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Localized Lyapunov Exponents and the Prediction of Predictability 3 July 2000 Physics Letters A 271Ž. 2000 237±251 www.elsevier.nlrlocaterpla Localized Lyapunov exponents and the prediction of predictability 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 Lyapunov exponent 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 <<k. << i Ž.x log 2 Ž.M Žx,k .zi Ž.x Lyapunov exponents stems from the fact that their k value does not depend upon the metricŽ or coordinate system. used; this is not the case for the exponents 1 s s Ž k. log 2 Ž.i Ž.x .2 Ž. based upon a finite length of trajectory, yet in prac- k tice only the latter are available. We return to this issue in Section 5. Finally, there is the question of Properties of these exponents have been noted by model error in nonlinear forecasting, either paramet- Lorenzwx 4 , Grassberger et al. w 16 x , Abarbanel wx 5 and ric or structuralwx 12 . Arguably, model error may be references thereof. By Oseledec's Theoremwx 1 , in ™ Ž k. more responsible for poor predictions of real nonlin- the limit k ` the li Ž.x converge to a unique set ear systems than `chaos.' Model imperfections are of values, the Lyapunov exponents Li, which are the C. Ziehmann et al.rPhysics Letters A 271() 2000 237±251 239 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 attractor 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.<<M Žx,k .Ž.lx << ,5 Ž. with 121k 6 and r Ž k.Ž.x is similarly defined using lx Ž.. ts0.4y mm22q q Both the r Žk.
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