be on thefirst aspect of extreme events, i.e., the mechanisms. We willalso discuss some aspect of the real-time prediction of theextremes, specially the quantification of the reliability of theindicators of extreme events. In section 2, we introduce thegeneral set-up and notation. Section 3 reviews some well-known mechanisms for extreme event formation in determin-istic and stochastic dynamical systems. In section 4, we re-view a variational method for discovering the mechanismsof extreme events and illustrate its application with two ex-amples: intermittent turbulent energy dissipation and rogueocean waves. In section 5, we discuss reliable indicators ofextreme events for their real-time prediction. Section 6 con-tains our concluding remarks.

2 Setup and notationIn this section, we lay out the setup of the problem that

allows for a dynamical systems framework for extreme eventanalysis. We consider systems that are governed by an initialvalue problem of the form

∂tu = N(u), (1a)

u(x,0) = u0(x), ∀x ∈Ω, (1b)

where the state u(t) , u(·, t) ∈U belongs to an appropriatefunction space U for all times t ≥ 0. The initial state ofthe system is specified by u0 : Ω→ Rd , where Ω ⊆ Rd andd ∈ N. The operator N is a potentially nonlinear operatorthat is provided by the physics. The PDE (1) should also besupplied with appropriate boundary conditions u|∂Ω where∂Ω denotes the boundary of Ω.

System (1) generates a solution map

S t :U→Uu0 7→ u(t) (2)

that maps the initial state u0 to its image u(t) at a later time t.The solution map has the semi-group property, i.e., S0(u0) =u0 and S t+s(u0) = S t(S s(u0)) = S s(S t(u0)) for all u0 ∈U.

We equip the space U with further structure. In particu-lar, we assume that (U,B,µ) is a probability space and thatthe probability measure µ is S t -invariant. We refer to a mea-surable function f : U→R as an observable. Note that for anobservable f , Xt = f S t is a continuous stochastic processwhose realizations are made by choosing an initial conditionu0 drawn in a fashion compatible with the probability mea-sure µ.

The observable f is a quantity whose statistics and dy-namical evolution is of interest. For instance, in the waterwave problem considered in section 4.3 below, the observ-able is the wave height. In meteorology, the observable ofinterest could be temperature or precipitation. Here, we arein particular interested in the extreme values of the observ-able f . In practice, the extreme values are often defined bysetting a threshold fe. The observable values that are largerthan this threshold constitute an extreme event. This moti-vates the following definition of extreme events.

Definition 1 (Extreme Events). For an observable f :U → R, the extreme event set E( fe), corresponding to theprescribed extreme event threshold fe ∈ R, is given by

E f ( fe) = u ∈U : f (u)> fe= f−1(( fe,∞)

). (3)

The extreme event sets within the state space U are de-picted in figure 2. As the system trajectory S t(u0) passesthrough the extreme event set E f ( fe), the time series of theobservable f exhibit a sudden burst. In this figure, the ex-treme event set is depicted as a collection of patches, but inprinciple this set can have an extremely complex geometry.

In certain problems, the extreme events may correspondto unusually small values of the observable f . In that case,Definition 1 is still operative by studying the observable− f instead of f . A third type of rare events (which arenot necessarily extreme) is the rare transition between long-lived states (see figure 3). In this case, the system evolvesfor long times around a particular state before it is sud-denly ejected to the neighborhood of a different state aroundwhich the system evolves for a long time before being ejectedagain [58, 59, 60, 61, 62, 63, 64]. Although such rare transi-tions do not necessarily fall under Definition 1, we return tothem in section 3.3 and review the mechanisms that cause thetransitions.

Finally, we point out that, although the governing equa-tions (1) are formulated as a partial differential equation

3

u0

U

f

St(u0)t

f(u(t))

fe

Fig. 2: Geometry of the state space U and the time history of the observable f . As the trajectory passes through the extremeevent set E f ( fe)– marked in red- a burst in the observable time series appears.

V(u

)

u1 u2U

f(u1)

f(u2)

f

t

Fig. 3: Rare transitions between two stable states made possible through noise.

(PDE), we will also consider systems that are described bya set of ordinary differential equations (ODEs), u = N(u),where u(t) ∈ Rn denotes the state of the system at time t.This ODE could also arise from a finite-dimensional approx-imation of a PDE model as is common in numerical dis-cretization of PDEs.

3 Routes to extreme eventsThere are certain classes of dynamical systems exhibit-

ing extreme events for which the mechanisms that triggerthese events are well-understood. In this section, we reviewthree such systems and discuss the underlying mechanismsof extreme events in them.

3.1 Multiscale systemsAn interesting type of extreme events appear in slow-

fast dynamical systems where the motion is separated intodistinct timescales. The extreme events in such systems ap-pear as bursts when a system trajectory is dominated by thefast timescales of the system. The early work on this subjectwas motivated by the observation of relaxation oscillationsin electrical circuits [65, 66, 67]. Later, slow-fast dynam-ics found applications in a wide range of problems such aschemical reactions [68,69,70,71,72], excitable systems (e.g.,neural networks) [73, 74, 75, 76, 77, 78, 79], extreme weatherpatterns [80, 81, 82], and dynamics of finite-size particles influid flows [83, 84].

We first discuss the phenomenology of bursting in slow-fast systems and then demonstrate its implications on a con-crete example. Figure 4 sketches the phase space geome-

try of a slow-fast system. It has an invariant slow manifoldwhere the trajectories follow the slow time scale. In the di-rections transverse to the slow manifold, the dynamics followthe fast time scales. We assume that the slow manifold is nor-mally hyperbolic. Loosely speaking, normal hyperbolicitymeans that the transverse attraction to and repulsion from themanifold is stronger than its internal dynamics [85, 86]. Wealso assume that the slow manifold is of the saddle type, thatis, it consists of two components: attracting and repelling.Normal perturbations to the manifold on its attracting com-ponent decay over time while the perturbations over the re-pelling component grow. Due to invariance of the slow man-ifold, trajectories starting on the manifold remain on it for alltimes unless they exit the manifold through its boundaries.

Now consider a trajectory that starts slightly off the slowmanifold over its attracting component (the black curve infigure 4). Initially this trajectory converges towards the slowmanifold until it approaches its repelling component. At thispoint, the normal repulsion pushes the trajectory away fromthe slow manifold where the fast time scales are manifest.This rapid repulsion continues until the trajectory leaves theneighborhood of the repelling component and is pulled backtowards the attracting components.

If the normal repulsion is strong enough, the episodeswhere the trajectory travels away from the slow manifold ap-pear as rapid bursts. The repelling subset of the slow mani-fold can be a very complex set as a result of which the burstcan appear chaotic and sporadic.

We demonstrate the bursting in slow-fast systems on a

4

Slow Manifold

Fig. 4: A schematic picture of the a trajectory of a slow-fast system. The slow manifold is of the saddle type, thatis, it consists of a normally attracting component Ma (blue)and a normally repelling component Mr (red). The trajectorydiverges rapidly away from the slow manifold when it visitsa repelling subset. Subsequently the trajectory approachesthe slow manifold along its attracting component.

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

x

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

y

Ma

Ma

Mr

p1

p2

Fig. 5: Orbits of the rescaled system (7) with ε = 0 projectedon the x−y plane. The s-shaped curve marks the slow mani-fold y = x2 +x3 which consists of three connected segments:two segments are attracting (blue) and one segment is re-pelling (red).

normal form of a singular Hopf bifurcation [87],

ε x = y− x2− x3

y = z− x

z =−ν−ax−by− cz, (4)

where ε ≥ 0 is a small parameter. For our discussion wefix the remaining parameters, a = −0.3872, b = −0.3251,c = 1.17 and ν = 0.0072168.

We first discuss the singular limit where ε = 0. In thislimit, system (4) reduces to the differential-algebraic equa-tions

0 = y− x2− x3

y = z− x

z =−ν−ax−by− cz. (5)

This reduced system describes the slow flow on the criticalmanifold

M0 = (x,y,z) : y = x2 + x3. (6)

In order to discern the dynamics outside the critical manifold,we use a blow-up construction by rescaling time accordingto t = ετ. The derivative with respect to the fast time τ isgiven by d

dτ= ε

ddt . With this change of variable, equation (4)

becomes

x′ = y− x2− x3

y′ = ε(z− x)

z′ = ε(−ν−ax−by− cz), (7)

where the prime denotes derivative with respect to the fasttime τ. In the singular limit, ε = 0, we have y′ = 0 and z′ = 0.Moreover, in the rescaled system, every point on the criticalmanifold is a fixed point since x′ = y− x2− x3 = 0. This isan artifact of the rescaling t = ετ which is singular at ε =0. More precisely, points on the critical manifolds are fixedpoints with respect to the fast time scale. In turn, the slowdynamics on the critical manifold is given by the reducedsystem (5). The combination of the reduced system (5) andthe rescaled system (7) describes the motion on the criticalmanifold and away from it.

Of particular relevance to us is the behavior of trajecto-ries in a small neighborhood of the critical manifold. Thecritical manifold consists of three connected components(see figures 5 and 6). Two of these components, denoted byMa, are normally attracting, meaning that trajectories start-ing away from them in a transverse direction converge to-wards the critical manifold. In contrast, transverse perturba-tions to the normally repelling segment Mr diverge rapidlyfrom the critical manifold. As a result, trajectories startingnear the repelling submanifold Mr are repelled to a neigh-borhood of the attracting manifold Ma where they follow theslow time scales along the critical manifold until they reachone of the fold points p1 or p2 (see figure 5). At the folds,located on the boundary between the attracting and repellingsubmanifolds, the trajectory is repelled again from Mr to-ward the second segment of the attracting submanifold Ma.This cycle continues indefinitely, creating bursting trajecto-ries that are repelled away from the repelling submanifoldand attracted back towards the critical manifold along its at-tracting segment.

Now we turn our attention to the nonsingular case whereε > 0. The above analysis of the singular flow (ε = 0) bearssome relevance to the nonsingular case (ε > 0). For suffi-ciently small perturbations, 0 < ε 1, the Geometric Singu-lar Perturbation Theory (GSPT) [88] guarantees, under cer-tain conditions, that the critical manifold M0 survives as aperturbed invariant manifold Mε, that Mε is as smooth asthe critical manifold, and that Mε is O(ε) close to the crit-ical manifold M0. Furthermore, the normally attracting or

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x

-0.05

0

0.05

0.1

0.15

0.2

0.25

y

Ma

Ma

Mr

(a) (b)

(c)

Ma

Ma

Mr

0 5 10 15

t

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

y

Fig. 6: The state space geometry of system (4) with parameters (a,b,c,ν,ε) = (−0.3872,−0.3251,1.17,0.0072168,0.001).(a) A stable periodic orbit of the system (black curve) is shown together with the critical manifold (6). The attracting parts ofthe manifold are colored in blue and the repelling part is colored in red. (b) Projection of panel (a) onto the x− y plane. Theinset shows a closeup view of the region enclosed in a gray box. (c) Time series of the y-coordinate along the periodic orbit.

repelling properties of the perturbed manifold Mε is similarto that of the critical manifold M0.

In particular, for system (4), the critical manifold M0 de-forms into a nearby slow manifold Mε. The perturbed slowmanifold has its own repelling and attracting submanifoldssimilar to those of M0 which create bursting repulsion fromand attraction towards the slow manifold. Figure 6 showsa trajectory of the system for ε = 10−3. At this parametervalues, the system has undergone a supercritical Hopf bifur-cation [89] giving birth to a stable periodic orbit (the blackcurve). This periodic orbit carries much of the bursting prop-erties described above for the singular system. Most of thetime, the trajectory spirals outwards near the fold p1. Atsome point, the trajectory approaches the repelling segmentof the slow manifold Mε whereby it is repelled away towardsits attracting segment. The trajectory follows the attractingsegment until it reaches the fold p2 where the repelling seg-ment again repels the trajectory towards the second attractingsegment. Following the attracting segment, the trajectoriesreturns towards the fold p1 and the small spiral motion re-peats. This cycle continues indefinitely. Figure 6(c) showsthe time series of the y-coordinate along the periodic orbitshowing the bursts resulting from repulsion away from theslow manifold.

For illustrative purposes, we presented in figure 6 a pa-rameter set where the asymptotic motion of the system isrelatively simple, dictated by a single stable periodic orbit.The dynamics is not always so predictable. There are param-eter values (a,b,c,ν,ε) where the system undergoes perioddoubling bifurcations resulting in several co-existing unsta-ble periodic orbits. As a result, a generic trajectory neversettles down to a particular periodic orbit. Instead, it indefi-nitely bounces back and forth between unstable periodic or-

bits. As a result, the bursting time series appear chaotic andless predictable.

We illustrate this on a system which exhibits chaoticbursts for a wide range of parameters. Consider the coupledFitzHugh–Nagumo units [77],

xi =xi(ai− xi)(xi−1)− yi + kn

∑j=1

Ai j(x j− xi),

yi =bixi− ciyi, (8)

where n is the number of units and (ai,bi,ci) are constantparameters. The units are coupled to each other through thesummation term. The matrix A with entries Ai j ∈0,1 is theadjacency matrix that determines which units are coupled.The constant k determines the strength of the couplings.

Figure 7 shows a typical trajectory of the FitzHugh–Nagumo system with two units (n = 2). Also shown is thetime series of the mean of xi, i.e. x = 1

n ∑ni=1 xi. The mean x

mostly oscillates chaotically around 0 with a relatively smallvariance. Once in a while, however, it exhibits relativelylarge excursions away from 0 in the form of bursts. As op-posed to the periodic extreme events of figure 6, these burstsappear chaotically, with no regular recurrent pattern. Similarextreme events have been observed in the FitzHugh–Nagumosystem with larger number of units and various parametervalues [78].

In this chaotic regime, the geometry of the invariantsets and their stable and unstable manifolds can be incred-ibly complex. One of the recent contributions to the fieldof slow-fast systems has been the development of accu-rate numerical methods for computing such invariant man-ifolds [90, 91, 92, 93]. The computational cost of these man-

6

(a)

0 1 2 3 4 5

t ×104

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

(b)

1.65 1.7 1.75 1.8

t ×104

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

(c)

Fig. 7: The FitzHugh–Nagumo oscillators (8) with two units (n = 2). The parameters are a1 = a2 = −0.025794, c1 =c2 = 0.02, b1 = 0.0065, b2 = 0.0135 and k = 0.128. The adjacency matrix A is symmetric with entries A11 = A22 = 0 andA12 = A21 = 1. (a) A trajectory of the system projected onto the (x1,y1,y2) subspace. (b) Time series of the observablex = 1

n ∑ni=1 xi. (c) A closeup view of the first burst of x.

x

y

z

(a)

x

y

z

(b)

Fig. 8: Sketches of a homoclinic (a) and a heteroclinic (b)orbit.

ifolds increases with the dimension of the system such thattheir computation is currently limited to four or five dimen-sional systems [94]. Nonetheless, understanding the mecha-nism behind extreme events in prototypical low-dimensionalslow-fast systems has been greatly informative at the concep-tual level.

3.2 Homoclinic and heteroclinic burstingAnother geometric mechanism of generating extreme

events is through homoclinic and heteroclinic connections(see figure 8). Since these mechanisms share many of thecharacteristics of the slow-fast systems discussed in sec-tion 3.1, we limit this section to a brief discussion of themain ideas underlying homoclinic and heteroclinic bursting.

An example of a homoclinic connection is that of theShilnikov orbit of a saddle-focus fixed point. This is an un-stable fixed point with a two-dimensional stable manifoldand a one-dimensional unstable manifold (see figure 8(a)).Within the stable manifold, the trajectories spiral towards thefixed point while they are repelled from the fixed point in itsunstable direction. The Shilnikov orbit is the homoclinic tra-jectory that is asymptotic to the fixed point both in forwardtime and in backward time.

Shilnikov [95] proved that, if the attraction within thestable manifold is weaker than the repulsion along the un-stable manifold, small perturbations to the system give birth

to infinitely many unstable periodic orbits around the homo-clinic orbit. These periodic orbits resemble the shape of theoriginal Shilnikov orbit [96]. More precisely, the periodicorbits consist partly of spiral motion towards the fixed pointand partly of bursting motion away from it. Generic trajec-tories shadow these periodic orbits such that the time seriesof their z-component exhibits small scale oscillations, corre-sponding to the spiraling motion, and occasional bursts, cor-responding to repulsion along the unstable manifold [97,98].Since the periodic orbits are all unstable, the motion alonggeneric trajectories can be chaotic resulting in very complexdynamics. A classical example of such chaotic motion is theRossler attractor [99, 100].

Although the Shilnikov bifurcation was first studied asa route to chaotic motion in simple systems, it has foundmany applications in explaining the self-sustained burstingphenomena observed in nature. These include, for instance,sudden variations in geophysical flow patterns [101, 102],spiking and synchronization in neural networks [103,75,104]and chemical reactions [105].

A similar mechanism of bursting is through heteroclinicconnections. As opposed to the homoclinic case, the hete-roclinic orbit asymptotes to different fixed points in forwardand backward times. Figure 8(b), for instance, depicts a het-eroclinic connection corresponding to the phase space of athree-dimensional vector field introduced in Ref. [106]. Asin the homoclinic case, the heteroclinic bursting has beenuseful in explaining several spiking behavior observed in na-ture from nonlinear waves to turbulent fluid flow [107, 108,109, 110, 111, 112, 113].

3.3 Noise-induced transitionsSo far we have discussed deterministic systems which

possess a self-sustaining mechanism for generating extremeevents. However, an important class of rare extreme eventsare induced by noise [114, 115, 116, 117]; see also [118] foran excellent review for this form of transitions. Such sys-tems typically have equilibria that are stable in the absenceof noise. Noise, however, makes it possible to transition from

7

the neighborhood of one equilibrium to the other.In such systems the transition mechanism is the noise

and, as such, there is no ambiguity regarding what underliesthe rare events. However, the route the system takes duringeach transition is not as clear. In fact, due to the randomnature of the system, the transition routes can only be iden-tified probabilistically. In particular, one can inquire aboutthe most likely route the system takes in traveling betweentwo states. The answer facilitates the prediction of individ-ual transitions as well as the quantification of transition ratesin an ensemble of experiments. In this section, we brieflyreview the transition-path theory which is a framework foraddressing these questions.

The origins of the transition-path theory stem fromchemical physics where one is interested in computing therate of chemical reactions that lead to a transition from thereactant state to the product state [119,120,58,121,122,123].

The transition-path theory aims to go beyond computingthe transition rates and determines the most likely paths thatthe system may take during the transitions. To describe thistheory, we consider the Langevin equation,

mx =−∇V (x)− γx+√

2σ(x)η(t), (9)

where u = (x, x) ∈ R2n determines the state of the system, mis the mass matrix, V : Rn → R is the potential and γ is thefriction coefficient. The stochastic process η(t) is a whitenoise with mean zero and covariance 〈ηi(t)η j(s)〉= δi jδ(t−s) and a(x) = σ(x)σ(x)† ∈ Rn×n is the diffusion matrix.

For our introductory purposes it is helpful to considerthe over-damped case γ 1 where equation (9) reduces to

u =−∇V (u)+√

2σ(u)η(t), (10)

where u = x ∈ Rn determines the state of the system. Forsimplicity, we have assumed m = Id and rescaled time toeliminate the dependence on the friction coefficient γ.

Figure 3 sketches the over-damped system (10) in theone-dimensional case (n = 1). In absence of noise η(t), thesystem converges asymptotically to one of the two local min-ima of the potential V . The noise, however, nudges the tra-jectory away from these equilibria. In rare instances, thetrajectory can even pass over the saddle separating the twoequilibria causing a transition from equilibrium u1 towardsequilibrium u2 and vice versa.

In this simple example, there is no ambiguity about thepath these rare transitions take since there is only one degreeof freedom available. However, in higher dimensional prob-lems (n≥ 2), the rare transitions have more options for trav-eling between local minima of the potential V . Figure 9(a),for instance, shows the so-called rugged Mueller potential intwo dimensions (n = 2) with infinitely many possible pathsbetween any pair of local minima.

In this case, the following question arises: Given twosets D1,D2 ⊂ U, what is the most likely path the systemmay take for transitioning from set D1 to set D2? Figure 9(b)

(a) (b)

(c) (d)

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Fig. 9: Two-dimensional rugged Mueller potential. (a) Con-tour lines of the potential V . Darker colors mark smaller val-ues. (b) Contour lines of a committor function correspondingto the rugged Muller potential. (c) Contour lines of the cor-responding probability density ρ12 of transition trajectories.(d) The flow lines of the probability current J12 of the transi-tion trajectories. Figure reproduced from Ref. [124]

shows two such sets that cover the the lowest valleys of thepotential V . To answer this question, we assume that sys-tem (10) is ergodic with a unique invariant probability den-sity ρ : U→ R+ such that the probability density of observ-ing the state u is ρ(u). We would like to find the probabilitydensity ρ12(u) which corresponds to the probability that atrajectory passing through u has come from D1 and will begoing to D2. This probability density is given by

ρ12(u) = q+(u)q−(u)ρ(u), (11)

where q−,q+ : U→ [0,1] are the so-called committor func-tions. The committor function q−(u) measures the probabil-ity that a trajectory passing through u came from D1. On theother hand, q+(u) measures the probability that the trajectorypassing through u will arrive at D2 before arriving at D1.

One can show that the committor functions satisfy thebackward Kolmogorov equations

Lq+ = 0, u ∈U\(D1∪D2),

q+ = 0, u ∈ D1,

q+ = 1, u ∈ D2,

(12)

and L†q− = 0, u ∈U\(D1∪D2),

q− = 1, u ∈ D1,

q− = 0, u ∈ D2,

(13)

where L = −∇V (u) ·∇+ a(u) : ∇∇ is a linear operator andL† is its adjoint with respect to the inner product 〈α,β〉 =

8

∫U α(u)β(u)ρ(u)du (see, e.g., Refs. [125, 62]). In addition,

the density ρ satisfies the forward Kolmogorov (or Fokker–Planck) equation,

∇ · (ρ∇V )+∇∇ : (ρa) = 0, (14)

where the time derivative vanishes since the density is invari-ant.

In order to evaluate the transition probability densityρ12, one needs to solve equations (12), (13) and (14) for q+,q− and ρ, respectively. Then the transition probability den-sity is computed from (11). Figure 9(c) shows the transitionprobability density ρ12 corresponding to the rugged Muellerpotential.

Recall that the probability density ρ12(u) corresponds tothe probability that a trajectory passing through u has comefrom D1 and will be going to D2. Although useful, this prob-ability density is still a pointwise quantity which does notimmediately inform us about the most likely path the systemwill take in going from D1 to D2.

To address this shortcoming, the transition-path theoryuses the probability current J12 : U\(D1∪D2)→ Rn associ-ated with the transition probability density ρ12. The vectorfield J12 is defined such that for any codimension-one sur-face S ∈U\(D1∪D2) the integral of J12 over the surface, i.e.∫

S J12(u) ·dS(u), equals the probability flux of transition tra-jectory through S . The current J12 can be expressed explic-itly in terms of the quantities introduced previously [125,62]as

J12 = q+q−J+ρq−a∇q+−ρq+a∇q−, (15)

where J = −ρ∇V −∇ · (ρa) is the probability current asso-ciated with the probability density ρ.

Figure 9(d) shows the streamlines of the transition cur-rent J12. The color encodes the probability of the transitionalong each path such that the darker colors mark a highertransition probability. This figure finally shows the mostprobable path the transitions trajectories take in going fromD1 to D2.

Therefore, for noise-driven rare transitions, thetransition-path theory provides a rigorous framework forcomputing the most likely mechanism for the rare events. Werecall, however, that computing the transition paths in thisframework requires the solutions to three PDEs (12), (13)and (14). Solving these equations in higher dimensions arequite costly such that the applications of transition-path the-ory have been limited to two- and three-dimensional sys-tems [63]. We finally point out that a number of numeri-cal methods for approximating the rare transition paths havebeen developed in order to partially remedy this high com-putational cost [126, 127, 128, 129, 130, 131, 132].

4 Variational method for physics-based probing of ex-treme eventsIn this section, we review a recent variational method for

discovering the mechanisms that cause the extreme events.This method exploits the physics given by the governingequations (1) together with the statistical information fromthe system attractor in order to find initial states u0 that overa prescribed time interval develop into an extreme event.The hope is to learn about the mechanism that causes theextremes by examining the states that precede the extremeevents. We first introduce the variational method in a generalframework and then present two specific applications of themethod.

4.1 The variational methodConsider an observable f : U → R whose time series

along the system (1) is known to exhibit extreme events (seeDefinition 1). Also assume that there is a typical timescale τ

over which the observable grows from its typical values andincreases past its extreme value threshold fe. We thereforeseek initial states u0 ∈U such that f (Sτ(u0))> fe. This mo-tivates the definition of the domain of attraction of extremeevents as follows.

Definition 2 (Extreme Event Domain of Attraction).For an extreme event set E f ( fe) and a prescribed time τ > 0,the corresponding finite-time domain of attraction to theextreme events is the set

A f (τ, fe) = u ∈U\E f ( fe) : ∃ t ∈ (0,τ], S t(u) ∈ E f ( fe)

=

[ ⋃0<t≤τ

S−t(E f ( fe))

]\E f ( fe). (16)

Here S−t(B) is shorthand for the pre-image (S t)−1(B)of a set B ∈ B . The set A f (τ, fe) contains the states u thatat some future time t, with t ≤ τ, enter the extreme event setE f ( fe). We remove the extreme event set E f ( fe) from thedomain of attraction to exclude the states that are extreme atthe initial time.

The extreme event domain of attraction A f can be an ex-tremely complex set whose numerical estimation is a daunt-ing task. In addition, determination of the entire set may beunnecessary for deciphering the mechanisms that give rise toextreme events. Instead, one representative state from thisset may suffice in discovering the extreme event generatingmechanism.

We proposed in [133] to obtain the desired represen-tative states as the solutions of a constrained optimizationproblem. In this approach, we seek states u0 ∈U that max-imize the growth of the observable f over a prescribed timeinterval of length τ > 0. More precisely, we seek the solu-tions to the maximization problem

supu0∈A

[ f (Sτ(u0))− f (u0)] , (17)

9

where A is a subset of U to be discussed shortly. There aretwo constraints that are embedded in the optimization prob-lem (17). One constraint is enforced through S t generatedby the governing equations (1). In other words, it is implic-itly implied that u(t) = S t(u0) is a solution of the governingequations.

A second constraint is implied by requiring the state u0to belong to the subset A . We envision A to approximatethe attractor of the system (1). This constraint is essentialfor discarding exotic states that belong to the state space Ubut have negligible probability of being observed under thenatural dynamics generated by the governing equations. Itis known that dissipative differential equations often possesan attractor which is a subset of the state space [134, 135].While the system can be initialized from any arbitrary statesu0 ∈U, its trajectories quickly converges to the attractor andremain on it. As a result, much of the function space U isunexplored; the only states relevant to long term dynamics ofthe system are the ones belonging to the attractor or a smallneighborhood of it. To this end, this additional constraint notonly leads to more relevant states as precursors, but it alsoreduces the computational cost of the optimization problem,since we explore only the physically relevant solutions. Forinstance, the state space of the FitzHugh–Nagumo systemshown in figure 7 is R4. However, it is visually appreciablethat its trajectories converge to a small subset of R4.

Constraining the optimal states u0 in equation (17) tobelong to the attractor A eliminates the states that may leadto a large growth of the observable but are dynamically ir-relevant. The FitzHugh–Nagumo system, for instance, hastransient trajectories along which x becomes larger than 1.5which is much larger than the typical bursts shown in fig-ure 7. These unusually large bursts, however, occur alongtrajectories that are away from the attractor and therefore arenot sustained.

The attractor can be a very complex set whose estima-tion is quite difficult. In fact, numerical approximation of theattractors even in low-dimensional systems is an active areaof research (see, e.g., Refs. [136, 137]). For our purposesan approximate representation of the attractor is sufficient.Here, we assume that the attractor can be approximated bythe set

A = u0 ∈U : ci ≤Ci(u0)≤ ci, i = 1,2, · · · ,k, (18)

where k ∈ N determines the number of constraints, the mapsCi : U → R are smooth enough and ci,ci ∈ R are the lowerand upper bounds of Ci. The choice of the maps Ci and theirbounds depends on the problem and is elaborated in the fol-lowing sections.

With the two constraints discussed above, the optimiza-tion problem (17) can be written more explicitly as

supu0∈U

[ f (u(τ))− f (u0)] , (19a)

S (O)

O

Ef (fe)

f

fe

t

S (u0)

u0

Fig. 10: Nearby trajectories to the optimal solution also giverise to extreme events. The upper panel shows a solution u0of the optimization problem (19) and the ensuing trajectory(red curve). Trajectories passing through a sufficiently smallopen neighborhood O of u0 also give rise to extreme events.The lower panel depicts the evolution of the observable falong these trajectories.

∂tu = N(u), u(0) = u0, (19b)

ci ≤Ci(u0)≤ ci, i = 1,2, · · · ,k, (19c)

where u(t) is the shorthand notation for a trajectory of thesystem (1). If the set A is compact in U and the observ-able f and the solution map S t are smooth enough, thenthere exist solutions to problem (19). These solutions arenot necessarily unique. In fact, often there are multiple lo-cal maxima which may or may not be informative as to theorigins of the extreme events. The relevance of the localminimizers can only be determined a posteriori. There arestandard numerical methods for approximating the solutionsof the constrained optimization problems of the form (19)that we do not review here but refer the interested reader toRefs. [138, 139, 140, 141].

Let u0 denote a solution of the problem (19) correspond-ing to an extreme event, i.e., f (Sτ(u0)) > fe. We point outthat a generic trajectory of the system (1) may never exactlypass through the state u0. However, if the solution map S t

is continuous, any trajectory passing through a sufficientlysmall neighborhood of u0 will also develop into an extremeevent. This is illustrated in figure 10.

We demonstrate the application of this variationalmethod on two examples. The first example involves the dis-covery of internal energy transfers that lead to the extremeenergy dissipation episodes in a turbulent fluid flow. The sec-ond examples involves prediction of unusually large oceansurface waves, commonly known as rogue waves.

10

4.2 Application to a turbulent fluid flowIn this section, we present the application of the varia-

tional method to the extreme energy dissipation in a turbulentfluid flow. This flow is analyzed in detail in Ref. [133]; here,we reiterate our main findings and add a number of com-plementary comments. Consider the solutions to the two-dimensional incompressible Navier–Stokes equation

∂tu =−u ·∇u−∇p+ν∆u+F, ∇ ·u = 0, (20)

where u : T2×R+ → R2 is the velocity filed, p : T2 → Ris the pressure field, ν is the kinematic viscosity and thetorus T2 = [0,2π]× [0,2π] is the fluid domain with periodicboundary conditions. The velocity u(x, t) and pressure p(x, t)are functions of the spatial variables x = (x1,x2) ∈ T2 andtime t ∈ R+. The flow is driven by the deterministic Kol-mogorov forcing F = sin(k f x2)e1 where k f = 4 is the forcingwavenumber and e1 = (1 0)>. The simulations start froma random initial condition u(x,0) which is in turn propa-gated forward in time by numerically integrating the Navier–Stokes equation (20). We allow enough time elapse beforecollecting data in order to ensure that the initial transientshave decayed and the trajectory has settled to the system at-tractor.

Because of the simplicity of the forcing F and theboundary conditions, the Kolmogorov flow (i.e. the Navier–Stokes equations driven by the Kolmogorov forcing) hasbeen studied extensively both by numerical and analyticalmethods [142, 143, 144, 145, 146, 113]. Similar variants ofthe Kolmogorov flow have also been investigated experimen-tally [147, 148, 149, 150].

In spite of the simplicity of the forcing and the boundaryconditions, the Kolmogorov flow exhibits complex chaoticdynamics when the Reynolds number Re = ν−1 is suffi-ciently large. In particular, the Kolmogorov flow is knownto undergo intermittent bursts in this chaotic regime [113].The bursts are detected by monitoring certain system observ-ables such as the energy dissipation rate D : U→R+ and theenergy input rate I : U→ R,

D(u) =ν

(2π)2

∫T2|∇u|2dx, I(u) =

1(2π)2

∫T2

u ·Fdx.

(21)The energy input rate I measures the rate at which the ex-ternal forcing pumps energy into the system. The energydissipation rate D measures the rate at which the system dis-sipates energy through diffusion.

Figure 11(a) shows the time series of the energy dissipa-tion rate along a typical trajectory of the Kolmogorov flow atRe= 40. This time series clearly exhibits chaotic, short-livedbursts. The bursts of the energy dissipation are almost con-current with the bursts of the energy input rate I. This canbe inferred from figure 11(b) showing the joint probabilitydensity p

I,Dassociated with the joint probability distribution

FI,D

(I0,D0) = µ(u ∈U : I(u)≤ I0, D(u)≤ D0) , (22)

where µ is the invariant probability measure induced by thesolution map S t of the Kolmogorov flow (cf. section 2). Inpractice, the density p

I,Dis approximated from data sampled

from long-time simulations along several trajectories [151].Since the large values of I correlate strongly with the

large values of D (figure 11(b)), it is reasonable to assumethat the same mechanism instigates the bursts of both quan-tities. From a physical point of view, one is interested in theburst of the energy dissipation rate D. However, since theenergy input rate I is linear in the velocity field u, it is math-ematically more convenient to work with this quantity.

Given the simple form of the Kolmogorov forcing F =sin(k f y)e1, the energy input rate (21) can be written moreexplicitly as I(u(t)) =−Im[a(0,k f , t)] where a(k1,k2, t) ∈Care the Fourier coefficients such that

u(x, t) = ∑k∈Z2

a(k, t)|k|

(k2−k1

)eik·x, (23)

where k = (k1,k2). This Fourier series is written in adivergence-free form so that the incompressibility condition∇ · u = 0 is ensured. The energy input rate can be writ-ten in terms of the modulus r(k, t) and phase φ(k, t) of theFourier coefficients as I(u(t)) = −r(0,k f , t)sin(φ(0,k f , t))where a(k, t) = r(k, t)exp(iφ(k, t)). Therefore, there are twoscenarios through which the energy input rate I can increase:(i) For a fixed r(0,k f , t), the phase φ(0,k f , t) approaches−π/2 resulting in −sin(φ(0,k f , t)) 1 and subsequentlyincreasing I. (ii) For a fixed phase φ(0,k f , t), the modulusr(0,k f , t) increases resulting in the growth of I.

Scenario (i) implies the alignment of the external forcingF and the velocity field u(t) in the L2 function space. Thisscenario, although appearing a priori more likely, is rejectedbased on numerical observations (see Ref. [133] for moredetails). Instead, it is the increase in the modulus r(0,k f , t)that in turn leads to the increase in I during its bursts (sce-nario (ii)). The growth of r(0,k f , t) is only possible throughthe internal energy transfers operated by the nonlinear termu ·∇u. It is known that the nonlinear term redistributes theenergy (injected by the external forcing) among the Fouriermodes a(k, t) in such a way that the total transfer of energyamong modes is zero [152,153]. Note that both the nonlinearterm and the pressure gradient conserve energy since

∫T2

u · (u ·∇u)dx = 0,∫T2

u ·∇pdx = 0. (24)

Examining the structure of the Navier–Stokes in theFourier space reveals that Fourier modes are coupled to-gether in triads such that the mode a(k, t) is affected by pairsof modes a(k′, t) and a(k′′, t) with k = k′+ k′′ [152]. Eachset of modes whose wavenumbers satisfy k = k′+ k′′ are re-ferred to as a triad. Since each mode may belong to severaltriads [153], they form a complex network of triad interac-tions that continuously redistributes the energy among vari-ous modes. As a result, it is not straightforward to discern

11

(a) (b)

Fig. 11: Intermittent bursts in the Kolmogorov flow at Reynolds number Re = 40. (a) Time series of the energy dissipationrate D. (b) Logarithm of the joint probability density p

I,Dof the energy input rate I and the energy dissipation rate D.

(a)

Forcing

a(0, kf )

a(1, kf )a(1, 0)(b)

Fig. 12: The optimal solution for the Kolmogorov flow atReynolds number Re = 40 and forcing wave number k f = 4.(a) The optimal solution in the Fourier space. The colorrefers to the modulus of the Fourier modes, |a(k1,k2)|. Mostmodes are vanishingly small (white color). (b) A sketch ofthe main triad that is obtained from the optimal solution. Theother modes (−1,0), (1,k f ), etc. that are present in the op-timal solution are repetitions of these three modes due to thecomplex conjugate relation a(−k, t) = −a(k, t)∗. The redwavy arrows represent the energy dissipated by each mode.The dashed lines represent the coupling to other triads thatnot shown here.

the mode(s) responsible for the growth of the modulus of themode a(0,k f ), resulting in the bursts of the energy input I.

In Ref. [133], we employed a constrained optimizationsimilar to (19) to discover the modal interactions that causethe extreme events in the Kolmogorov flow. Skipping thedetails, figure 12 shows the obtained optimal solution in theFourier space. This optimal solution essentially consists ofthree Fourier modes with wavenumbers (0,k f ), (1,0) and(1,k f ). Interestingly, these three modes form a triad since(1,k f ) = (1,0)+ (0,k f ). Moreover, the wavenumber (0,k f )is present in this triad supporting scenario (ii) that postulatedthat the internal transfers of energy to mode a(0,k f ) are re-

0

0.2

0.4

0.6

0.8

|a(1,0)|

0 500 1000 1500 2000

t

0

0.1

0.2

0.3

0.4

|a(0,4)|

Fig. 13: Times series of the modulus of the modes a(0,k f )and a(1,0) for the Kolmogorov flow at Reynolds num-ber Re = 40 and forcing wavenumber k f = 4. Note that|a(0,k f )|= r(0,k f ).

sponsible for extreme events in the Kolmogorov flow.

Figure 13 shows the evolution of the moduli |a(0,k f )|and |a(1,0)| along a typical trajectory of the Kolmogorovflow. First, we notice that |a(0,k f )| has bursts similar tothose of the energy dissipation rate (see figure 11). Secondly,the modulus |a(1,0)| has sharp dips which are almost con-current with the bursts of |a(0,k f )|. This observation showsthat, during extreme events, the mode a(1,0) loses its en-ergy and transfers most of it to mode a(0,k f ) through thetriad interaction (1,k f ) = (1,0) + (0,k f ). The increase in|a(0,k f )|, in turn, leads to an increase in the energy inputrate I = −Im[a(0,k f )] causing the observed bursts in I (seefigure 11).

How does this transfer of energy from a low wavenum-ber (1,0) to a higher wavenumber (0,k f ) cause the bursts inthe energy dissipation rate D? To answer this question we

12

z = 0

z = 1

xz

Free Surface

Fluid Domain

(x, t)

Fig. 14: A sketch of the water wave problem. At any timet, the free surface z = η(x, t) is given as a graph over thehorizontal coordinate x.

observe that

D(u) = ν ∑k∈Z2

|k|2|a(k)|2 (25)

which follows directly from the definition of the energy dissi-pation (21) and the Fourier series (23). The transfer of energyfrom the mode a(1,0) to the mode a(0,k f ) will significantlyincrease the energy dissipation rate since the term |a(0,k f )|2is multiplied by a larger prefactor k2

f = 16 compared to theterm |a(1,0)| whose prefactor is 1.

4.3 Application to oceanic rogue wavesIn this section, we consider the real-time prediction of

rogue water waves. Rogue waves refer to unusually largewaves when compared to the surrounding waves. Whilethere is no rigorous definition of a rogue wave, it is custom-ary to define it as a wave whose height exceeds twice thesignificant wave height. For a given sea state, the significantwave height refers to four times the standard deviation of thesurface elevation [1].

As a starting point, we consider the free-surface, as anunidirectional, irrotational flow in deep seas. The surfaceelevation η : (x, t) 7→ η(x, t) is a function of the horizontalspatial variable x and time t (see figure 14). The vertical co-ordinates are denoted by the variable z, such that the velocitypotential is given by φ : (x,z, t) 7→ φ(x,z, t). In this setting,the water waves are governed by the set of equations [154],

∂φ

∂t+

12|∇φ|2 +gz = 0, z = η(x, t), (26a)

∆φ = 0, −∞ < z < η(x, t), (26b)

∂φ

∂z= 0, z =−∞, (26c)

∂η

∂t+

∂φ

∂x∂η

∂x− ∂φ

∂z= 0, z = η(x, t). (26d)

(a) (b)

(c)

Fig. 15: Breather solutions of the NLS equation. (a) TheMa breather [159] is periodic in time and localized in space.(b) The Akhmediev breather [160] is localized in time andperiodic in space. (c) The Peregrine breather [161] is doublylocalized in both space and time.

Equation (26a) is the Bernoulli equation for irrotational flowswith a free surface. Equation (26b) follows from the conser-vation of mass. Equations (26c) and (26d) are the boundaryconditions at the bottom of the sea and the surface, respec-tively. The constant g denotes the gravitational acceleration.

To solve the water wave equations (26) numerically, weneed the initial surface elevation η(x,0) and the initial veloc-ity potential φ(x,z,0). While the practical measurement ofthe surface elevation is possible [155,156,157,158], measur-ing the entire velocity potential beneath the surface remainsa challenging task. Therefore, it is highly desirable to decou-ple the surface evolution from the velocity potential. Thismotivates the use of the so-called envelope equations, an ap-proximation to the water wave equations that only involvesthe surface elevation η.

The envelope equations govern the evolution of pertur-bations to the linear waves

φ =agω0

ek0z sin(k0x−ω0t), η = acos(k0x−ω0t), (27)

where a≥ 0 is the wave amplitude, λ0 = 2πk−10 is the wave-

length and T0 = 2πω−10 is the wave period. The pair (27) is

referred to as a linear wave since it is an exact solution ofthe linear part of the water wave equation (26). The wavefrequency ω0 and the wave number k0 satisfy the linear dis-persion relation ω0(k0) =

√gk0. The envelope equations

describe the evolution of small perturbations to the linearwave (27). These perturbations are of the modulation form

η(x, t) = Re

u(x, t)ei(k0x−ω0t), (28)

13

where u ∈ C is the complex wave envelope. Perturbationanalysis shows that, to the first order, u(x, t) satisfies the non-linear Schrodinger (NLS) equation [162, 163, 164],

∂u∂t

+12

∂u∂x

+i8

∂2u∂x2 +

i2|u|2u = 0, (29)

where we have normalized the space and time variables withthe wavelength and wave period of the underlying periodicwave train so that x 7→ k0x and t 7→ ω0t. This perturbationanalysis is valid under certain assumptions [164, 165], in-cluding that the wave steepness ε = ak0 is small, i.e., ε 1.These assumptions can be relaxed by considering higher-order terms in the perturbation analysis [163, 166, 167].

Several exact solutions of the NLS equation have beenfound over the years. Figure 15 shows three types of theso-called breather solutions of the NLS equation. These so-lutions are localized in time or space or both. Of particularinterest to us is the Peregrine breather (panel c) since it mim-ics the rogue waves in the sense that it starts from a planewave, develops into a localized large wave and again decaysto a plane wave.

The breather solutions have been observed in carefullycontrolled experiments [165, 168, 169, 170, 171]. However,real ocean waves are irregular wave fields consisting of manydispersive wave groups so that the detection of breathersfrom a given wave field becomes a difficult task [172]. Moreimportantly, these exact breather solutions are not the onlypossible mechanism for rogue wave formation. For instance,Cousins and Sapsis [173] studied the evolution of initialwave groups of the form |u|=A0 sinh(x/L0) for various com-binations of wave amplitude A0 and length scale L0. Theyfind a range of parameters (A0,L0) where the initially smallwave groups develop into a rogue wave at a later time whenevolved under the NLS equation.

One can approximate an irregular wave field as a super-position of localized wave groups with sech envelopes,

|u(x)| 'n

∑i=1

Aisech(

x− xi

Li

), (30)

where the parameters (Ai,Li,xi) are chosen so that the ap-proximation error is minimized. An example of such a de-composition is shown in figure 16(a). Figure 16(b) showsthe joint probability density function (PDF) of the parame-ters (Ai,Li). This PDF is computed by approximating manyrealizations of random waves with the superposition (30).

This joint PDF contains several interesting pieces of in-formation. In particular, it indicates the most likely com-bination of length scale and amplitude of wave groups ina given random sea (marked with a white square). Thesewave groups, however, do not necessarily develop into roguewaves. The solid black curve in figure 16(b) marks theboundary between wave groups that develop into a roguewave at some point in the future (the wave groups abovethe curve) and those that do not (the wave groups below the

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Most likely wave group

Most likely wave group to develop into a rouge wave

(a)

(b)

Fig. 16: Reduced-order prediction of rogue waves. (a) An ir-regular wave field (solid blue) is approximated by the super-position of localized wave groups (dotted red). (b) The jointprobability density function of the length scales L and am-plitudes A obtained from decomposing many realizations ofrandom wave fields into localized wave groups. Darker col-ors mark higher probability. The solid black curve mark theboundary above which the wave groups develop into a roguewave at some point in the future when evolved under the NLSequation. The dashed red curve marks the same boundarybut under the modified NLS (MNLS) equation [166]. Figurereproduced from Ref. [174].

curve). The intersection of this curve with the joint PDF de-termines the most ‘dangerous’ waves (marked by a white cir-cle), i.e., the most likely wave groups that will develop intoa rogue waves at some point in the future.

Cousins and Sapsis [174] used this information to de-velop a reduced-order method for prediction of rogue wavesin unidirectional water waves in deep sea. This method doesnot require the numerical integration of the NLS equationsand, as a result, is computationally much less expensive. Inaddition, the reduced-order model only requires the knowl-edge of localized wave groups that form the wave field. Assuch, this method can be applied to cases where the wavefield is only partially known or when the measurement res-olution is low. Later, Farazmand and Sapsis [175] gener-alized the reduced-order prediction of rogue waves to two-dimensional water waves.

We finally point out that the most ‘dangerous’ wavegroups could alternatively be found as solutions to a con-strained optimization problem similar to (17). However,since the computational cost of generating figure 16 is notprohibitive, the most dangerous waves were estimated di-rectly from the joint PDF. Later, in the context of large de-viation theory, Dematteis et al. [176] obtained similar resultsby solving a constrained optimization problem.

14

U

f

g

t + t1 t + t2t Time

Trajectory

u

St1(u)

St2(u)

Fig. 17: The evolution of the observable f and the indicatorg along a trajectory in the state space U. The large value ofthe observable g at a state u signals an upcoming large valueof the observable f over the future time interval [t1, t2].

5 Prediction of extreme eventsIn this section, we turn our attention to the discovery of

indicators of extreme events. Given an observable f : U →R of the system (1), we seek indicators g : U → R whoseevolution along a trajectory u(t) signal an upcoming extremevalue of the observable f . This is sketched in figure 17 wherethe indicator g attains a relatively large value at time t justbefore the observable f attains a large value over the futuretime interval [t + t1, t + t2]. Note that the indicator g is itselfan observable of the system, but it is carefully chosen suchthat it predicts the extreme events associated with f .

As a first step, we need to quantify the predictive skill ofan observable g. To this end, we define a number of quanti-ties. In particular, we define the maximum observable valuesover a future time interval,

fm(u; t1, t2) = maxt1≤τ≤t2

f (Sτ(u)) , (31)

where u ∈ U is a state and 0 < t1 ≤ t2. We refer to t1 asthe time horizon of the prediction. In the special case wheret1 = t2, we have

fm(u; t1, t1) = f (S t1(u)), (32)

where fm(u; t1, t1) is the value of the observable at t1 timeunits in the future if the current state of the system is u. If t1 6=t2, then fm(u; t1, t2) is the maximum of the observable f overthe future time interval [t1, t2] along the trajectory passingthrough the state u. Our goal therefore is to find an indicatorg : U → R whose large values correlate strongly with the

large values of fm(·; t1, t2) for appropriate choices of t1 andt2. We quantify this correlation through conditional statistics.

5.1 Conditional statistics for extreme eventsLet p

fm,g∈ L1(R2) denote the probability density asso-

ciated with the joint probability distribution,

Ffm,g

( f0,g0) = µ(u ∈U : fm(u; t1, t2)≤ f0,g(u)≤ g0)

=∫ f0

−∞

∫ g0

−∞

pfm,g

(a,b)dbda, (33)

for given 0 < t1 < t2. Similarly, let pg∈ L1(R) denote the

probability density associated with the distribution

Fg(g0) = µ(u ∈U : g(u)≤ g0)

=∫ g0

−∞

pg(b)db. (34)

Therefore, the conditional probability density pfm|g is given

by

pfm|g =

pfm,g

pg

. (35)

Roughly speaking, the conditional probability densityp

fm|g( f0,g0) measures the probability of fm(u; t1, t2) = f0

given that g(u) = g0.Recall from Definition 1 that an extreme event corre-

sponds to f > fe. Therefore, an extreme event takes placeover the future time interval [t1, t2] if fm(u; t1, t2) > fe. Anideal indicator g of extreme events should have a correspond-ing threshold ge such that g(u)> ge implies fm(u; t1, t2)> fe.Conversely, g(u) < ge indicates that no upcoming extremeevents are expected, that is fm(u; t1, t2) < fe. The corre-sponding conditional PDF p

fm|g of such an ideal indicator isshown in figure 18(a). Unsuccessful predictions correspondto the cases where either g(u)> ge and fm(u; t1, t2)< fe org(u)< ge and fm(u; t1, t2)> fe.

These four possibilities are summarized below:

(I) Correct Rejections: g < ge and fm < fe.The indicator correctly predicts that no extreme eventsare upcoming.

(II) Correct Predictions: g > ge and fm > fe.The indicator correctly predicts an upcoming extremeevent.

(III) False Negatives: g < ge and fm > fe.The indicator fails to predict an upcoming extremeevent.

(IV) False Positives: g > ge and fm < fe.The indicator falsely predicts an upcoming extremeevent.

15

(a) (b)

g gge

1

0.5

Pee Pee

fm

(I)

(II)

(III)

(IV)

fm

g

(I)

(II)

(III)

(IV)

g

g = ge

fm = fe

Fig. 18: Two possible conditional PDFs for the predictor g(t) of a future extreme event fm(t; t1, t2) = maxτ∈[t+t1,t+t2] f (τ) ofan observable f . (a) A skillful predictor characterized by low false positives and low false negatives. (b) A ‘bad’ predictorthat returns high false negatives. (c) A ‘bad’ predictor that returns high false positives and high false negatives.

These possibilities divide the conditional PDF plots ofp

fm|g into four quadrants (see figure 18). Figure 18(a)sketches the conditional PDF corresponding to a reliable in-dicator: there is a threshold ge for which negligible false pos-itives and false negatives are recorded (low density in quad-rants III and IV). Figure 18(b), on the other hand, sketchesan unreliable predictor. For this indicator there is no choiceof the threshold ge that leads to negligible amount of falsepositives and false negatives. The sketched threshold, for in-stance, returns no false positives but at the same time, doesnot predict any of the extreme events, hence, returning highfalse negatives.

The conditional PDF pfm|g also enables us to quantify

the probability that an extreme event will take place over thefuture time interval [t1, t2], given the value of the indicator atthe present time. More precisely, we can measure the proba-bility that fm(u; t1, t2)> fe, given that g(u) = g0. We refer tothis quantity as the probability of upcoming extreme events(or probability of extreme events, for short).

Definition 3 (Probability of Upcoming Extremes).For a given observable f : U → R, its associated futuremaximum fm(·; t1, t2) : U→ R and an indicator g : U→ R,we define the probability of an upcoming extreme event as

Pee(g0) =∫

∞

fep

fm|g(a,g0)da, (36)

where pfm|g is the conditional PDF defined in (35) and fe is

the threshold of extreme events (see Definition 1).

Roughly speaking, in terms of the invariant probabilitymeasure µ, Pee(g0) measures

µ(u ∈U : fm(u; t1, t2)> fe | g(u) = g0) . (37)

For a reliable indicator g, we have Pee(g0)' 0 if g0 < ge andPee(g0)' 1 if g0 > ge, with a sharp transition in between (seefigure 18).

5.2 ApplicationsNow we demonstrate how these quantities are applied

in practice by returning to the examples discussed in sec-tions 4.2 and 4.3. Recall from section 4.2 that the extremeevents in the Kolmogorov flow (i.e., large values of the en-ergy dissipation rate) occur when a significant amount of en-ergy is transfered from the mode a(1,0) to the forcing modea(0,k f ). As a result, during the extreme events, the modea(1,0) loses energy, resulting in relatively small values of|a(1,0)|. Visual examination of the time series of the energydissipation rate D and modulus |a(1,0)| suggest that this en-ergy loss takes place shortly before the extreme values of theenergy dissipation rate are registered. This observation sug-gests that small values of |a(1,0)| can be used for short-termprediction of the extreme events.

The conditional statistics discussed above allows us toquantify the extent to which such predictions are feasible.

16

(a)

-0.8 -0.6 -0.4 -0.2 0

g

0

0.2

0.4

0.6

0.8

1

Pee

(b)

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

Pee

Prediction of an upcoming extreme

t

f

(c)

Fig. 19: Prediction of extreme events in the Kolmogorov flow at Reynolds number Re = 40 and forcing wave number k f = 4.(a) Conditional probability density p

fm|g where fm(u; t1, t2) = maxτ∈[t1,t2] D(u(τ)) is the maximum future values of the energydissipation rate to be predicted and the indicator g(u) = −|a(1,0)| is the indicator. (b) Probability of future extreme eventsPee as a function of the indicator g(u) =−|a(1,0)|. (c) An instance of an extreme event and its short-term prediction signaledby Pee = 0.5. The observable being predicted is the energy dissipation rate, i.e., f (u(t)) = D(u(t)).

Figure 19(a) shows the conditional PDF pfm|g where the ob-

servable f is the energy dissipation rate (21), i.e, f (u) =D(u). The indicator is chosen to be g = −|a(1,0)|. Theminus sign ensures that relatively large values (although neg-ative) of the indicator correlate with the large values of theobservable.

We point out a number of the important features of thisfigure. Most importantly, very small portion of the condi-tional probability density resides in the false positive or falsenegative regions (quadrants III and IV). Since the extremeevents are rare, most of the density is concentrated in fm < feregion. This region correlates strongly with g < ge whichmeans the indicator successfully rules out the non-extremedynamics (quadrant I). Conversely, we see also a high cor-relation between fm > fe and g > ge which means that theindicator successfully identifies upcoming extreme events.This is better captured through the resulting probability ofupcoming extreme events Pee shown in figure 19(b). Forg < ge = −0.5, we have Pee ' 0 that means the probabil-ity of upcoming extreme events is almost zero. Conversely,for ge > −0.3, we have Pee ' 1, that is an extreme event isalmost certainly upcoming. Due to the monotonicity of Pee,there is a point where Pee = 0.5 corresponding to an indicatorthreshold g = ge which in this case is approximately −0.39.

Figure 19(c) shows the application of the indicator topredicting extreme events along a trajectory of the Kol-mogorov flow. Along this trajectory, the indicator g is mea-sured and the resulting Pee(g) is computed. Most of the time,the probability of extreme events is almost zero. At aroundtime t = 30, however, this probability increases rapidly andeventually passes the threshold Pee = 0.5, signaling an immi-nent extreme event in the near future.

Similar results are obtained for the prediction of roguewaves. Recall from section 4.3 that rogue waves developfrom localized wave groups with certain range of lengthscales and amplitudes. Cousins and Sapsis [174] proposedan indicator g of upcoming rogue waves by projecting the

wave envelope u unto a subspace which captures this ‘dan-gerous’ range of length scales and amplitudes (see Ref. [174]for further details). The larger the projection, the more likelyis the occurrence of a future rogue wave.

Figure 20 shows the resulting conditional PDF pfm|g and

the probability of upcoming rogue waves Pee. Here, the ob-servable is the maximum wave amplitude over the entire do-main, i.e., f = maxx |u(x, t)|. This conditional PDF has asimilar structure to that of the Kolmogorov flow shown infigure 19: strong correlation between small (resp. large) val-ues of the indicator g and the small (resp. large) values of thefuture observable fm. However, the conditional probabilitydensity in figure 20(a) has a more significant density in quad-rant III, i.e., there is a higher probability of false negatives.This is also reflected in the probability of upcoming extremesPee shown in figure 20(b). Note that even for small values ofthe indicator, g < 0.1, there is a non-negligible probability ofextremes, 0.05 < Pee < 0.2. Contrast this with figure 19(b)where for small indicator values, the probability of future ex-tremes is almost zero.

Nonetheless, the false negatives comprise only 5.9% ofthe predictions which is relatively low. A more reliable in-dicator of future rogue waves would have an even lower rateof false negatives (as well as false positives). In the nextsection, we discuss possible methods for discovering mostreliable indicators of extreme events for a given dynamicalsystem.

5.3 Data-driven discovery of indicatorsIn section 5.2, we demonstrated that the analysis of the

structure of the governing equations assisted with the varia-tional method of section 4 can lead to the discovery of reli-able indicators of extreme events. In the Kolmogorov flow(section 4.2), for instance, we showed that such a reliableindicator is the modulus of a particular Fourier mode.

This approach relies on the solution of a solutions ofa constrained optimization problem involving the governing

17

fm g

Peeg

Fig. 20: Prediction of rogue waves. (a) The conditional PDF pfm|g where the observable is the maximum wave height,

f (u) = maxx |u(x, t)|. (b) The resulting probability of extreme events Pee as defined in Definition 3. Figure reproducedfrom [174].

equations of the system. One may wonder whether there isa purely data-driven method for discovery of reliable indica-tors of extreme events. For Kolmogorov flow, for instance,it is quite possible that a carefully customized data analysistechnique, applied to a long term simulation data, could haveled to the discovery of the same indicator.

To date, a systematic framework for discovery of indica-tors of extreme events from data is missing. In the remainderof this section, we briefly sketch properties that such an ap-proach should have. Recall that a reliable indicator of ex-treme events should return low rates of false positive andfalse negative predictions. An indicator that constantly is-sues alarms of upcoming extremes will correctly ‘predict’the extreme events. However, this indicator is not desirablesince it also returns a large number of false alarms. Con-versely, an indicator that never issues an alarm, will have nofalse alarms but will also miss all the extreme events. There-fore, a reliable indicator is one that returns minimal numberof combined false positives and false negatives.

The false positive and false negative predictions can becombined into a quantity called the failure rate. For an ob-servable f : U→R with the extreme event threshold fe ∈R,the failure rate of an indicator g : U→ R is

L(g;ge, t1, t2) :=µ(u ∈U : fm(u; t1, t2)> fe|g(u)< ge)+

µ(u ∈U : fm(u; t1, t2)< fe|g(u)> ge) ,(38)

where fm(·; t1, t2) : U→ R is the future maximum of the ob-servable f as defined in (31) and ge ∈R is the alarm thresholdsuch that g > ge signals an upcoming extreme event. Notethat L(·;ge, t1, t2) : L∞(U)→ [0,1] measures the probabilityof false negative ( fm > fe given that g< ge) and false positive( fm < fe given that g > ge) predictions.

It follows from the definition of the conditional PDF

pfm|g that the failure rate is equal to

L(g;ge, t1, t2) =∫

∞

fe

∫ ge

−∞

pfm|g(a,b)dbda︸ ︷︷ ︸

False Negatives

+

∫ fe

−∞

∫∞

ge

pfm|g(a,b)dbda︸ ︷︷ ︸

False Positives

. (39)

This expression measures the conditional density residing inquadrants (III) and (IV) of figure 18, measuring the false neg-atives and the false positives, respectively.

The failure rate depends on the indicator g ∈ L∞(U) andthree parameters, ge, t1 and t2. The main objective is to findan indicator g that minimizes the failure rate. However, oneshould simultaneously search for the appropriate parameters(ge, t1, t2). In figure 19, for instance, the predictions corre-spond to t1 = 1 and t2 = t1 +1. If we gradually increase theprediction horizon t1 the prediction skill of the indicator de-teriorates such that for t1 > 4 the indicator loses any predic-tive power. This finite-time predictability is expected sincein chaotic systems the observables tend to have finite corre-lation times. A similar observation is valid for the indicatorthreshold ge. Therefore, the minimization of the failure rateL should be carried out simultaneously over the measurableobservables g and the parameters (ge, t1, t2).

The resulting minimizer is a reliable indicator of ex-treme events. Solving this optimization problem, however,is not straightforward because of the nonlinear and non-smooth dependence of the failure rate L on the function g(see Eqs. (35) and (39)). Treatment of this optimization prob-lem will be addressed elsewhere.

6 Summary and conclusionsThe study of extreme events can be divided into four

components: mechanisms, real-time prediction, mitigationand statistics. Compared to the last topic that has been stud-ied deeply for certain systems [55, 177], the other three as-pects have received less attention. In this review, we focused

18

on two of these aspects, namely mechanisms and real-timeprediction, and reviewed the quantitative treatment of them.

Mechanisms that lead to the formation of extreme eventsare not unique. Depending on the system, they can be, forinstance, a result of multiscale instabilities, driven by noiseor the consequence of nonlinear energy transfers. Yet, ourreview suggests that there might be a unified mathematicalframework for discovering these mechanisms.

In high-dimensional chaotic systems, the mechanismsunderlying the extreme events are usually difficult to discernby relying solely on observation (or simulation) data. A suc-cessful method for discovering the underlying mechanismsshould take a blended approach combining the governingequations of the system with the observation data or somelow-order statistics. For instance, the variational method ofSection 4 seeks the extreme event mechanisms as the solu-tions of a constrained optimization problem. Here the gov-erning equations are used to form an appropriate objectivefunctional and the observation data is used to form the ap-propriate constraints.

Prediction of individual extreme events is another aspectreviewed here. The prediction problem consists of designinga reliable indicator function whose behavior (e.g., large val-ues) signals an upcoming extreme event. A reliable indicatoris one that returns relatively low rates of false positive andfalse negative predictions. We saw that even partial knowl-edge of the mechanisms that lead to the extremes can informthe choice of a reliable indicator. While the discovery of themechanisms relies on the governing equations, the predic-tions can be performed in a purely data-driven fashion. Thisof course assumes that the derived indicator can be measuredor observed in practice.

Discovery of reliable indicators of extreme events di-rectly from observed data is highly desirable. This is spe-cially the case for problems, such as earthquakes, epilepticseizures, and social dynamics, where the governing equa-tions are unknown. In section 5.3, we sketched some de-sirable properties that such a reliable indicator should have.We also outlined several technical problems surrounding thisapproach that remain unresolved and should be addressed infuture work.

As for the mitigation of extreme events, very little hasbeen done. The existing studies are narrow in scope and re-volve around arbitrary perturbations that may nudge the sys-tem away from extreme events. Therefore, a control theo-retic study of the mitigation of extreme events merits fur-ther investigation. This includes formulating the problem interms of observable quantities that can be measured in prac-tice, as well as control variables that can be adjusted. Thisgreatly limits the admissible perturbations to the system andsheds light on the practical limitations of mitigating extremeevents.

Finally, we point out that our discussion of extremeevents was limited to autonomous systems. These are sys-tems governed by a fixed set of principles. Our discus-sion does not apply to non-autonomous systems, such asstock markets [178] or social networks [179, 180], wherethe rules of engagement can change over time. While non-

autonomous dynamical systems have been studied exten-sively [181, 182], the literature on extreme events in thesesystems is vanishingly small and remains an attractive areato be investigated.

AcknowledgementsThis work has been supported through the ARO MURI

grant W911NF-17-1-0306, the ONR MURI grant N00014-17-1-2676 and the AFOSR grant FA9550-16-1-0231.

References[1] Dysthe, K., Krogstad, H. E., and Muller, P., 2008.

“Oceanic rogue waves”. Annu. Rev. Fluid Mech., 40,pp. 287–310.

[2] Donelan, M. A., and Magnusson, A.-K., 2017. “Themaking of the Andrea wave and other rogues”. Scien-tific Reports, 7, p. 44124.

[3] Ropelewski, C. F., and Halpert, M. S., 1987. “Globaland regional scale precipitation patterns associatedwith the El Nino/Southern Oscillation”. MonthlyWeather Review, 115(8), pp. 1606–1626.

[4] Easterling, D. R., Evans, J., Groisman, P. Y., Karl,T. R., Kunkel, K. E., and Ambenje, P., 2000.“Observed variability and trends in extreme climateevents: a brief review”. Bulletin of the American Me-teorological Society, 81(3), pp. 417–425.

[5] Moy, C. M., Seltzer, G. O., Rodbell, D. T., and An-derson, D. M., 2002. “Variability of El Nino/SouthernOscillation activity at millennial timescales during theHolocene epoch”. Nature, 420(6912), p. 162.

[6] Dakos, V., Scheffer, M., van Nes, E. H., Brovkin,V., Petoukhov, V., and Held, H., 2008. “Slowingdown as an early warning signal for abrupt climatechange”. Proceedings of the National Academy of Sci-ences, 105(38), pp. 14308–14312.

[7] United States Geological Survey, 2018.https://earthquake.usgs.gov/.

[8] Crucitti, P., Latora, V., and Marchiori, M., 2004.“Model for cascading failures in complex networks”.Phys. Rev. E, 69, p. 045104.

[9] Fang, X., Misra, S., Xue, G., and Yang, D., 2012.“Smart gridthe new and improved power grid: Asurvey”. IEEE communications surveys & tutorials,14(4), pp. 944–980.

[10] Scheffer, M., Bascompte, J., Brock, W. A., Brovkin,V., Carpenter, S. R., Dakos, V., Held, H., Van Nes,E. H., Rietkerk, M., and Sugihara, G., 2009. “Early-warning signals for critical transitions”. Nature,461(7260), pp. 53–59.

[11] Ghil, M., Yiou, P., Hallegatte, S., Malamud, B.,Naveau, P., Soloviev, A., Friederichs, P., Keilis-Borok, V., Kondrashov, D., Kossobokov, V., et al.,2011. “Extreme events: dynamics, statistics and pre-diction”. Nonlinear Processes in Geophysics, 18(3),pp. 295–350.

[12] Dakos, V., Carpenter, S. R., Brock, W. A., Ellison,

19

A. M., Guttal, V., Ives, A. R., Kefi, S., Livina, V.,Seekell, D. A., van Nes, E. H., et al., 2012. “Meth-ods for detecting early warnings of critical transitionsin time series illustrated using simulated ecologicaldata”. PloS one, 7(7), p. e41010.

[13] Ben-Menahem, A., and Singh, S. J., 2012. Seismicwaves and sources. Springer Science & Business Me-dia.

[14] Murphy, J. M., Sexton, D. M., Barnett, D. N., Jones,G. S., Webb, M. J., Collins, M., and Stainforth, D. A.,2004. “Quantification of modelling uncertainties in alarge ensemble of climate change simulations”. Na-ture, 430(7001), p. 768.

[15] Scarrott, C., and MacDonald, A., 2012. “A reviewof extreme value threshold estimation and uncertaintyquantification”. REVSTAT–Statistical Journal, 10(1),pp. 33–60.

[16] Mohamad, M. A., and Sapsis, T. P., 2015. “Prob-abilistic description of extreme events in intermit-tently unstable dynamical systems excited by corre-lated stochastic processes”. SIAM/ASA Journal onUncertainty Quantification, 3(1), pp. 709–736.

[17] Doucet, A., De Freitas, N., and Gordon, N., 2001.“An introduction to sequential Monte Carlo meth-ods”. In Sequential Monte Carlo methods in practice.Springer, pp. 3–14.

[18] Majda, A. J., and Harlim, J., 2012. Filtering complexturbulent systems. Cambridge University Press.

[19] Vanden-Eijnden, E., and Weare, J., 2013. “Data as-similation in the low noise regime with applicationto the Kuroshio”. Monthly Weather Review, 141(6),pp. 1822–1841.

[20] Altwegg, R., Visser, V., Bailey, L. D., and Erni, B.,2017. “Learning from single extreme events”. Phil.Trans. R. Soc. B, 372(1723), p. 20160141.

[21] Alligood, K. T., Sauer, T. D., and Yorke, J. A.,1996. Chaos: An Introduction to Dynamical Systems.Springer.

[22] Hirsch, M. W., Smale, S., and Devaney, R. L., 2012.Differential Equations, Dynamical Systems, and anIntroduction to Chaos. Academic Press.

[23] Turitsyn, K., Sulc, P., Backhaus, S., and Chertkov, M.,2011. “Options for control of reactive power by dis-tributed photovoltaic generators”. Proceedings of theIEEE, 99(6), pp. 1063–1073.

[24] Susuki, Y., and Mezic, I., 2012. “Nonlinear Koopmanmodes and a precursor to power system swing insta-bilities”. IEEE Transactions on Power Systems, 27(3),pp. 1182–1191.

[25] Belk, J. A., Inam, W., Perreault, D. J., and Turitsyn,K., 2016. “Stability and control of ad hoc dc micro-grids”. In Decision and Control (CDC), 2016 IEEE55th Conference on, IEEE, pp. 3271–3278.

[26] Cavalcante, H. L. d. S., Oria, M., Sornette, D., Ott, E.,and Gauthier, D. J., 2013. “Predictability and suppres-sion of extreme events in a chaotic system”. PhysicalReview Letters, 111(19), p. 198701.

[27] Galuzio, P. P., Viana, R. L., and Lopes, S. R., 2014.

“Control of extreme events in the bubbling onsetof wave turbulence”. Physical Review E, 89(4),p. 040901.

[28] Chen, Y.-Z., Huang, Z.-G., and Lai, Y.-C., 2014.“Controlling extreme events on complex networks”.Nature Scientific Reports, 4, p. 6121.

[29] Chen, Y.-Z., Huang, Z.-G., Zhang, H.-F., Eisen-berg, D., Seager, T. P., and Lai, Y.-C., 2015. “Ex-treme events in multilayer, interdependent complexnetworks and control”. Nature Scientific Reports, 5,p. 17277.

[30] Bialonski, S., Ansmann, G., and Kantz, H., 2015.“Data-driven prediction and prevention of extremeevents in a spatially extended excitable system”. Phys-ical Review E, 92(4), p. 042910.

[31] Joo, H. K., Mohamad, M. A., and Sapsis, T. P., 2017.“Extreme events and their optimal mitigation in non-linear structural systems excited by stochastic loads:Application to ocean engineering systems”. OceanEngineering, 142, pp. 145 – 160.

[32] Morgan, M. G., Henrion, M., and Small, M., 1990.Uncertainty: a guide to dealing with uncertainty inquantitative risk and policy analysis. Cambridge Uni-versity Press.

[33] Wilmott, P., 2007. Paul Wilmott introduces quantita-tive finance. John Wiley & Sons.

[34] McNeil, A. J., Frey, R., and Embrechts, P., 2015.Quantitative Risk Management: Concepts, Tech-niques and Tools. Princeton University Press.

[35] Longin, F., 2017. Extreme Events in Finance: AHandbook of Extreme Value Theory and its Applica-tions. Wiley handbooks in financial engineering andeconometrics. John Wiley & Sons, Inc.

[36] de Haan, L., and Ferreira, A., 2007. Extreme ValueTheory: An Introduction. Springer Science & Busi-ness Media.

[37] Frechet, M. “Sur la loi de probabilite de l’ecart maxi-mum”. In Ann. Soc. Polon. Math, Vol. 6, pp. 93–116.

[38] Fisher, R. A., and Tippett, L. H. C., 1928. “Limit-ing forms of the frequency distribution of the largestor smallest member of a sample”. In MathematicalProceedings of the Cambridge Philosophical Society,Vol. 24, pp. 180–190.

[39] Gnedenko, B., 1943. “Sur la distribution limite duterme maximum d’une serie aleatoire”. Annals ofMathematics, 44(3), pp. 423–453.

[40] Watson, G. S., 1954. “Extreme values in samples fromm-dependent stationary stochastic processes”. Annalsof Mathematical Statistics, 25(4), pp. 798–800.

[41] Loynes, R. M., 1965. “Extreme values in uniformlymixing stationary stochastic processes”. Annals ofMathematical Statistics, 36(3), pp. 993–999.

[42] Leadbetter, M. R., 1974. “On extreme values instationary sequences”. Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete, 28(4), Dec,pp. 289–303.

[43] Leadbetter, M. R., 1983. “Extremes and local depen-dence in stationary sequences”. Probability Theory

20

and Related Fields, 65(2), pp. 291–306.[44] Hsing, T., Husler, J., and Leadbetter, M. R., 1988.

“On the exceedance point process for a stationarysequence”. Probability Theory and Related Fields,78(1), pp. 97–112.

[45] Leadbetter, M. R., and Nandagopalan, S., 1989. “Onexceedance point processes for stationary sequencesunder mild oscillation restrictions”. In Extreme valuetheory. pp. 69–80.

[46] Chernick, M. R., Hsing, T., and McCormick, W. P.,1991. “Calculating the extremal index for a class ofstationary sequences”. Advances in Applied Probabil-ity, 23(4), p. 835850.

[47] Freitas, A. C. M., and Freitas, J. M., 2008. “Onthe link between dependence and independence in ex-treme value theory for dynamical systems”. Statistics& Probability Letters, 78(9), pp. 1088 – 1093.

[48] Freitas, A. C. M., Freitas, J. M., and Todd, M., 2015.“Speed of convergence for laws of rare events and es-cape rates”. Stochastic Processes and their Applica-tions, 125(4), pp. 1653 – 1687.

[49] Lucarini, V., Faranda, D., de Freitas, A. C. G. M. M.,de Freitas, J. M. M., Holland, M., Kuna, T., Nicol, M.,Todd, M., and Vaienti, S., 2016. Extremes and Recur-rence in Dynamical Systems. John Wiley & Sons.

[50] Cramer, H., 1938. “Sur un nouveau theoreme-limitede la theorie des probabilites”. Actual. Sci. Ind., 736,pp. 5–23.

[51] Donsker, M. D., and Varadhan, S. R. S., 1975.“Asymptotic evaluation of certain Markov process ex-pectations for large time, I”. Communications on Pureand Applied Mathematics, 28(1), pp. 1–47.

[52] Donsker, M. D., and Varadhan, S. R. S., 1975.“Asymptotic evaluation of certain Markov process ex-pectations for large time, II”. Communications onPure and Applied Mathematics, 28(2), pp. 279–301.

[53] Donsker, M. D., and Varadhan, S. R. S., 1976.“Asymptotic evaluation of certain Markov process ex-pectations for large time, III”. Communications onpure and applied Mathematics, 29(4), pp. 389–461.

[54] Donsker, M. D., and Varadhan, S. R. S., 1983.“Asymptotic evaluation of certain Markov process ex-pectations for large time. IV”. Communications onPure and Applied Mathematics, 36(2), pp. 183–212.

[55] Varadhan, S. R. S., 2008. “Large deviations”. Ann.Probab., 36(2), pp. 397–419.

[56] Touchette, H., 2009. “The large deviation approachto statistical mechanics”. Physics Reports, 478(1),pp. 1–69.

[57] Touchette, H., 2011. “A basic introduction to largedeviations: Theory, applications, simulations”. arXivpreprint arXiv:1106.4146.

[58] Eyring, H., 1935. “The activated complex in chemicalreactions”. The Journal of Chemical Physics, 3(2),pp. 107–115.

[59] Evans, M. G., and Polanyi, M., 1935. “Some applica-tions of the transition state method to the calculationof reaction velocities, especially in solution”. Trans-

actions of the Faraday Society, 31, pp. 875–894.[60] Laidler, K. J., and King, M. C., 1983. “Development

of transition-state theory”. The Journal of physicalchemistry, 87(15), pp. 2657–2664.

[61] Truhlar, D. G., Garrett, B. C., and Klippenstein, S. J.,1996. “Current status of transition-state theory”. TheJournal of physical chemistry, 100(31), pp. 12771–12800.

[62] Vanden-Eijnden, E., 2006. “Transition path the-ory”. Computer Simulations in Condensed MatterSystems: From Materials to Chemical Biology Volume1, pp. 453–493.

[63] Metzner, P., Schtte, C., and Vanden-Eijnden, E., 2006.“Illustration of transition path theory on a collection ofsimple examples”. The Journal of Chemical Physics,125(8), p. 084110.

[64] Weinan, E., and Vanden-Eijnden, E., 2010.“Transition-path theory and path-finding algo-rithms for the study of rare events”. Annual Review ofPhysical Chemistry, 61(1), pp. 391–420.

[65] van der Pol, B., 1926. “LXXXVIII. on “relaxation-oscillations””. The London, Edinburgh, and DublinPhilosophical Magazine and Journal of Science,2(11), pp. 978–992.

[66] van der Pol, B., 1934. “The nonlinear theory of elec-tric oscillations”. Proceedings of the Institute of RadioEngineers, 22(9), pp. 1051–1086.

[67] Benoıt, E., 1983. “Systemes lents-rapides dans r3 etleurs canards”. Asterisque, 109-110, pp. 159–19.

[68] Field, R. J., and Noyes, R. M., 1974. “Oscillations inchemical systems. IV. Limit cycle behavior in a modelof a real chemical reaction”. The Journal of ChemicalPhysics, 60(5), pp. 1877–1884.

[69] Gillespie, D. T., 1977. “Exact stochastic simulation ofcoupled chemical reactions”. The Journal of PhysicalChemistry, 81(25), pp. 2340–2361.

[70] Connors, K. A., 1990. Chemical kinetics: the study ofreaction rates in solution. John Wiley & Sons.

[71] Koper, M. T. M., and Gaspard, P., 1991. “Mixed-mode and chaotic oscillations in a simple model of anelectrochemical oscillator”. The Journal of PhysicalChemistry, 95(13), pp. 4945–4947.

[72] Arneodo, A., and Elezgaray, J., 1995. “Modelingfront pattern formation and intermittent bursting phe-nomena in the couette flow reactor”. In ChemicalWaves and Patterns, R. Kapral and K. Showalter, eds.Springer Netherlands, pp. 517–570.

[73] Ermentrout, G. B., and Kopell, N., 1986. “Parabolicbursting in an excitable system coupled with a slowoscillation”. SIAM Journal on Applied Mathematics,46(2), pp. 233–253.

[74] Rinzel, J., 1987. “A formal classification of burstingmechanisms in excitable systems”. In Mathematicaltopics in population biology, morphogenesis and neu-rosciences. Springer, pp. 267–281.

[75] Izhikevich, E. M., 2000. “Neural excitability, spikingand bursting”. International Journal of Bifurcationand Chaos, 10(06), pp. 1171–1266.

21

[76] Guckenheimer, J., and Oliva, R. A., 2002. “Chaos inthe Hodgkin–Huxley model”. SIAM Journal on Ap-plied Dynamical Systems, 1(1), pp. 105–114.

[77] Ansmann, G., Karnatak, R., Lehnertz, K., and Feudel,U., 2013. “Extreme events in excitable systems andmechanisms of their generation”. Phys. Rev. E, 88(5),p. 052911.

[78] Karnatak, R., Ansmann, G., Feudel, U., and Lehnertz,K., 2014. “Route to extreme events in excitable sys-tems”. Physical Review E, 90(2), p. 022917.

[79] Saha, A., and Feudel, U., 2017. “Extreme events inFitzHugh-Nagumo oscillators coupled with two timedelays”. Physical Review E, 95(6), p. 062219.

[80] Latif, M., and Keenlyside, N. S., 2009. “ElNino/Southern Oscillation response to global warm-ing”. Proceedings of the National Academy of Sci-ences, 106(49), pp. 20578–20583.

[81] Dijkstra, H. A., 2013. Nonlinear climate dynamics.Cambridge University Press.

[82] Roberts, A., Guckenheimer, J., Widiasih, E., Timmer-mann, A., and Jones, C. K. R. T., 2016. “Mixed-modeoscillations of el nino–southern oscillation”. Journalof the Atmospheric Sciences, 73(4), pp. 1755–1766.

[83] Haller, G., and Sapsis, T., 2010. “Localized instabilityand attraction along invariant manifolds”. SIAM Jour-nal on Applied Dynamical Systems, 9(2), pp. 611–633.

[84] Sapsis, T., and Haller, G., 2008. “Instabilities in thedynamics of neutrally buoyant particles”. Physics offluids, 20(1), p. 017102.

[85] Wiggins, S., 1994. Normally hyperbolic invariantmanifolds in dynamical systems. No. 105 in AppliedMathematical Sciences. App. Math. Sci., Springer.

[86] Jones, C. K. R. T., 1995. “Geometric singular per-turbation theory”. In Dynamical Systems. Springer,pp. 44–118.

[87] Desroches, M., Guckenheimer, J., Krauskopf, B.,Kuehn, C., Osinga, H. M., and Wechselberger, M.,2012. “Mixed-mode oscillations with multiple timescales”. SIAM Review, 54(2), pp. 211–288.

[88] Fenichel, N., 1979. “Geometric singular perturbationtheory for ordinary differential equations”. Journal ofDifferential Equations, 31(1), pp. 53–98.

[89] Guckenheimer, J., 2008. “Singular Hopf bifurcationin systems with two slow variables”. SIAM Journalon Applied Dynamical Systems, 7(4), pp. 1355–1377.

[90] Guckenheimer, J., and Vladimirsky, A., 2004. “Afast method for approximating invariant manifolds”.SIAM Journal on Applied Dynamical Systems, 3(3),pp. 232–260.

[91] Krauskopf, B., Osinga, H. M., Doedel, E. J., Hen-derson, M. E., Guckenheimer, J., Vladimirsky, A.,Dellnitz, M., and Junge, O., 2005. “A survey ofmethods for computing (un)stable manifolds of vec-tor fields”. International Journal of Bifurcation andChaos, 15(03), pp. 763–791.

[92] Lebiedz, D., Siehr, J., and Unger, J., 2011. “A vari-ational principle for computing slow invariant mani-folds in dissipative dynamical systems”. SIAM Jour-

nal on Scientific Computing, 33(2), pp. 703–720.[93] Castelli, R., Lessard, J.-P., and James, J. D. M., 2015.

“Parameterization of invariant manifolds for periodicorbits I: Efficient numerics via the Floquet normalform”. SIAM Journal on Applied Dynamical Systems,14(1), pp. 132–167.

[94] Babaee, H., Farazmand, M., Haller, G., and Sapsis,T. P., 2017. “Reduced-order description of transientinstabilities and computation of finite-time Lyapunovexponents”. Chaos, 27(6), p. 063103.

[95] Shilnikov, L. P., 1965. “A case of the existence of adenumerable set of periodic motions”. In Sov. Math.Dokl., Vol. 6, pp. 163–166.

[96] Gaspard, P., and Nicolis, G., 1983. “What can welearn from homoclinic orbits in chaotic dynamics?”.Journal of statistical physics, 31(3), pp. 499–518.

[97] Cvitanovic, P., and Eckhardt, B., 1989. “Periodic-orbit quantization of chaotic systems”. Physical Re-view Letters, 63(8), p. 823.

[98] Cvitanovic, P., 1991. “Periodic orbits as the skeletonof classical and quantum chaos”. Physica D: Nonlin-ear Phenomena, 51(1-3), pp. 138–151.

[99] Rossler, O., 1976. “An equation for continuouschaos”. Physics Letters A, 57(5), pp. 397 – 398.

[100] Letellier, C., Dutertre, P., and Maheu, B., 1995. “Un-stable periodic orbits and templates of the Rosslersystem: toward a systematic topological characteriza-tion”. Chaos, 5(1), pp. 271–282.

[101] Meacham, S. P., 2000. “Low-frequency variabilityin the wind-driven circulation”. Journal of PhysicalOceanography, 30(2), pp. 269–293.

[102] Timmermann, A., Jin, F.-F., and Abshagen, J., 2003.“A nonlinear theory for El Nino bursting”. Journal ofthe Atmospheric Sciences, 60(1), pp. 152–165.

[103] Ermentrout, B., 1998. “Neural networks as spatio-temporal pattern-forming systems”. Reports onprogress in physics, 61(4), p. 353.

[104] Izhikevich, E. M., 2003. “Simple model of spikingneurons”. IEEE Transactions on neural networks,14(6), pp. 1569–1572.

[105] Elezgaray, J., and Arneodo, A., 1992. “Crisis-inducedintermittent bursting in reaction-diffusion chemicalsystems”. Physical review letters, 68(5), p. 714.

[106] Farazmand, M., and Sapsis, T. P., 2016. “Dynami-cal indicators for the prediction of bursting phenom-ena in high-dimensional systems”. Phys. Rev. E, 94,p. 032212.

[107] Coller, B., Holmes, P., and Lumley, J., 1994. “Inter-action of adjacent bursts in the wall region”. Physicsof Fluids, 6(2), pp. 954–961.

[108] Jones, C., and Kopell, N., 1994. “Tracking invariantmanifolds with differential forms in singularly per-turbed systems”. Journal of Differential Equations,108(1), pp. 64 – 88.

[109] Han, S. K., Kurrer, C., and Kuramoto, Y., 1995. “De-phasing and bursting in coupled neural oscillators”.Physical Review Letters, 75(17), p. 3190.

[110] Haller, G., and Wiggins, S., 1995. “Multi-pulse

22

jumping orbits and homoclinic trees in a modal trun-cation of the damped-forced nonlinear Schrodingerequation”. Physica D: Nonlinear Phenomena, 85(3),pp. 311–347.

[111] Coller, B., and Holmes, P., 1997. “Suppression ofbursting”. Automatica, 33(1), pp. 1–11.

[112] Kawahara, G., and Kida, S., 2001. “Periodic motionembedded in plane Couette turbulence: regenerationcycle and burst”. Journal of Fluid Mechanics, 449,pp. 291–300.

[113] Farazmand, M., 2016. “An adjoint-based approachfor finding invariant solutions of Navier-Stokes equa-tions”. J. Fluid Mech., 795, pp. 278–312.

[114] Horsthemke, W., 1984. “Noise induced transitions”.In Non-Equilibrium Dynamics in Chemical Systems.Springer, pp. 150–160.

[115] Van den Broeck, C., Parrondo, J. M. R., and Toral,R., 1994. “Noise-induced nonequilibrium phase tran-sition”. Physical review letters, 73(25), p. 3395.

[116] Neiman, A. B., and Russell, D. F., 2002. “Syn-chronization of noise-induced bursts in noncoupledsensory neurons”. Physical review letters, 88(13),p. 138103.

[117] Moore, R. O., Biondini, G., and Kath, W. L., 2008.“A method to compute statistics of large, noise-induced perturbations of nonlinear schrodinger soli-tons”. SIAM review, 50(3), pp. 523–549.

[118] Forgoston, E., and Moore, R. O., 2017. “A primeron noise-induced transitions in applied dynamical sys-tems”. Under review by SIAM Review.

[119] Wigner, E., 1938. “The transition state method”.Transactions of the Faraday Society, 34, pp. 29–41.

[120] Horiuti, J., 1938. “On the statistical mechanical treat-ment of the absolute rate of chemical reaction”. Bul-letin of the Chemical Society of Japan, 13(1), pp. 210–216.

[121] Yamamoto, T., 1960. “Quantum statistical mechani-cal theory of the rate of exchange chemical reactionsin the gas phase”. The Journal of Chemical Physics,33(1), pp. 281–289.

[122] Chandler, D., 1978. “Statistical mechanics of iso-merization dynamics in liquids and the transition stateapproximation”. The Journal of Chemical Physics,68(6), pp. 2959–2970.

[123] Pratt, L. R., 1986. “A statistical method for identifyingtransition states in high dimensional problems”. TheJournal of Chemical Physics, 85(9), pp. 5045–5048.

[124] Weinan, E., and Vanden-Eijnden, E., 2010.“Transition-path theory and path-finding algo-rithms for the study of rare events”. Annual review ofphysical chemistry, 61.

[125] Weinan, E., and Vanden-Eijnden, E., 2006. “Towardsa theory of transition paths”. Journal of statisticalphysics, 123(3), p. 503.

[126] Gonzalez, C., and Schlegel, H. B., 1989. “An im-proved algorithm for reaction path following”. TheJournal of Chemical Physics, 90(4), pp. 2154–2161.

[127] Bolhuis, P. G., Chandler, D., Dellago, C., and

Geissler, P. L., 2002. “Transition path sampling:Throwing ropes over rough mountain passes, in thedark”. Annual Review of Physical Chemistry, 53(1),pp. 291–318.

[128] Dellago, C., Bolhuis, P., and Geissler, P. L., 2002.“Transition path sampling”. Advances in chemicalphysics, 123(1).

[129] Weinan, E., Ren, W., and Vanden-Eijnden, E., 2002.“String method for the study of rare events”. PhysicalReview B, 66(5), p. 052301.

[130] Maragliano, L., Fischer, A., Vanden-Eijnden, E., andCiccotti, G., 2006. “String method in collective vari-ables: Minimum free energy paths and isocommittorsurfaces”. The Journal of chemical physics, 125(2),p. 024106.

[131] Weinan, E., Ren, W., and Vanden-Eijnden, E., 2007.“Simplified and improved string method for com-puting the minimum energy paths in barrier-crossingevents”. Journal of Chemical Physics, 126(16),p. 164103.

[132] Pan, A. C., Sezer, D., and Roux, B., 2008. “Find-ing transition pathways using the string method withswarms of trajectories”. The journal of physical chem-istry B, 112(11), pp. 3432–3440.

[133] Farazmand, M., and Sapsis, T. P., 2017. “A varia-tional approach to probing extreme events in turbu-lent dynamical systems”. Science Advances, 3(9),p. e1701533.

[134] Ruelle, D., 1989. Chaotic evolution and strange at-tractors, Vol. 1. Cambridge University Press.

[135] Constantin, P., Foias, C., Nicolaenko, B., and Temam,R., 1989. Integral manifolds and inertial manifoldsfor dissipative partial differential equations, Vol. 70of Applied Mathematical Sciences.

[136] Dellnitz, M., and Junge, O., 2004. On the Approxima-tion of Complicated Dynamical Behavior. SpringerNew York, New York, NY, pp. 400–424.

[137] Chen, K. K., Tu, J. H., and Rowley, C. W., 2012.“Variants of dynamic mode decomposition: boundarycondition, Koopman, and Fourier analyses”. Journalof Nonlinear Science, 22(6), pp. 887–915.

[138] Hinze, M., Pinnau, R., Ulbrich, M., and Ulbrich, S.,2008. “Optimization with PDE constraints”. Vol. 23of Mathematical Modeling: Theory and Applications.Springer Science & Business Media.

[139] Herzog, R., and Kunisch, K., 2010. “Algo-rithms for PDE-constrained optimization”. GAMM-Mitteilungen, 33(2), pp. 163–176.

[140] Farazmand, M., Kevlahan, N. K.-R., and Protas,B., 2011. “Controlling the dual cascade of two-dimensional turbulence”. J. Fluid Mech., 668,pp. 202–222.

[141] Bertsekas, D. P., 2014. Constrained optimization andLagrange multiplier methods. Academic press.

[142] Obukhov, A. M., 1983. “Kolmogorov flow and labora-tory simulation of it”. Russian Mathematical Surveys,38(4), pp. 113–126.

[143] Marchioro, C., 1986. “An example of absence of tur-

23

bulence for any Reynolds number”. Commun. Math.Phys., 105, pp. 99–106.

[144] Platt, N., Sirovich, L., and Fitzmaurice, N., 1991. “Aninvestigation of chaotic Kolmogorov flows”. Phys.Fluids A, 3, pp. 681–696.

[145] Foias, C., Manley, O., Rosa, R., and Temam, R., 2001.Navier–Stokes Equations and Turbulence. CambridgeUniv. Press, Cambridge.

[146] Chandler, G. J., and Kerswell, R. R., 2013. “Invari-ant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow”. J. Fluid Mech., 722,pp. 554–595.

[147] Batchaev, A., and Dovzhenko, V., 1983. “Labora-tory simulation of the stability loss of periodic zonalflows”. In Akademiia Nauk SSSR Doklady, Vol. 273,pp. 582–584.

[148] Burgess, J. M., Bizon, C., McCormick, W., Swift, J.,and Swinney, H. L., 1999. “Instability of the Kol-mogorov flow in a soap film”. Physical Review E,60(1), p. 715.

[149] Ouellette, N. T., and Gollub, J. P., 2008. “Dynamictopology in spatiotemporal chaos”. Physics of Fluids,20(6), p. 064104.

[150] Suri, B., Tithof, J., Grigoriev, R. O., and Schatz, M. F.,2017. “Forecasting fluid flows using the geometry ofturbulence”. Phys. Rev. Lett., 118, p. 114501.

[151] Majda, A., Abramov, R. V., and Grote, M. J., 2005.Information theory and stochastics for multiscale non-linear systems, Vol. 25. American Mathematical Soc.

[152] Kraichnan, R. H., 1971. “Inertial-range transfer intwo- and three-dimensional turbulence”. J. FluidMech., 47, pp. 525–535.

[153] Moffatt, H. K., 2014. “Note on the triad interactionsof homogeneous turbulence”. J. Fluid Mech., 741 R3,pp. 1–11.

[154] Stoker, J. J., 1958. Water waves: The mathematicaltheory with applications. John Wiley & Sons.

[155] Nieto Borge, J. C., RodrIguez, G. R., Hessner, K.,and Gonzalez, P. I., 2004. “Inversion of marine radarimages for surface wave analysis”. Journal of At-mospheric and Oceanic Technology, 21(8), pp. 1291–1300.

[156] Fu, T. C., Fullerton, A. M., Hackett, E. E., and Merrill,C., 2011. “Shipboard measurments of ocean waves”.In OMAE 2011, pp. 1–8.

[157] Story, W. R., Fu, T. C., and Hackett, E. E., 2011.“Radar measurement of ocean waves”. In ASME 201130th International Conference on Ocean, Offshore andArctic Engineering, pp. 707–717.

[158] Nieto Borge, J. C., Reichert, K., and Hessner, K.,2013. “Detection of spatio-temporal wave groupingproperties by using temporal sequences of X-bandradar images of the sea surface”. Ocean Modelling,61, pp. 21–37.

[159] Ma, Y.-C., 1979. “The perturbed plane-wave solutionsof the cubic Schrodinger equation”. Studies in AppliedMathematics, 60(1), pp. 43–58.

[160] Akhmediev, N. N., and Korneev, V. I., 1986. “Modula-

tion instability and periodic solutions of the nonlinearSchrodinger equation”. Theoretical and Mathemati-cal Physics, 69(2), pp. 1089–1093.

[161] Peregrine, D., 1983. “Water waves, nonlinearSchrodinger equations and their solutions”. The Jour-nal of the Australian Mathematical Society. Series B.Applied Mathematics, 25(01), pp. 16–43.

[162] Benney, D. J., and Newell, A. C., 1967. “The propaga-tion of nonlinear wave envelopes”. Journal of Mathe-matics and Physics, 46(1-4), pp. 133–139.

[163] Zakharov, V. E., 1968. “Stability of periodic waves offinite amplitude on the surface of a deep fluid”. Jour-nal of Applied Mechanics and Technical Physics, 9(2),pp. 190–194.

[164] Hasimoto, H., and Ono, H., 1972. “Nonlinear modula-tion of gravity waves”. Journal of the Physical Societyof Japan, 33(3), pp. 805–811.

[165] Yuen, H. C., and Lake, B. M., 1975. “Nonlinear deepwater waves: Theory and experiment”. The Physics ofFluids, 18(8), pp. 956–960.

[166] Dysthe, K. B., 1979. “Note on a modification tothe nonlinear Schrodinger equation for applicationto deep water waves”. Proc. R. Soc. A, 369(1736),pp. 105–114.

[167] Trulsen, K., Kliakhandler, I., Dysthe, K. B., and Ve-larde, M. G., 2000. “On weakly nonlinear modula-tion of waves on deep water”. Phys. Fluids, 12(10),pp. 2432–2437.

[168] Chabchoub, A., Hoffmann, N. P., and Akhmediev,N., 2011. “Rogue wave observation in a water wavetank”. Phys. Rev. Lett., 106(20), p. 204502.

[169] Chabchoub, A., Hoffmann, N., Onorato, M., andAkhmediev, N., 2012. “Super rogue waves: observa-tion of a higher-order breather in water waves”. Phys-ical Review X, 2(1), p. 011015.

[170] Chabchoub, A., Akhmediev, N., and Hoffmann, N.,2012. “Experimental study of spatiotemporally local-ized surface gravity water waves”. Physical Review E,86(1), p. 016311.

[171] Narhi, M., Wetzel, B., Billet, C., Toenger, S.,Sylvestre, T., Merolla, J.-M., Morandotti, R., Dias, F.,Genty, G., and Dudley, J. M., 2016. “Real-time mea-surements of spontaneous breathers and rogue waveevents in optical fibre modulation instability”. Naturecommunications, 7, p. 13675.

[172] Chabchoub, A., 2016. “Tracking breather dynamics inirregular sea state conditions”. Phys. Rev. Lett., 117,p. 144103.

[173] Cousins, W., and Sapsis, T. P., 2015. “Unsteadyevolution of localized unidirectional deep-water wavegroups”. Phys. Rev. E, 91(6), p. 063204.

[174] Cousins, W., and Sapsis, T. P., 2016. “Reduced-orderprecursors of rare events in unidirectional nonlinearwater waves”. J. Fluid Mech, 790, 3, pp. 368–388.

[175] Farazmand, M., and Sapsis, T. P., 2017. “Reduced-order prediction of rogue waves in two-dimensionaldeep-water waves”. J. Comput. Phys., 340, pp. 418 –434.

24

[176] Dematteis, G., Grafke, T., and Vanden-Eijnden, E.,2018. “Rogue waves and large deviations in deepsea”. Proceedings of the National Academy of Sci-ences, 115(5), pp. 855–860.

[177] Davison, A. C., and Huser, R., 2015. “Statistics ofextremes”. Annual Review of Statistics and its Appli-cation, 2, pp. 203–235.

[178] Drazen, A., 2000. Political economy in macroeco-nomics. Princeton University Press.

[179] Keizer, K., Lindenberg, S., and Steg, L., 2008. “Thespreading of disorder”. Science, 322(5908), pp. 1681–1685.

[180] Rand, D. G., Arbesman, S., and Christakis, N. A.,2011. “Dynamic social networks promote coopera-tion in experiments with humans”. Proceedings of theNational Academy of Sciences, 108(48), pp. 19193–19198.

[181] Kloeden, P. E., and Rasmussen, M., 2011. Nonau-tonomous dynamical systems. No. 176. AmericanMathematical Soc.

[182] Carvalho, A., Langa, J. A., and Robinson, J., 2012. At-tractors for infinite-dimensional non-autonomous dy-namical systems, Vol. 182. Springer Science & Busi-ness Media.

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