Extreme events and dynamical complexity

Chaos, Solitons & Fractals 74 (2015) 46–54

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Review

Extreme events and dynamical complexity C. Nicolis a,⇑, G. Nicolis b a

Institut Royal Météorologique de Belgique, 3 Avenue Circulaire, 1180 Brussels, Belgium Interdisciplinary Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Campus Plaine, CP 231, bd du Triomphe, 1050 Brussels, Belgium b

a r t i c l e

i n f o

Article history: Available online 13 January 2015

a b s t r a c t The principal signatures of deterministic dynamics in the probabilistic properties of extremes are identified. Analytical expressions for n-fold cumulative distributions and their associated densities are derived. Substantial differences from the classical statistical theory of extreme values are found and illustrated on generic classes of dynamical systems giving rise to fully developed chaos and to quasi-periodic behavior. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Extreme events in the form of large deviations from the long-term average of a representative variable are of great importance in a variety of contexts since they can signal phenomena like the breakdown of a mechanical structure, an earthquake, a severe thunderstorm, flooding, or a financial crisis. Information on the probability of their occurrence and the capability to predict the time and place at which this occurrence may be expected are thus of great value. While the probability of such events decreases with their magnitude, the damage that they may bring increases rapidly with the magnitude as does the cost of protection against them. These opposing trends make the task of prediction extremely challenging. Ordinarily, in view of their unexpectedness and variability extremes are considered to be governed by statistical laws. There exists a powerful statistical theory of extremes, which in its most familiar version stipulates that the successive values X 0 ; . . . ; X n1 of the variable recorded are independent and identically distributed random variables (i.i.d.r.v.’s), irrespectively of the number and nature of other variables that may be involved in the system of interest. In the asymptotic limit of infinite observational window n it leads then under suitable linear scaling to

⇑ Corresponding author. http://dx.doi.org/10.1016/j.chaos.2014.12.006 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.

three types of universal extreme value distributions - the Gumbel, Frechet and Weibull distributions – that can be combined into a single generalized extreme value distribution (GEV) involving just three parameters [1]. On the other hand, the fundamental laws of nature are deterministic. What is more, there is increasing awareness that deterministic systems can generate complex behaviors in, for instance, the form of abrupt transitions, a multiplicity of states, or spatio-temporal chaos, at the basis of a wide variety of phenomena encountered in everyday experience [2]. A key point for our purposes is that under the above conditions deterministic dynamics induces nontrivial probabilistic properties, such as the existence of invariant probability measures possessing smoothness along at least certain directions in the phase space or the stronger property of mixing i.e., the irreversible approach to these invariant measures starting from sufficiently smooth initial probability distributions [3]. The present paper reviews how, based on work of the present authors and coworkers in the last 20 years, one can bridge the gap between the stochastic character of extreme events as perceived in our everyday experience and the deterministic character of the laws of nature. To this end deterministic dynamics will be embedded into a self-consistent probabilistic description, in which probabilities emerge as natural outcomes of the underlying evolution laws. This will enable us to construct from first

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C. Nicolis, G. Nicolis / Chaos, Solitons & Fractals 74 (2015) 46–54

principles the principal quantities of interest in extreme value theory, free of phenomenological assumptions. Special emphasis will be placed on dynamical systems with few degrees of freedom, for which non-classical behavior of extreme value properties will be revealed for finite observational windows. The general formulation is presented in Section 2. Section 3 is devoted to the derivation of the probability distributions of extreme values generated by one-dimensional maps in the interval, a generic class of models giving rise to complex behavior. Further extreme value-related properties such as the probabilistic properties of ranges of sums are summarized in Section 4. Extreme events in multivariate systems are considered in Section 5 and the main conclusions are summarized in Section 6. 2. Formulation The basic question asked in connection with extreme events is, given a sequence X 0 ; . . . ; X n1 constituted by the successive values of an observable monitored at regularly spaced time intervals 0; s; . . . ; ðn  1Þs, what is the probability distribution of the largest value x found in the sequence, M n ¼ maxðX 0 ; . . . ; X n1 Þ:

F n ðxÞ ¼ Prob ðX 0 6 x; . . . ; X n1 6 xÞ

ð1Þ

Obviously, F n ðaÞ ¼ 0 and F n ðbÞ ¼ 1; a and b being the upper and lower ends (not necessarily finite) of the domain of variation of x. In actual fact F n ðxÞ is therefore a cumulative probability, related to the probability density qn ðxÞ (in so far as the latter exists) by

F n ðxÞ ¼

Z a

x

Eq. (3) is given by the invariant probability density, qðX 0 Þ. In contrast, the two classes differ by the nature of the conditional probabilities. In stochastic systems these quantities are typically smooth. But in deterministic systems the successive values of X are related by a set of evolution laws which we write schematically in the form

X kþ1 ¼ f ðX k Þ

ð4Þ

In a multivariate system the time dependence of a given observable such as the quantity monitored in an experimental record depends, typically, on the full set of variables present. This case will be addressed in Section 5. In the present section we will assume that the conditions for a reduced description in which the observable of interest satisfies an autonomous dynamics are satisfied. As shown in dynamical systems theory [3] this may necessitate to project the dynamics on the most unstable direction, or to follow the traces of the full phase space trajectory on a Poincaré surface of section. As a result, the mapping f ðxÞ in Eq. (4) will typically be non-invertible. Eq. (3) takes the general form

qn ðX 0 ; . . . ; X n1 Þ ¼ qðX 0 Þ

n Y dðX k  f ðX k1 ÞÞ

ð5Þ

k¼1

By definition the cumulative probability distribution F n ðxÞ – the relevant quantity in a theory of extremes – is the n-fold integral of Eq. (5) over X 0 ; . . . ; X n1 from the lower bound a up to the level x of interest. This converts the delta functions into Heaviside theta functions, yielding

F n ðxÞ ¼

Z

x

dX 0 qðX 0 Þ hðx  f ðX 0 ÞÞ . . . hðx  f

ðn1Þ

ðX 0 ÞÞ

ð6Þ

a 0

dx qn ðx0 Þ

ð2Þ

Clearly as n increases F n ðxÞ will shift toward increasing values of x, being practically zero in the remaining part of the domain of variation. The possibility to elaborate a systematic theory of extremes depends on the cumulative probability FðxÞ of the underlying process (FðxÞ ¼ ProbðX 6 xÞÞ. If the latter satisfies certain conditions, then one can zoom the vicinity of the upper limit of x through appropriate scaling transformations and explore the possibility of universal behaviors. On the other hand, taking the infinite n limit is at the expense of erasing information on the variability of extreme value related properties, as captured by (finite time) fluctuations around the asymptotic averages. In particular, as we see presently, the n-fold probabilities like F n ðxÞ exhibit singularities which are quite pronounced for small or even moderate n’s in a wide range of threshold values x. To evaluate the n-fold cumulative probability F n ðxÞ in Eq. (1), we start with the multivariate probability density qn ðX 0 ; . . . ; X n1 Þ to realize the sequence X 0 ; . . . ; X n1 (not to be confused with qn ðxÞ in Eq. (2)). We express this quantity as

qn ðX 0 ; . . . ; X n1 Þ ¼ qðX 0 ÞWðX n1 ; X n2 ; . . . ; X 1 jX 0 Þ

ð3Þ

where W stands for the conditional probability to follow the path X 1 ; . . . ; X n1 starting initially on state X 0 . In both stochastic and deterministic systems the first factor in

where the superscript in f denotes an iterate of the mapping. In other words, F n ðxÞ is obtained by integrating n qðX 0 Þ over those ranges of X 0 in which x P f ðX 0 Þ; . . . ; o ðn1Þ ðX 0 Þ . This structure, involving a product of step funcf tions, is to be contrasted from the one obtained in the i.i.d.r.v. case, where F n ðxÞ would simply be the nth power of FðxÞ. Let now x be moved upwards. As shown previously by Balakrishnan and the present authors [4,5] new integration ranges will then be added, since the slopes of the succesðkÞ sive iterates f with respect to X 0 are, typically, both different from each other and X 0 dependent. Each of these ranges will open up past a threshold value where either the values of two different iterates will cross, or an iterate will cross the manifold x ¼ X 0 . This latter type of crossing will occur at x values belonging to the set of periodic orbits of all periods up to n  1 of the dynamical system. At the level of F n ðxÞ and of its associated probability density qn ðxÞ these properties entail the following consequences. (i) Since a new integration range can only open up by increasing x and the resulting contribution is necessarily non-negative F n ðxÞ is a monotonically increasing function of x, as indeed expected. (ii) More unexpectedly, the slope of F n ðxÞ with respect to x will be subjected to abrupt changes at a discrete set of x values corresponding to the successive crossing thresholds. At these values it may increase or decrease, depending on the structure of ðkÞ the branches f ðX 0 Þ involved in the particular crossing

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C. Nicolis, G. Nicolis / Chaos, Solitons & Fractals 74 (2015) 46–54

f ðXÞ ¼ 1  j1  2Xj 0 6 X 6 1

configuration considered. (iii) Being the derivative of F n ðxÞ with respect to x, the probability density qn ðxÞ will possess discontinuities at the points of non-differentiability of F n ðxÞ and will in general be non-monotonic. Properties (ii) and (iii) are fundamentally different from those familiar from the statistical theory of extremes, where the corresponding distributions are smooth functions of x. In particular, the discontinuous non-monotonic character of qn ðxÞ complicates considerably the already delicate issue of prediction of extreme events.

ð7Þ

The results confirm entirely the theoretical predictions. The dashed lines in the figure indicate the results of the statistical approach for the same system for which qðxÞ happens to be a constant and FðxÞ ¼ x, leading eventually to the Weibull (here simply exponential) distribution. One would think that this approach should be applicable since this system’s correlation function happens to vanish from the very first time step. Yet we see substantial differences, both qualitative and quantitative, associated to considerable fluctuations, discontinuities and a nonmonotonic behavior which persist for any value of window n, even though the invariant probability qðxÞ of the system defined by Eq. (7) is perfectly smooth. These differences subsist in the more typical case where the correlation function decays exponentially. They underlie the fact that in a deterministic system the condition of statistical independence is much more stringent than the rapid vanishing of the autocorrelation function.

3. One-dimensional maps in the interval Having identified some universal signatures of the deterministic character of the dynamics on the properties of extremes we now turn to the derivation of the detailed structure of F n ðxÞ and qn ðxÞ for some prototypical classes of dynamical systems. 3.1. Fully developed chaotic maps in the interval

3.2. Quasi-periodic behavior A dynamical system exhibiting this kind of behavior possesses a mean expansion rate larger than one and an exponentially large number of points belonging to unstable periodic trajectories. In view of the comments following Eq. (6), this will show up through the property that the set of points in which F n ðxÞ changes slope and the set of discontinuity points of the associated probability density qn ðxÞ become dense in some interval as n is increased. One may refer to this peculiar property as a ‘‘generalized devil’s staircase’’. It follows that the first smooth segment of F n ðxÞ will have a support of Oð1Þ and the last one an exponentially small support, delimited by the rightmost ðn1Þ fixed point of the iterate f and the right boundary b of the interval. Since F n ðxÞ is monotonic and F n ðbÞ ¼ 1, the slopes will be exponentially small in the first segments and will gradually increase as x approaches b. These properties differ markedly from the structure found in the classical statistical theory of extremes [5]. Figs. 1(a) and (b) depict (full lines) the functions q20 ðxÞ and F 20 ðxÞ as deduced by direct numerical simulation of the tent map, an example of (4) defined by the iterative function

The canonical form for this behavior is uniform quasiperiodic motion on an invariant set in the form of a twodimensional torus, which is topologically equivalent to a unit square subjected to periodic boundary conditions. It can be described by two angle variables /1 and /2 evolving according to the equations

/1  /10 ¼ x1 t /2  /20 ¼ x2 t

mod 1

ð8aÞ

x1 ; x2 being the two characteristic frequencies. If x1 and x2 are incommensurate the evolution is ergodic, possessing a smooth invariant density equal to unity. Choosing a Poincaré surface of section transversal to /2 one is lead to the circle map, a one-dimensional recurrence of the form

/nþ1 ¼ a þ /n

mod 1

ð8bÞ

where 0 6 /0 < 1 and a is irrational. The fundamental difference with case 3.1 above is the breakdown of the mixing 1

ρ20

F

a

b

20

0.8

15

0.6 10

0.4 5

0.2

0 0.8

0 0.85

0.9

0.95

x

1

0.8

0.85

0.9

0.95

x

1

Fig. 1. Probability density (a) and cumulative probability distribution (b) of extremes for the tent map as obtained numerically using 106 realizations. Dashed curves represent the prediction of the classical statistical theory of extremes. The irregular succession of plateaus in q20 ðxÞ and the increase of the slope of F 20 ðxÞ in the final part of the interval are in full agreement with the theory. The irregularity increases rapidly with the window and there is no saturation and convergence to a smooth behavior in the limit of infinite window.

C. Nicolis, G. Nicolis / Chaos, Solitons & Fractals 74 (2015) 46–54

property and the absence of dynamical instability (all expansion rates are constant and equal to unity). Furthermore, there are neither fixed points nor periodic orbits. Yet the structure of F n ðxÞ, depicted in Fig. 2, is similar to that of Fig. 1(b), displaying again slope breaks [4,6]. These arise from the second mechanism invoked in Section 2, namely, intersections between different iterates of the mapping functions. Remarkably, these intersections give rise to integration domains in the computation of F n ðxÞ that can only have three sizes a; b and a þ b, where a and b are determined by parameter a in Eq. (8b). 4. Probabilistic properties of ranges of sums In many situations of interest, especially in environment-related problems such as the regime of river discharges, the variability is so pronounced that no underlying regularity seems to be present. An ingenious way to handle such records, suggested originally by Hurst [7], is to monitor the distance, r between the largest and the smallest value recorded in a certain time window s and analyze its statistical properties as a function of s. We start by deducing from the record X 0 ; . . . ; X n1 the sample mean X n and the associated standard deviation C n (assuming that the X i ’s have a finite variance). Subtracting X n from each of the values of the record leads then to a new sequence of variables that have by construction zero sample mean,

x0 ¼ X 0  X n ; . . . ;

xn1 ¼ X n1  X n

ð9Þ

Next, one introduces the cumulative sums of all x values up to some xk ,

S1 ¼ x0 ;

S2 ¼ x0 þ x1 ; . . .

ð10Þ

This set will have a maximum and a minimum value,

M n ¼ max Sk ;

mn ¼ min Sk

ð11Þ

where the k’s run up to n. The range r n of the phenomenon described by the sequence is then quite naturally defined as

r n ¼ M n  mn

ð12aÞ

or, in rescaled form,

r n ¼

rn Cn

ð12bÞ

Since rn is still expected to display a high variability, to sort out systematic trends one should evaluate its average value or variance. The basic quantity involved in such an average is the (cumulative) probability distribution of the event

Fðu; v ; nÞ ¼ ProbðMn 6 u; mn P v ; nÞ

ð13Þ

where u; v are taken to be positive, M n > 0 and mn < 0. Since r ¼ M  m, the probability density associated to Fðu; v ; nÞ should be integrated over all values of u in the interval from 0 to r prior to the calculation of the moments of r:

Pðrn Þ ¼

Z

rn

duf ðu; r n  u; nÞ

ð14aÞ

@2F @u@ v

ð14bÞ

0

with

f ðu; v ; nÞ ¼

We now outline the evaluation of the quantity in Eq. (13) for deterministic dynamical systems along the lines of Section 2, following recent work by H. Van de Vyver and one of the present authors [8]. This task is complicated by the fact that contrary to Section 2 where the successive values X k were related to the initial one X 0 by Eq. (4), the dependence of the cumulative sums Sk on S1 in the present case is much more involved:

S1 ¼ X 0  X n ¼ X 0  1

Fn

49



n1 1X ðrÞ f ðX 0 Þ  Wnð1Þ ðX 0 Þ n r¼0

ð15aÞ



0.75

Sk ¼ kðS1  X 0 Þ þ 0.5

k1 X ðrÞ f ðX 0 Þ  WnðkÞ ðX 0 Þ

ð15bÞ

r¼0

To express Sk entirely in terms of S1 in (15b) one thus needs to express X 0 in terms of S1 from (15a). Since f – and thus Wð1Þ n – are non-invertible this latter dependence is not oneto-one,

0.25

0

1Þ

ðkÞ ðk1Þ X 0;a ¼ ðWð1Þ ðS1 Þ n Þa ðS1 Þ and Sk;a ¼ Wn ðX 0;a Þ  g a

0.6

0.7

0.8

0.9

x

1

Fig. 2. Cumulative probability distribution pffiffiffiF n ðxÞ of extremes for uniform quasi-periodic motion with x1 =x2 ¼ ð 5  1Þ=3. Upper curve corresponds to n ¼ 4 and lower to n ¼ 10. Notice that in both cases the number of slope changes in F n ðxÞ, whose positions are indicated in the figure by vertical dashed lines, is equal to 3.

ð16Þ

Sk is thus a multivalued function of S1 depending on the domain Ia of X 0 values associated to branch a of the inverse of mapping Wð1Þ n . The procedure is, then, to partition domain I of map f into m subintervals each containing one of the pre-images X 0;a of S1 , as S1 runs over its interval of variation. The probability density of realizing the sequence S1 ; . . . ; Sn is then (cf. Eq. 5)

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C. Nicolis, G. Nicolis / Chaos, Solitons & Fractals 74 (2015) 46–54

qn ðS1 ; .. . ; Sn Þ ¼

m n1 X Y PðX 0 2 Ia ÞqðS1 jX 0 2 Ia Þ  d½Sjþ1  g ðjÞ a ðS1 Þ

a¼1

ð17Þ

j¼1

and

Fðu; v ; nÞ ¼

Z m X PðX 0 2 Ia Þ

v

a¼1

Z

u

v

u

dSn

dS1 qa ðS1 Þ

n1 Y d½Sjþ1  g ðjÞ a ðS1 Þ

Z

u v

dS2    ð18Þ

j¼1

Fig. 3 depicts the dependence of F as obtained analytically from Eq. (18) for the tent map (Eq. (7)) as a function of v for u fixed and as a function of u for v fixed, for window n ¼ 3. We observe again a broken line-like structure, much like the one found in Fig. 1. We next turn to the probability density of the range, Eq. (14). The probability density of the non-normalized range r for the tent map and for window n ¼ 3 as obtained analytically from this equation is displayed in Fig. 4. We observe a complex, stair-like structure reminiscent of the one observed in Fig. 1(a). Fig. 5(a) and (b) display the probability densities of the normalized ranges r n for the same

a

dynamical system as obtained numerically for windows n ¼ 3 and n ¼ 100. In addition to the jumps present in Fig. 4 one observes here evidence of delta-like singularities, arising from the fact that the single integration over u in Eq. (14a) may not be sufficient to smooth the singularities arising when taking the second (mixed) derivative of Fðu; v ; nÞ in Eq. (14b). For comparison, in Fig. 5(b) we plot the probability density associated to a case of i.i.d.r.v. possessing a uniform density as in the tent map originally evaluated by Feller [9]. We observe that the actual probability density follows the main body of the i.i.d.r.v. density. Still, the irregular pronounced oscillations around the overall smooth envelope persist on small scales and remain present for much higher window values, e.g. n ¼ 1000. Closely related to the foregoing is the Hurst phenomenon. Specifically [7], in a wide range of environmental records it turns out that the mean normalized range < rn > varies with the window n as nH , where H (referred as Hurst exponent) is close to 0.7. To put this in perspective, Feller proved that for the reference case of i.i.d.r.v.’s H is bound to be 0.5. This implies in turn that Sk has somehow a persistent effect of X k , i.e., highs tend to be followed by highs, lows by lows. The question is, whether dynamical systems like those considered here may give rise to this kind of behavior. There is as yet no satisfactory answer to it, owing to a slow drift of the exponent H deduced from numerical experiments with respect to the total length of the time series generated by the system.

5. Multivariate systems Typically, iterative mappings in the interval arise when monitoring the successive intersections of the phase space trajectory of a continuous time dynamical system on a Poincaré surface of section. Their merit is to capture in a compact way the processes of expansion and reinjection, which constitute the principal ingredients behind the complexity of the dynamics. Now, real world systems are as a

b

Fig. 3. Cumulative probability distribution Fðu; v ; 3Þ, (Eq. (18)), for the tent map as a function of v with u = 0.1 (a); and as a function of u with v ¼ 0:1 (b) as obtained numerically using 106 realizations and a mesh size equal to 0.01. Dashed lines indicate the nondifferentiable points.

Fig. 4. Probability density of the non-normalized range for the tent map with n ¼ 3 as obtained numerically. Number of realizations and mesh size as in Fig. 3.

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C. Nicolis, G. Nicolis / Chaos, Solitons & Fractals 74 (2015) 46–54

a

celebrated example of systems of this kind is the baker transformation, defined in the phase space 0 6 x 6 1; 0 6 y 6 1 by the two-dimensional mapping [11]

  x y

 ¼

 ;

f ðx; yÞ gðx; yÞ

ð19Þ

or, more explicitly:

x1 ¼ 2x;

y1 ¼

x1 ¼ 2x  1;

b

Fig. 5. Probability density of the normalized range for the tent map with n ¼ 3 (a); and n ¼ 100 (b) as obtained numerically. Dashed line in (b) stands for the i.i.d.r.v. case. Number of realizations and mesh size as in Fig. 3.

rule multivariate continuous time dynamical systems. Furthermore, their evolution involves an intricate intertwining between local expansion and contraction stages. The question thus arises whether the singular behavior seen in the contracted description afforded by the one-dimensional mappings generated by the above mechanisms subsists at the level of the extremes associated to the individual variables fxi ðtÞg of such systems or, rather, it is smoothed out by the process of averaging over all the remaining variables and the associated unstable and stable directions. To the observer the role of dynamical complexity in the extreme value statistics would then be blurred, and applicability of the classical statistical theory of extremes would at first sight appear to be legitimate. In the present section we outline the main effects arising from the presence of several coupled variables on extreme value properties, following recent work by the present authors [10]. In order to sort out as directly as possible the new features arising from the coexistence of stable and unstable manifolds and of the reinjection of trajectories, we consider the class of K-flows, which are highly unstable dynamical systems in which each phase space point lies at the intersection of stable and unstable manifolds. A

y 2

06x<

y1 ¼

yþ1 2

1 2 1 6x61 2

ð20Þ

x and y span thus, respectively, unstable and stable directions, and the part of the transformation defined by x in the right half of the unit interval is associated with the process of reinjection. This system possesses a positive and a negative Lyapunov exponent equal to r1 ¼ ln 2; r2 ¼  ln 2 and a uniform invariant density qðx; yÞ ¼ 1. Furthermore, projecting onto the x axis one obtains a closed evolution equation for x identical to the one-dimensional Bernoulli mapping. This closure property does not apply to the projection onto the y axis, since according to (19) the evolution of y is conditioned according to whether x is in the left or in the right part of the unit interval. In the sequel we evaluate successively the bivariate cumulative distribution F n ðu; v Þ of x and y, its univariate contractions F n ðuÞ F n ðv Þ for x and y, as well as the univariate cumulative distribution of a linear combination of x and y through which the roles of the stable and unstable manifolds become intertwined. 5.1. The bivariate cumulative distribution of extremes of the original variables and its one-dimensional contractions Consider the first non-trivial case of window n ¼ 2. Keeping in mind the formulation in Section 2 we write

F 2 ðu; v Þ ¼

Z

u

Z v

dxdyqðx; yÞhðu  f ðx; yÞÞhðv  gðx; yÞÞ

0

0

Utilizing Eqs. (19) and (20) one obtains straightforwardly the following expressions. (1) For u < 12:

F 2 ðu; v Þ ¼

Z

u

dx

0

Z v

dyhðu  2xÞh



v

0

y 2

Upon introducing the new variables x0 ¼ 2x; y0 ¼ y=2 and changing the integration limits accordingly, one arrives at

F 2 ðu; v Þ ¼

uv 2

ð21aÞ

(2) For u > 12:

F 2 ðu; v Þ ¼

Z 0



1 2

dx

Z v 0

Z v 0

dyhðu  2xÞh



v

y þ 2

Z

u

dx

1 2

  yþ1 dyhðu  2x þ 1Þh v  2

The first integral can be handled as in the u < 1=2 case. The value of the second integral depends on whether v is smaller or larger than 1/2. This yields

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C. Nicolis, G. Nicolis / Chaos, Solitons & Fractals 74 (2015) 46–54

uv 1 1 ; u> ; v< 2 2 2   uv 1 ; F 2 ðu; v Þ ¼ þ ð2u  1Þ v  2 2 F 2 ðu; v Þ ¼

ð21bÞ u>

1 ; 2

v>

1 2

ð21cÞ

To obtain the one-dimensional cumulative distributions of extremes for x and y we set, respectively, v ¼ 1 and u ¼ 1 in Eq. (20). We obtain:

u ; 2 3u  1 ¼ ; 2

F 2 ðuÞ ¼

06x6

1 2

v

ð22Þ

1 2

1
1 0u 2 1
1 2 3 ¼ ; 2

q2 ðv Þ ¼ ;

1 q2 ðu; v Þ ¼ ; 2 5 ¼ ; 2

0v

F 2 ðv Þ ¼ v 2 ;

ð24aÞ

1 0

 x þ y dy d z  2

Z

1 2

  1 0 dx hðz  x0 Þh  z þ x0 2

Using the constraints introduced by the step functions one obtains

1 2 1 z> 2

z<

¼ 4ð1  zÞ;

ð26Þ

which is to be contrasted with the uniform invariant density of the original variables. Consider now the distribution of extremes for window n ¼ 2. Using Eq. (20) one has

   2x þ 2y x þ y dy h u  h u 2 2 0 0 ! Z 1 Z 1  2x  1 þ yþ1 x þ y 2 þ dx dy h u  h u 1 2 2 0 2 Z

1 2

dx

Z

1

ð27Þ Actually it is more informative to evaluate the density

ð24bÞ

1 1 1 0  u  ; 0  v  1; < u  1; 0  v  2 2 2 1 1 < u  1;
q2 ðu; v Þ ¼ 4uv

q2 ðuÞ ¼ 2u q2 ðv Þ ¼ 2v

Z

or, introducing the variables x0 ¼ x=2; y0 ¼ y=2,

F2 ¼

These results are fully corroborated by the numerical evaluation of the different F 2 ’s and q2 ’s starting from a time series of x and y generated by the baker map. They are also in agreement with the results summarized in Section 3, since x samples the dynamics along the unstable manifold and integrates therefore in an explicit way the role of the dynamical complexity in the properties of the extremes. For reference, if x and y were independent uniformly distributed random variables in the unit interval, the statistical theory of extremes would lead to the quite different expressions

F 2 ðuÞ ¼ u2 ;

dx

0

ð23Þ

1 2

1
F 2 ðu; v Þ ¼ u2 v 2 ;

1

qðzÞ ¼ 4z;

We conclude that F 2 ðuÞ and F 2 ðv Þ are non-differentiable at u ¼ 1=2 and v ¼ 1=2, and F 2 ðu; v Þ is non-differentiable along the manifold ð0; 1=2Þ [ ð0; 1=2Þ of the unit square. The corresponding densities, q2 ðuÞ ¼ dF 2 =du; q2 ðv Þ ¼ dF 2 =dv and q2 ðu; v Þ ¼ @ 2 F 2 ðu; v Þ=@u@ v possess discontinuities on the corresponding values,

1 q2 ðuÞ ¼ ; 2 3 ¼ ; 2

Z

0

1
0y

qðzÞ ¼

qðzÞ ¼ 4

and

F 2 ðv Þ ¼ ; 2 3v  1 ¼ ; 2

unstable manifolds. For illustrative purposes we carry out the calculation in some detail for the case z ¼ ðx þ yÞ=2. We first evaluate the invariant density of z

ð25Þ

all of which are smooth differentiable and monotonic functions of their arguments. 5.2. The cumulative distribution of extremes of a linear combination of the original variables We next turn to the extreme value properties of a variable z sampling the combined effect of the stable and

q2 ðuÞ associated to F 2 ; q2 ðuÞ ¼ dF 2 ðuÞ=du. This quantity is the sum of four terms in each of which one of the step functions appearing in Eq. (27) is replaced by a delta function. Introducing the same change of variables as in the calculation leading to Eq. (26), performing the delta function and evaluating the remaining parts by introducing the constraints imposed by the Heaviside function one arrives after a rather laborious calculation to the result [10]

8u ; 3 16u  2 ¼ ; 3 10  8u ¼ ; 3 16ð1  uÞ ; ¼ 3

q2 ðuÞ ¼

1 4 1 1
ð28Þ

Fig. 6(a) and (b) depict this quantity along with the invariant density qðzÞ associated to the monitored variable ðx þ yÞ=2. Both results are indistinguishable from those of a numerical evaluation starting from the time series of the original variables x and y. The point is that, contrary to the extreme value densities of the original variables displayed in Eq. (24), the density of extreme values of the variable z ¼ ðx þ yÞ=2 for window 2 possess no discontinuity. Alternatively, the corresponding cumulative probability is now a (once-) differentiable function (Fig. 6(c)). Clearly, this smoothing arises from the intertwining of the actions of the unstable and stable manifolds on the variable z, just as speculated in the beginning of this section. Still,

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a

a

b

b

Fig. 7. As in Fig. 6(a) (full line) but for a ¼ 0:05 (a); and a ¼ 0:1 (b); as obtained numerically.

c

Fig. 6. Probability density of extremes associated to the variable ½ax þ ð1  aÞy of system (20) with a ¼ 0:5 and for a time window n ¼ 2 (full line) (a); associated invariant probability density (b); and cumulative probability with n ¼ 2 (c); as obtained analytically and numerically using 2  106 realizations. Dashed line in (a) stands for an i.i.d.r.v. process.

distribution (28) remains different from the distribution corresponding to the classical statistical theory, also depicted in Fig. 6(a) (dashed line):

q2;cl ðuÞ ¼ 4

Z 0

1

dx

Z 0

1

 x þ y dy xy d u  2

Notice that the invariant density of z in Fig. 6b is identical to that obtained from the classical statistical theory of i.i.d.r.v.’s.

Fig. 7(a) and (b) depict the numerically evaluated q2 ðuÞ for two cases where z is a weighted average of x and y of the form z ¼ ax þ ð1  aÞy, with weights a corresponding to the x direction a ¼ 0:05 and a ¼ 0:1. Again, the discontinuities in Eq. (24) are smoothed out although the overall shape is less smooth than that of q2 depicted in Fig. 6(a) for a ¼ 1=2, being a succession of plateau-like parts joined by lines of high slopes. Furthermore, the number of points of non-differentiability increases with respect to Fig. 6(a). Analytic evaluations, though more involved, remain feasible. They are in full agreement with the numerically obtained results and show that the discontinuities actually arise only in the limit a ¼ 0 or a ¼ 1: a mixing, however unbalanced, of the stable and unstable directions suffices to transform q2 ðuÞ from a discontinuous to a continuous non-differentiable function. These trends persist in the case of observational windows larger than 2. 6. Conclusions The central theme of this review has been that the correlations and the selection rules inherent in deterministic dynamics are at the origin of distinct features of extreme value distributions, as compared to those derived in the classical statistical approach. Most prominent among them

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are the broken line-like structure of the n-fold cumulative distributions F n ðxÞ for finite (possibly large) windows n, the associated staircase-like structure of the probability densities qn ðxÞ and their repercussions in the statistics of a number of relevant quantities such as ranges of sums. These features have potentially serious implications on the issue of prediction, beyond the fact that in view of the unexpectedness and variability of extremes traditional predictors based on averages are bound to fail. A second major result presented in this paper is that in the presence of several variables the effect of dynamical complexity on the probabilistic properties of extreme values tends to be masked by the intertwining between locally ongoing expansions, contractions and reinjections in phase space. Still, behind the apparent smoothness of the resulting probability distributions there exists an ’’organizing center’’ in the form of an appropriate combination of the variables probing the dominant unstable modes, which display singularities in e.g. the form of non-differentiability of the cumulative distributions of extremes on certain sets of phase space points. We have shown on representative systems both analytically and by numerical simulations how these singularities are gradually smoothed out depending on the nature of variables involved. As it turns out smoothing is typically reflected by the regain of continuity on the probability density of extremes or of differentiability of the associated cumulative probability, but as a rule higher order singularities are bound to subsist. In addition to their interest in the fundamental understanding of the origin of extreme value distributions in deterministic systems, our results have some potentially important applications. We suggest that smooth-looking extreme value distributions deduced from the data may actually contain higher order singularities inherited from their deterministic character and thus belong to a class of functions different from the GEV distributions. These differences should be more apparent for the probability distributions of certain privileged types of variables, and for not too long observational windows. Furthermore, quantities of operational interest for prediction purposes like e.g. the return times of extremes [12] might prove to be more appropriate for revealing these differences. It would undoubtedly be interesting to conduct data analyses in this perspective and reassess the validity of some time-honored conclusions on the nature and the prediction of extreme events.

The dynamical systems approach to extreme events is still in its infancy. In particular, the possibility that for the type of observables considered here the extreme value distributions induced by deterministic dynamics obey to universal laws in the asymptotic limit of large observational windows under suitable scaling remains a challenging, open question. In this respect it is worth noting that even within the framework of the classical theory, finite size corrections to the limiting distributions display some highly non-trivial properties [13]. Further open problems include the statistics of repeated crossings, memory effects, non-stationary systems and finally, spatially extended systems [14]. Acknowledgments We are pleased to acknowledge enlightening discussions over the years with our colleague V. Balakrishnan. This work is supported, in part, by the European Space Agency and the Belgian Federal Science Policy Office. References [1] Embrechts P, Klüppelberg P, Mikosch T. Modelling extremal events. New York: Springer; 1997. [2] Nicolis G, Nicolis C. Foundations of complex systems. 2nd ed. Singapore: World Scientific; 2012. [3] Gaspard P. Chaos, scattering and statistical mechanics. Cambridge: Cambridge University Press; 1998. [4] Nicolis C, Balakrishnan V, Nicolis G. Extreme events in deterministic dynamical systems. Phys Rev Lett 2006;97:210602. [5] Balakrishnan V, Nicolis C, Nicolis G. Extreme value distributions in chaotic dynamics. J Stat Phys 1995;80:307–36. [6] Nicolis G, Balakrishnan V, Nicolis C. Probabilistic aspects of extreme events generated by periodic and quasiperiodic deterministic dynamics. Stochastics Dyn 2008;8:115–25. [7] Hurst H. Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng 1951;116:770–808. [8] Van de Vyver H, Nicolis C. Probabilistic properties of ranges of sums in dynamical systems. Phys rev 2010;82:031107. [9] Feller W. The asymptotic distribution of the ranges of sums of independent random variables. Ann Math Stat 1951;22:427–32. [10] Nicolis C, Nicolis G. Extreme events in multivariate deterministic systems. Phys Rev E 2012;85:056217. [11] Arnold V, Avez A. Ergodic problems of classical mechanics. New York: Benjamin; 1968. [12] Nicolis C, Nicolis SC. Return time statistics of extreme events in deterministic dynamical systems. EPL 2007;80:40003. [13] Györgyi G, Maloney N, Ozogány K, Rácz Z. Finite-size scaling in extreme statistics. Phys Rev Lett 2008;100:210601. [14] Nicolis C, Nicolis SC. Propagation of extremes in space. Phys Rev E 2009;80:026201.