3 July 2000

Physics Letters A 271Ž. 2000 237±251 www.elsevier.nlrlocaterpla

Localized Lyapunov exponents and the prediction of

Christine Ziehmann a,), Leonard A. Smith a,b,c, JurgenÈ Kurths a a UniÕersitatÈÈ Potsdam, Institut fur Physik-Nichtlineare Dynamik, Postfach 60 15 53, D-14469, Potsdam, Germany b Mathematical Institute,UniÕersity of Oxford, Oxford, OX1 3LB, UK c London School of Economics, London WC2A 2AE, UK

Received 9 December 1999; received in revised form 19 April 2000; accepted 3 May 2000 Communicated by C.R. Doering

Abstract

Every forecast should include an estimate of its likely accuracy, a current measure of predictability. Two distinct types of localized Lyapunov exponents based on infinitesimal uncertainty dynamics are investigated to reflect this predictability. Regions of high predictability within which any initial uncertainty will decrease are proven to exist in two common chaotic systems; potential implications of these regions are considered. The relevance of these results for finite size uncertainties is discussed and illustrated numerically. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 05.45.-a; 05.10.-a; 05.45.Tp Keywords: Lyapunov exponents; Predictability; Chaos; Weather forecasting; Ensemble prediction; Nonlinear dynamics

1. Introduction difficulty in quantifying predictability:Ž. 1 the depen- dence of measures of predictability upon the particu- lar metric adoptedŽ. 2 the dependence of uncertainty Prediction of predictability refers to the quantita- dynamics upon the magnitude of the uncertainty in tive attempt to assess the likely error in a particular the initial condition, andŽ. 3 the fact that errors in the forecast a priori. There areŽ. at least three sources of model are often unknown until after predictions are observed to fail. Lyapunov exponentswx 1±3 quantify predictability through globally averaged effective growth rates of uncertainty in the limits of large time and small uncertainty; thus by construction they are ) Corresponding author. Tel.: q49-331-977-1302, fax: q49- of limited use. To obtain a quantitative estimate of 331-977-1142. the accuracy of a particular forecast, the local dy- E-mail address: [email protected]Ž. C. Ziehmann . namics of uncertainties about that initial condition

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S0375-9601Ž. 00 00336-4 238 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 are more relevantwx 4±9 . By allowing better risk not considered in this Letter as there is no systematic assessment, a prediction of predictability is of value manner to include systemrmodel mismatches, thus it in any field from physics to economics; weather is assumed throughout the Letter that the perfect forecasting provides a particular example relevant to model is known. both fields. EffectiÕe growth rates defined over a fixed dura- tion are also used to quantify predictability; they are employed daily in the operational weather forecast 2. Localized Lyapunov exponents centers of Europe and North Americawx 10,11 . In Section 2, the distinction between what will be called The dynamics of infinitesimal uncertainties about finite time exponents and finite sample exponents is a point x in an m-dimensional state space are shown to lie in the particular initial orientation of 0 governed by the linear propagator, Ž.x ,Dt , which the perturbation each considers for a given initial M 0 evolves any infinitesimal initial uncertainty e gR m condition; this can result in dramatically different 0 about x forward for a time Dt along the system's effectiÕe growth rates. Both types of exponent are 0 trajectory to x : called `Local Lyapunov exponents', and recognizing Dt the distinction between them resolves some confu- e s e Dt MŽ.x,Dt 0 .1 Ž. sion in the literature. In terms of predicting the forecast accuracy, the finite time exponents are shown In discrete time maps, the linear propagator over k to be the more relevant quantities in Section 3, where iterations is simply the product of Jacobians along it is also proven that the mean of the largest finite s the trajectory, that is MŽ.Ž.x0 ,k J xky1 ... time exponent does not provide an unbiased estimate JŽ.Ž.x10J x . of the corresponding global exponent, and similarly For high dimensional systems, interest tends to be for the mean of the smallest finite time exponent. In focused on subspaces which are likely to contain the addition, in regions of state space where the largest fastest growing perturbationswx 4,10,11 . Two orienta- finite time exponent is less than zero all perturba- tions of particular interest areŽ. i that which will tions will shrink independent of their orientation; this have grown the most under the linearized dynamics is investigated in Section 4 where such regions are Ž k. after k steps, z1 Ž.x , and Ž. ii the local orientation of proven to exist in two common chaotic maps. the globally fastest growing direction, lx1Ž., which Each class of discussed in is sometimes called the Lyapunov vectorwx 13 . The this Letter assumes the observational uncertainty is first of these orientations is defined by the singular infinitesimal; of course as long as it remains in- value decompositionwx 14 of the propagator: the finitesimal it cannot limit predictability, and once it Ž k. zi Ž.x are simply the right singular vectors of is finite its growth is no longer quantified by Lya- MŽ.x,k . Each is associated with a singular value, punov exponents. Therefore, the rigorous results re- Ž k. G siiiŽ.x ; by convention, s s q1. The finite time stricted to infinitesimal uncertainties are contrasted LyapunoÕ exponents wx15 are with numerical demonstrations for finite uncertain- ties in Section 4. Part of the popularity of global 1 lŽk.Žs <

Ž k. same for almost all x with respect to an ergodic and l11approach L , yet for the relatively small k measure. over which forecasts are typically made their proper- 1 ties are quite different and neither is constrained by L slŽ`.Ž' s k. L ii lim log 2 Ž.i , the value of 1. k™` k is1,2,...,m Ž.3 3. Properties of localized Lyapunov exponents If the sum of the Li is negative, a volume element in state space will shrink, on average, as it evolves along a trajectory, and most initial condi- General constraints on the relative magnitudes of tions will evolve towards an of dimension the largest and the smallest finite time and finite less than m wx17 . At each point x on such an sample exponents are now derived from the defini- tions above, and then illustrated below. By construc- attractor, the orientation lx1Ž.denotes the orienta- tion corresponding to L , that is the orientation tion, maximum growth corresponds to z1, thus 1 r Ž k. FlŽk. towards which almost every uncertainty e in the 11Ž.x Ž.x for each x, and therefore this in- r Ž k. F sufficiently distant past would have evolved, when equality also holds for the mean values ² 1 : lŽ k. P the trajectory reaches x. Similarly, define lxŽ.as ² 1 :²:, where denotes an arithmetic average m 2 taken with respect to the natural measure , and ²:P N the orientation corresponding to Lm Žfor details, see wx a numerical approximation with sample size N.Ex- 17.Ž.Ž. . Numerically, lx10and lxm 0 can be approx- imated by evolving the singular vectors of amples from two chaotic systems are given in Fig. 1. The Henon mapwx 24 is MŽ.ŽxŽyj.,2 j , that is, the 2 j step propagator about the j th pre-image of x ., forward j steps until the s y 2 q s 0 xiq1 1 axiiy , y iq1 bxi ,6Ž. trajectory reaches x . Thus 0 where as1.4 and bs0.3; the Jacobian is indepen- Ž j. Ž2 j. l x s xy , j z xy , is1,m.4 y i Ž.0 M Ž.ji Žj . Ž. dent of yi and has constant determinant equal to b. wx ™ Ž j. The Ikeda map 25 As j `, we expect lxi Ž.0 to approach the orien- Ž. s s s q y tations of lxi 0 for i 1 and i m, leading to the xiq1 1 m xiicosŽ.t y sin Ž.t , 1 definition of the finite sample LyapunoÕ exponents s q yiq1 m xiisinŽ.t y cos Ž.t ,7 Ž. 1 r Žk. Ž.x s log Ž.<

1 If a specific x is not of interest, approximation of lx1ŽŽ..t at xŽ.t along a numerical trajectory can be simply approximated by a very long integration of the system and tangent equations for an 2 We assume throughout that there exists a unique natural arbitrary initial uncertainty. The quality of this approximation may measure which is well approximated by the numerical iteration of be unknown, however, seewx 18 and the discussion in Section 5. the system. 240 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251

Ž k. w x Ž k. w x s s s Fig. 1. Distributions of l11Ž.Ž.a , b and r Ž.Ž.c , d in the Henon and the Ikeda systems for k 1Ž. thick , k 4 Ž. thin , and k 64 Ž.dashed , each with Ns4096. The arrows at the top axis indicate ²:N . Larger bin widths have been used in the lower panels.

Žk. 3 C F B A The mean values ²l1 :N do not increase with matrices A and B, thus s111s s . The equal- Ž2 k. F Žk. A increasing k. In fact ²l11:²l : for any k,as ity holds only if the first left singular vector u1 of B can be seen by considering the matrix C, the product A is aligned with the first right singular vector z1 s A P B s of Jacobians JŽ.xi ,i 1,2, . . . ,2 k. Divide C into of B Ži.e. u11z 1. , as in the uniform Baker's two sub-products A and B, each of length k: mapwx 17 and Baker's Apprentice Maps wx 7. . From Eq.Ž. 2 , the largest finite time Lyapunov exponent defined by C must be less than or equal to the

3 This follows immediately from the singular value decomposi- T The first singular value s A of reflects the tionŽ. SVD of a square matrix, AsUS V , where the super- 1 A T S maximum possible growth over the first k steps; the script denotes the transpose of a matrix. is a diagonal matrix whose largest entry is s1 and U and V are orthonormal rotation first singular value of matrix C must be less than or matrices. Noting that rotation matrices cannot enhance growth equal to the product of the first singular values of the yields the desired result. A brief proof is given in the Appendix. C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 241

Ž2 k. wx aÕerage of those defined by A and B: l11Ž.x sequences of functions 26,27 . While this does not s 11s C F ssB A s 1wlŽ k. q lŽk. 2k log 11112 k log Ž.2 Ž.xkq1 guarantee a monotonic decrease of ² 1 :Žor in- Žk. x Ž k. l11Ž.x . Similarly, the smallest finite time expo- crease of ²lm :. with increasing k, it does imply Ž2 k. G nent defined by C must satisfy lm Ž.x1 - 1 wlŽ k.ŽŽ.ql k. Ž.x 2 mkx q1 m x1 . As this is true for each ²lŽ k.:²GL lŽk.:FL Ž2 k. Ž2 k. 11and mmfor all k,10Ž. individual l1 Ž.Žximialternatively l Ž..x , the mean of the largestŽ. smallest finite time Lyapunov lŽ k. exponent will not increaseŽ. not decrease as k in- proving that the mean of the 1 distribution is not creases by a factor of two: an unbiased estimate of the global Lyapunov expo- nent L for any finite k wx5,28 . ²lŽ2 k.:²F lŽ k.:²lŽ2 k.:²G lŽ k.: 1 11and mm.8Ž. The mean of the distribution of finite sample L When A and B are of different lengths, k1 and exponents is equal to 1 by definition, independent ²lŽ k.: k2 , then of k. While in Fig. 1 each 1 N decreases with Žk. q increasing k Ž.²as indicated by the arrows , the r1 :N k qk ²lŽk1 k 2 .:²Fk lŽ k 2 .:²qk lŽ k1.: 9 Ž.121 21 11 Ž.coincide, providing a consistency check as to whether q Žk1 k 2 . Žk. and a similar relation is obtained for ²lm :, i.e., N might be large enough so that ² r1 :N approxi- Žk. Ž k. Ž k. the kl1 Žklm .Ž.are sub-additive super-additive mates the limiting value ² r1 :.

Fig. 2. Contrasting finite sample Lyapunov exponentsŽ. abscissa with the corresponding finite time Lyapunov exponents Ž. ordinate in the Ikeda system for ks1Ž.Ž.Ž. light grey , ks4 grey , and ks256 black iterations. Note, that in the Ikeda system Det Ž.J sm2. 242 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251

If the Jacobian determinant is constant, then for been determined analytically in the Lorenzwx 30 sys- all x temŽ see alsowx 20,31. ; we now present new results m for the Ikeda system and the Henon system. This lŽ k. s a Ý i Ž.x log 2 Ž.,11 Ž . result is then demonstratedŽ. numerically to hold in is1 several cases for finite uncertainties, but the exact s<< Ž k. F Ž k. where a det J . Recalling that r11Ž.x l Ž.x results below are subject to the caveat of infinitesi- Žk. G Ž k. and rmmŽ.x l Ž.x , it follows that for both the mal uncertainties, as are all general arguments re- Ž k. F Žk. Henon system and the Ikeda system r11l , garding the prediction of deterministic chaotic sys- Ž k.ŽG k. s y Ž k. r12l log 2Ž.a l 1, and tems. lŽk. F lŽ1. s a y r Ž1. 1121max Ž.log Ž.min Ž .,12 Ž . xx defining a triangle which bounds the distribution of 4.1. Infinitesimal uncertainties Ž k. Žk. Ž r11Ž.x ,l Ž..x , as illustrated in Fig. 2 for the Ikeda map. With increasing k the distributions of Naturally, exact results are most easily obtained Ž k. s Ž k. s s points approach the line r11l . For k 256, the for small k. Therefore we consider only k 1 and < Žk. y Ž k. < s largest observed value of l11Ž.x r Ž.x was k 2 analytically; numerical results are given for Ž1. - 0.03. Yet the widths of each of these distributions larger values. In a map l1 Ž.x 0 implies that the exceeds 0.3, indicating that this variation stems from largest singular value of the Jacobian JŽ.x is less the different initial conditions on the attractor, not than one. For the Ikeda system with m in the range the initial orientation. Ž.'13 y3 r2-m-1, the one-step finite time Lya- Ž1. punov exponent l1 Ž.x passes through zero at two circles about the origin with radii 4. Regions of high predictability in chaotic maps r s 6cy1" 6cy1 2 y1 Lyapunov exponents are often said to reflect pre- o,i ( (Ž. dictability, and a positive global Lyapunov exponent s< mrŽ m2 y .< lŽ1.Ž.- is often said to destroy any hope of `long-term' where c 1 . In this case 1 x 0 for Ž predictability. But since they are defined via the all points either within the inner circle i.e. those 22q - linear propagator Lyapunov exponents need only with (x y ri .Žor outside the outer circle i.e. 22q ) quantify the growth of infinitesimal uncertainties in (x y ro.. Fig. 3 shows points on the Ikeda Ž1. the initial condition, this is a high price to pay for attractor where the sign of l1 Ž.x is indicated by the s s invariance under a smooth change of coordinate. grey scale. For m 0.9, the radii are ri 0.135 and s Both the Ikeda systemŽŽ.. Eq. 6 and the Henon roo7.404Ž thus this attractor lies well within r , and systemŽŽ.. Eq. 7 are considered chaotic for the pa- the outer circle is not visible in Fig. 3. . As <

Ž1. ) Fig. 3. Regions of decreasing uncertainty in the Ikeda system. Points on the attractor are colored grey if l1 Ž.x 0, black otherwise. Within Ž1. - the circular region near the origin, l1 Ž.x 0 for all x.

s s The preimage of the y-axis is the parabola, y nii, where n are the roots of the characteristic ax 2 y1, and the two step propagator for points polynomial of M TM. Thus Ž j j 2 y . ,a 1is n 2 y2b2 Ž.2 a2j 2 q1 nqb4 s0.Ž. 13 j j 2 y s s s sj We can now test whether s 2 -1 for any j ; M Ž.Ž.,a 1,k 2 J10Ž.Ž.x 0 J x alternatively, we can solve for ss1 to find 01y2 aj 1 s b2 y1 0.91 ž/b 0 ž/b 0 jsŽ.s1 s" s. f"1.083 Ž. 14 2 ab 0.84 b 0 s , for as1.4, bs0.3. Note that the parabola and the y2 abj b ž/ x-axis intersect at the pointŽ. 0,y1 . At this intersec- where we have used the fact that in the Henon tion xsjs0 and it follows from the equation Žks2. s Ž ks2. s - system JŽ.x is only a function of x. To locate above that s12s b 1 or, equivalently, lŽks2. - lŽ ks2.Žs 1 sl ks2. - those x with 1 0, we determine the singular that 1222 log Ž.b 0. Showing that a 2 y s 2 values siiof MŽŽ j ,aj 1,..k 2 , noting that s point is negative in the range given by Eq.Ž. 14 244 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 proves that all points on the parabola within these preimages of the y-axis. The first three preimages of Ž ks2. - limits have l1 0. By continuity, there exists a the x-axis can be obtained analytically. The y-axis is finite region in the neighborhood of the parabola for the preimage of the x-axis, while the parabola noted above is the first preimage of the y-axis. The first 1yb2 1yb2 y -x- , preimage of the parabola is 2 ab 2 ab 2 within which the largest two-step finite-time Lya- aj 22y1 Ž.aj y1 xs , ysjy1qa , punov exponent is negative. A numerical estimate of bbž/ this region is shown in Fig. 5a. What about larger values of k? Trajectories pass- 1yb2 1yb2 ing through this region of lŽ2. -0 are often found to y -j- . 1 2 ab 2 ab Ž k. - ) have l1 0 for k 2 as well; negative values of Ž4. l1 are clearly visible in Fig. 1a. In the following, The first 4 preimages of the x-axis are shown to- we will illustrate the relation between the trajectories gether with the attractor in each panel of Fig. 4, Žk. - ) Žk. - of points on the attractor with l1 0 for k 2 and while points on the attractor with l1 0 are shown the y-axis; namely that such points tend to lie near for the specific values ks2,3,4,5 in the four panels.

Fig. 4. All panels show the Henon attractor, the x-axis, the y-axis and its last three preimages: the parabolaŽ. solid line , its second preimage Ž k. - s s Ž.long dashed and its third preimage Ž short dashed . . The different panels show points on the attractor with l1 0 for k 2a,Ž.k 3b, Ž. ks4Ž. c , and ks5d. Ž. C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 245

Ž k. - It is clear that the regions with l1 0 are related to Next, we investigate the behavior for even larger the intersections of the attractor and preimages of the k in the Henon system. Grassberger et al.wx 16 showed Ž k. - y-axis. We next show that this is even more evident that one should expect l1 0 for arbitrarily large for initial conditions in the general vicinity of the k, assuming a behavior essentially like averages of attractor. As in Fig. 4, Fig. 5 shows both the preim- random variables correlated only over short times. ages of the x-axis and the attractor; in addition, test This general picture is correct although the details Ž k. - points for which l1 0 are plotted as well, where are sometimes important, as we have argued else- the test points were drawn at random from the region wherewx 15 . Here, we conjecture that, due to the shown in this figure. Points with decreasing uncer- deterministic nature of the Henon system, this frac- F Ž k. tainties for k 5 are found in the vicinity of preim- tion of l1 decreases more quickly than the random ages of the y-axis, i.e. they often have trajectories variable argument would suggest. As shown in Fig. which include a near approach to the y-axis. Typi- 6, the fraction of initial conditions on the attractor Ž k. - cally, this occurs towards the end of that trajectory: with l1 0 is observed to decrease exponentially Ž2. - points with l1 0 are close to the first preimage of with k, as are the corresponding fraction when linear Ž3. - the y-axis, points with l1 0 are close to its first propagators of the map are combined at random. The and second preimage and so forth. random case for ks1,2,4 and 8 are shown; for each,

Fig. 5. All panels show the Henon attractor, the x-axis, the y-axis and its last three preimages: the parabolaŽ. solid line , its second preimage Ž k. - s Ž.long dashed and its third preimage Ž short dashed . . The different panels show points in the vicinity of the attractor with l1 0 for k 2 Ž.a,ks3b, Ž.ks4 Ž. c , and ks5d. Ž. 246 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251

s s Ž k. - Fig. 6. For the Henon system Ž.a 1.4,b 0.3 , the fraction of points with l1 0 as a function of k in the deterministic caseŽ. squares . Also shown are the results for random matrices, where the matrices are drawn from the distribution of the linear propagators of the Henon map, that is M Ž j., for js1,2,4 and 8. The solid lines reflect the best fit to an exponential decay over the range 8FkF40. For large k, about 2 30 iterations of the map where considered in the deterministic case. Note, that determinism is a strong constraint reducing the likelihood of finding negative finite time exponents. the fraction decreases less quickly than in the deter- condition with zero mean and the same standard ministic case. Disregarding the determinism in the deviation e. Taking a point on the attractor at ran- series of Jacobians leads to frequencies of negative dom, 128 `observations' were generated and pre- Žk. l1 which for the larger k exceed those of the dicted forward k steps; if the distance from truth at deterministic case by orders of magnitudes. final time was less than the initial perturbation ap- plied in more than 50% of the 128 cases, then the 4.2. Finite uncertainties original point on the attractor was considered to be within a region of high predictabilityŽ for that value From the practical point of view of estimating of e .Ž.. For ks1 there is a region not shown of predictability, the knowledge of such points would high predictability centered on the circle derived be of limited utility for large k, since the shrinking above and shown in Fig. 3. Points of high pre- region around each point may be very small. Further, dictability have been observed for es2y9 and per- es 1 recall that all estimates of predictability based upon sist for 8 . While all are near the origin, many Lyapunov exponents assume an infinitesimal initial fall outside the circle, but this is not surprising as the error. Therefore, we next consider finite uncertainties definition of high predictability in this numerical explicitly, first exploring Gaussian distributed uncer- experiment is much less restrictive than requiring a tainties in the Ikeda map, and then uncertainties of negative finite time Lyapunov exponent, which guar- uniform magnitude in the Henon map. In each case antees 100% of the uncertainties to shrink if they we allow the expected magnitude of the error to vary have infinitesimal magnitudes. The same tests for and discuss the relation between regions of enhanced ks4 are shown in Fig. 7. The results based upon Ž k. - predictability and the regions where l1 0. infinitesimal uncertainties shown in the upper left In the Ikeda map, we consider normally dis- panel are seen to reflect the region of high pre- tributed uncertainties in each coordinate of the initial dictability very well up until es1r8, which is a fair C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 247

infinitesimals; of course, structures smaller than e cannot be detected. It becomes harder to identify regions with differing properties in predictability with increasing prediction time; sensitive dependence on initial condition will limit the prediction of pre- dictability as well as prediction itself. Thus far, we have only considered the value of an exponent at a particular value of ksK; alterna- tively one might consider regions in which the expo- nent is negative for all kFK, with corresponding uncertainties decreasing monotonically for the total duration of K iterations. While such subsets are of interest, they are not investigated here since the existence of small positive values at intermediate k are consistent with regions of high predictability; indeed the points omitted from the set of points for a particular k are those that are said to display `return of skill' in meteorologywx 32 . Although beyond the scope of this Letter, it would be interesting to exam- ine the spatial distribution and fraction of initial condition in these subsets, both as a function of k and the magnitude of the initial uncertainty.

4.3. Coexistence of chaos and regions of high pre- dictability Fig. 7. All panels show the Ikeda attractor; the dots represent the attractor. Initial conditions on the attractor with enhanced pre- dictability are marked with a `q' when more than 50% of 128 Another interesting aspect concerning the regions Žk. - initial uncertainties show decreased magnitudes at final time with l1 0 is the interplay between these regions ks4. The different panels reflecting different initial magnitudes and the location of unstable periodic orbits, which eps are contrasted with the linear dynamics in the upper left panel. are believed to form the skeleton of the attractor in many chaotic systemswx 33 . Clearly, an unstable pe- riod-k cannot contain a point within a region Žk. - fraction of the diameter of the attractor. We stress for which l1 0, since that point would then be that results of this kind will be extremely system stable. In short, each point on each unstable period k specific. orbit must avoid all regions of the state space in lŽ4. Žk. - For the Henon map the portrait of negative 1 which l1 0: it is not easy to see how this comes revealed in Fig. 5 is contrasted with regions of high about if the orbits are dense on the attractor and the predictability for finite uncertainties in initial condi- area of the regions does not vanish. If the regions do tion with a much sharper test than for the Ikeda map. not vanish, then this observation suggests a new In this case each observation is placed at random on angle from which to view the extreme sensitivity of a circle of radius e about the true state; 1000 such the structure of the attractor to small changes in `observations' were considered for each true state parameter value. and only if the final time distance of eÕery one of It is also interesting to consider the implications e Ž k. them was less than was the point recorded as positive l1 might have on numerical results when `high predictability.' This is shown for 3 different the true attractor is a stable attracting periodic orbit; initial magnitudes in Fig. 8. For small magnitudes, no point in a periodic orbit need lie in a region for s Ž k. - f s e 0.001, the cartography is quite similar to that of which l1 0. For a 1.37511006867, b 0.3, 248 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251

Fig. 8. All panels show the Henon attractor; the dots represent initial conditions with enhanced predictability for finite uncertainties. The different panels belong to different initial magnitudes of initial uncertainty:Ž. a epss0 Ž thus coinciding with Fig. 5c . . In panels Ž.Ž. b , c , and Ž.d , a dot at x indicates that the distance between the image of the true state and each one of 1000 inexact `observations' decreased at ks4. The observations were initially distributed on a circle of radius eps centered on x. the Henon system has a stable period 24 orbit. The This observation suggests an interesting possible Ž24. - majority of points on this orbit have l1 0, but parallel between these simple two dimensional maps Ž24. ) for several, l1 0; the largest observed value is and the onset of turbulence in laminar fluid flows. It Ž24. f l1 0.3, implying a magnification factor of more has long been known that shear flows can become than a hundred within one cycle. When M is non- turbulent at Reynolds numbers well below the criti- normal, the magnitude of the leading singular value cal value as defined by the classical linear stability may be quite large regardless of whether or not the theory based on eigenvaluesŽwx 34 and references orbit is asymptotically stable. With a slight increase thereof; for a recent overview seewx 35. . Perturbations in a the system appears chaotic; the dynamics still in the directions of the singular vectors may grow resemble those of the stable orbit, but the attractor rapidly for aŽ. finite time, exciting nonlinear terms now consists of 24 small, clearly separated chaotic and thereby dominating the onset of turbulence; the `regions,' each visited in turn. It would be interesting long term behavior described by the eigenvalues to examine the distribution of leading singular values becomes irrelevant. Non-normality might hold simi- about stable periodic points on the same orbit, as a lar consequences for the numerical iteration of non- function of parameter; is there a positive lower bound linear systems. The smallest nonzero numerical per- on the angle between these eigenvectors? turbation is finite, being defined by the numerical C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 249

grid; and it could be quite difficult to identify a certainties along lx1Ž.have the advantage to be free Ž k. ) stable period k orbit with l1 0 by numerically of transients, but if of finite magnitude they also may iterating the map. The smallest nonzero numerical lie `off the attractor'. And even for finite uncertain- perturbation might well grow sufficiently to bring ties `on the attractor', finite time growth is not bound the nonlinear terms into play, resulting in sustained, by r1. These facts imply that the `super-Lyapunov' complicated dynamics up to the time-scales at which growth found by Nicolis et al.wx 8 is to be expected: the numerical orbit closed exactly upon itself, as all after time t an uncertainty may be magnified by L r trajectories on digital computers willwx 36 . We do not more than the larger of 2 1t and 2 1t, even if the claim that this is the case in the maps considered initial uncertainty is infinitesimal. Over what dura- above, but merely note the dynamics might appear tion can realisticŽ. i.e. operational uncertainties be similar and thus stress the value of performing a treated as infinitesimal? Or equivalently, what is the bifurcation analysis in addition to numerical itera- extent of the linear regime? This is an interesting tion. and open question, even in numerical weather fore- castingwx 38 . Note that computing exponents for finite time is 5. Discussion and conclusions somewhat gratuitous in that any increase will yield a positive effectiÕe exponent; a positive exponent im- The results presented in this article hold implica- plies effectively exponential growth then only in the tions for two questions of general interest: the ap- limit of infinite time. For finite time, a positive proximation of largest Lyapunov exponent L1 and exponent implies growth, but not exponential growth; the estimation of likely forecast accuracy. Noting it only reflects the time dependence of the uncer- Ž k. that the finite time Lyapunov exponents ²li : can tainty under the additional assumption that the be computed accurately by standard methods, a lower growth was exponential. The width of the distribu- f Žk. bound on the error in assuming L11²l :N is tions in Figs. 1 and 2 does not indicate uniform Ž k. y Ž2 k. ' Ž k. given by ²l11:²NNl :²d, while l 1: pro- exponential growth on these time scales. An alterna- vides an upper bound on L1. When a good approxi- tive approach to quantify predictability by computing mation of the `Lyapunov vector' l1 is available, one the time required to reach an uncertainty threshold is can also require for the difference between finite contrasted with the use of effective rates inwx 7,9 , < r Ž k. y lŽ k. < sample and finite time exponents ² 11:² : where examples with both large L1 and large uncer- - d. Yet since both l11and L are multiplicative tainty doubling times are discussed. As proven in ergodic statisticswx 1 , uncertainty in numerical esti- Section 4, there are initial conditions for which no mates of lx1Ž.remains largely unquantified. Simi- perturbations grow for two paradigm ; it larly, inasmuch as matrix multiplication does not would be interesting to investigate the relative loca- L Ž k. - commute, attempts to estimate the uncertainty in 1 tion of regions within which l1 0 and unstable via the standard bootstrap approach must be treated period k orbits for large k, as a function of parame- with care for deterministic systemswx 15 . The goal of ter in a variety of low dimensional maps; the numer- L Ž k. a sufficient condition for the convergence of 1 ics near stable period k points with l1 40 may estimates remains allusive. Quantitative necessary also prove of interest. conditions, along with explicit tests for convergence In this Letter each system has been considered in in aleatoric systemsŽ stochastic systems with positive its natural state space, its original, physically rele- wx L1. are discussed elsewhere 28,37 . vant co-ordinate system. It should be noted, that In terms of identifying the `worst forecast bust', neither the typical measures of forecast error 4 nor the l11are more important than the r simply be- the finite time Lyapunov exponents nor the finite cause the l1 are larger. While it is sometimes argued sample Lyapunov exponents are invariant under co- that the corresponding singular vectors may point

`off the attractor', the l1 remain relevant, as possible uncertainties about the true initial state will also lie `off the attractor', almost certainly. Infinitesimal un- 4 Seewx 39 for an atypical approach. 250 C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 ordinate changesŽ or even changes in a Riemannian matrix, a diagonal matrix, and another rotation ma- metric. . In this study, the forecast error is simply the trix, hence the spectral norm of a matrix is identical <