Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

Disc e e Dynamics in Na u e and Socie y, 2002 VoL. 7 (1), pp. 41-52 Taylo & F ancis Taylo & F ancis G oup E ec o Pa ame e Calcula ion in Di ec Es ima ion o he Lyapuno Exponen in Sho Time Se ies A.M. L(3PEZ JIMINEZ a’*, C. CAMACHO MARTI’NEZ VARA DE REY" and A.R. GARC[A TORRES b aDepa men o Expe imen al Psychology, Uni e si y o Se ille, A da. Camilo Jos Cela s/n. 41005 Se ille, Spain; bphysics Semina , LE.S. Los Vi e os, A da. Bias In an e, s/n. Se ille, Spain (Recei ed 21 Ap il 2001) The li e a u e abou non-linea dynamics o e s a ew ecommenda ions, which some imes a e di e gen , abou he c i e ia o be used in o de o selec he op imal calculus pa ame e s in he es ima ion o Lyapuno exponen s by di ec me hods. These ew ecommenda ions a e ci cumsc ibed o he analysis o chao ic sys ems. We ha e ound no ecommenda ion o he es ima ion o A s a ing om he ime se ies o classic sys ems. The eason o his is he in e es in dis inguishing a iabili y due o a chao ic beha io o de e minis dynamic sys ems o a iabili y caused by whi e noise o linea s ochas ic p ocesses, and less in he iden i ica ion o non-linea e ms om he analysis o ime se ies. In his s udy we ha e cen e ed in he dependence o he Lyapuno exponen , ob ained by means o di ec es ima ion, o he ini ial dis ance and he ime e olu ion. We ha e used gene a ed se ies o chao ic sys ems and gene a ed se ies o classic sys ems wi h a ying complexi y. To gene a e he se ies we ha e used he logis ic map. Keywo ds: Non-linea dynamic; A ac o ; Chaos; Lyapuno exponen INTRODUCTION The disco e y o chao ic beha io in de e minis ic dynamical sys ems has changed some philosophical aspec s in he p e ailing scien i ic pa adigm and has opened new pe spec i es o he design and analysis o ime se ies (Ba ne and Choi, 1989; Casdagli, 1991; Casdagli e al., 1991; Saye s, 1991; Be line , 1992; McCa ey e al., 1992; Nychka e al., 1992; Ge and Allen, 1993; Takens, 1993). In he 1980s, he b eak h oughs in he analysis o ime se ies based on he Quali a i e Theo y o Dynamical Sys ems ha e yielded a se o indexes. These, in heo y, should allow us o de e mine i he appa en ly andom ime sequence obse a ions o a sys em s a e, can o canno be due o chao ic beha io gene a ed by a sys em o nonlinea de e minis ic equa ions (Ashley e al., 1986; B oomhead and King, 1986; Ashley and Pa e son, 1989; B own e al., 1991; G assbe ge e al., 1991; McCa ey e al., 1992; Aba banel e al., 1993; Palu e al., 1993; Takens, 1993). As a sub-p oduc , i is possible o de e mine he numbe o a iables which his se o unknown equa ions would b ing in o play, as well as o classi y sys ems in o uni e sal classes (linea -non-linea , s ochas ic-de e minis ic) and ela e he changes in he beha io quan i ie s wi h changes occu ed in he dynamical beha io o he sys em (bi u ca ions) (Sugiha a and May, 1990; Mon e o and Mo in, 1992). Al hough cha ac e izing dynamical sys ems using he analysis o uni-dimensional ime se ies has been a me hod widely de eloped since he 1980s, he e a e se e al ques ions ha need some conside a ion, a leas in he cases when he indica o s a e ob ained om ime se ies esul ing om beha io al in es iga ion. In his ype o in es iga ion, like in mos si ua ions in eal li e, he da a combine de e minis ic dynamics wi h noise o di e en na u e and magni ude; in addi ion o his, in Psychology i is di icul o main ain he same obse a ion si ua ion o a long ime and his leads o a educ ion in he leng h o he se ies, he e o e he eliabili y o such indexes can be ques ionable. In his s udy, we ha e ied o p o ide answe s o he ques ions a ising om he calcula ion o dominan Lyapuno exponen using di ec me hods. We ha e *Co esponding au ho . Tel.: +34-954557812. Fax: +34-954551784. E-mail: [email p o ec ed] ISSN 1026-0226 (C) 2002 Taylo & F ancis L d 42 A.M.L. JIMNEZ e al. cen e ed on analyzing he s eng h o he exponen o di e en alues o e olu ion imes and ini ial dis ances in sho ime se ies. LYAPUNOV EXPONENTS: DIRECT ESTIMATION The dominan Lyapuno exponen is one o he mos widely used indica o s o desc ibe he quali a i e beha io in a dynamical sys em using he analysis o uni- dimensional ime se ies. To de ine wha is unde s ood by Lyapuno exponen (A) we s a om an ini ial condi ion Yo, o a disc e e dynamical sys em and we conside a e y close poin , whe e he ini ial dis ance (do) is ex emely small. Le d be he dis ance a e i e a ions. I we assume ha in Eq. (3) when do d aws o 0, he e m wi hin he loga i hm is he de i a i e o he i e a e o e alua ed in y0(( )(y0)). Applying he chain ule o di e en ia ion, he de i a i e o can be w i en as a p oduc o de i a i es o y) e alua ed a he successi e ajec o y poin s yo,yl,y2,.., and so on. We can hen de ine Lyapuno exponen in a mo e in ui i e way wi h he ollowing 1 A lim- In -1 H (Yl) =0 liml (lnl ’(y0)l + lnl (y)l +... --,eo + lnl ’(y,-l)l) Id, Id01 exp ( A) (1) hen A is wha we call Lyapuno exponen (Packa d e al., 1980; Schus e , 1984; Mon e o and Mo an, 1992; Nychka e al., 1992; Aba banel e al., 1993; Simmons, 1993; S oga z, 1994; Hilbo n, 1994; Ma in e al., 1995). Tha is, he a e age exponen ial a e o di e gence o con e gence o ajec o ies which a e e y close in phase space (Wol e al., 1985; DeSouza-Machado e al., 1990; Zeng e al., 1991). The numbe o Lyapuno exponen s in a dynamical equals he numbe o s a e a iables conside ed. A uni- dimensional sys em is cha ac e ized by one single exponen . The signs o Lyapuno exponen s p o ide quali a i e in o ma ion abou he dynamics o a sys em. I he sign is posi i e, his is an indica ion o chaos. I i is nega i e, he e is con e gence be ween close ajec o ies and he e o e classic a ac o s exis . I he beha io o a dynamical sys em ep esen ed by unc ion con e ges o a ixed poin (y*) o o a limi cycle o a pe iod p which con ains a y* poin , hen i is easy o p o e ha Lyapuno exponen A < 0. When Lyapuno exponen Z 0 he ini ial pe u ba ion will emain wi h , i.e. ajec o ies nei he di e ge no con e ge, hei ini ial dis ance emains cons an . This kind o beha io is ypical o a cons an pe iodical o bi (Sano and Sawada, 1985; Wol e al., 1985; McCa ey e al., 1992). I he sys em is h ee-dimensional (i.e. i con ains h ee s a e a iables) he possible combina ion o signs and he a ac o s hey desc ibe a e: (+, 0, -), o a s ange a ac o ; (0, 0,-), a quasi-pe iodical a ac o known as o us; (0,-, -), limi cycle and ), a ixed poin . I in Eq. (1) we ake loga i hms and we eplace d, by i s ma hema ical exp ession we ob ain A -ln In (2) Y00 7 I he exp ession (2) has a limi as oo we de ine ha limi o be he Lyapuno exponen : A= -.oo lim-llnl lim l ln (yo + do) (yo) do (3) lim 1 lnl ’(y )l ---,oo =0 (4) Equa ion (4) ells us ha he Lyapuno exponen is he a e age o he na u al loga i hm o he absolu e alue o he de i a i es o he disc e e dynamical sys em e alua ed in he ajec o y poin s o ime se ies conside ed. I he applica ion o he disc e e dynamical sys em o wo close ajec o ies e en ually leads us o sepa a e poin s, hen he absolu e alue o he de i a i e o is g ea e han 1 when we e alua ed a hose ajec o y poin s. I he absolu e alue is g ea e han 1, i s co esponding loga i hm is also posi i e. I he poin s o he ajec o y con inue o di e ge, hen he a e age o he loga i hms o he de i a i es is posi i e. I we calcula e Lyapuno exponen o a sample o ini ial poin s and we a e age he esul s, we can de ine he a e age Lyapuno exponen (X) o he sys em. An uni- dimensional disc e e sys em has chao ic ajec o ies, o ce ain pa ame e alues on which i s beha io depend, i he a e age o Lyapuno exponen s (X) is posi i e. The exp ession (4) o calcula ing A equi es ha he shape o he disc e e dynamical sys em be known. Bu , wha happens when we do no know he sys em bu know one ime se ies o a ele an s a e a iable? S udies conce ning non-linea dynamics ha e sugges ed wo app oxima ions o he es ima ion o Lyapuno exponen : di ec me hods o di ec es ima ion and Jacobian me hods (Guckenheime , 1982; Eckmann and Ruelle, 1985; Wol e al., 1985; McCa ey e al., 1992; Nychka e al., 1992; Aba banel e al., 1993; Damming and Mi schke, 1993). Di ec me hods calcula e Lyapuno exponen di ec ly om he ime se ies, wi hou any addi ional assump ions o app oxima ions abou he subjacen dynamical sys em. Since he subjacen sys em and i s dimensions a e unknown, we calcula e, using he econs uc ion ec o me hod and o di e en dimensions, exponen s un il ha ceases o a y signi ican ly as he dimension o econs uc ed space inc eases. Al hough he disad an age o his p ocedu e is ha i only p o ides he la ges Lyapuno exponen , we will discuss his me hod below. LYAPUNOV EXPONENT IN SHORT TIME SERIES 43 ’J’o 20 FIGURE Scale egion o a se ies o dis ances ob ained om y +l 4y (1 y ). Le us de ine yl,Ym,...,Y , as he elemen s o a ime se ies and dmax as he maximum dis ance be ween wo poin s o he se ies equi ed o be conside ed as "in ini esimally" close. I he sys em beha es chao ically o poin s Yi and yj; wi h a dis ance do--< dmax, he di e gence o close ajec o ies will be shown in he sequence di e ences do lyj Yil d lYj+ Yi+ll d2 [Yj+2 Yi+m[ d ly+ yi+ l This will show an exponen ial inc ease, o a leas he mean, T. Wi h his me hod o calcula ing , we ind wo close ajec o ies in he s a e space and we calcula e he se ies o dis ances, which de i e om hese wo ini ial condi ions. Al hough in gene al, calcula ing Lyapuno ’s maximum exponen is easy, we belie e ha i is wo h conside ing a ew aspec s. We assume a sepa a ion a e o exponen ial app oxi- ma ion be ween wo close ajec o ies. Fo a gi en ime se ies i is necessa y o demons a e his assump ion. One way o doing his is by plo ing he na u al loga i hm o he di e ences (ln d ) as a unc ion o index T (see Fig. 1). I he di e gence is exponen ial, he poin s (ln d , T) will app oxima e a line gi en by he ollowing exp ession in d In do + AT The slope o he i ed line--gene ally by leas squa es--is hen he alue o Lyapuno exponen The dispe sion diag am is ne e exac ly a line because he exponen ial di e gence a ies along he a ac o and eaches a maximum when i is compa able o he a ac o ’s diame e , which is de ined as he maximum dis ance be ween he poin s o he ajec o y e ol ed o e he a ac o . Two egions a e usually dis inguished: one ha is called scale egion o which he line is i ed and ano he one in which ln d emains mo e o less cons an as T inc eases. Adjus ing a line wi h leas squa es o he scale egion p o ides he measu emen o he dominan Lyapuno exponen and he accu acy o he adjus men On he o he hand, he alues o ) can, and gene ally depend on Yi alues chosen o be he ini ial condi ions, o a he , he alues o he ini ial dis ance (Id01) be ween hem. To cha ac e ize he subjacen a ac o o a gi en ime se ies we ha e o calcula e he a e age co esponding alue o he se o Lyapuno exponen s ob ained om a numbe o ajec o ies which ollow he condi ion (5). do lyi yjl <- dmax A a p ac ical le el, se e al ques ions a ise conce ning he ime span equi ed be ween poin s Yi and yj so ha hey can be conside ed as ini ial condi ions o wo ajec o ies, he leng h o he se ies, he numbe o ini ial condi ions o dis ances, he numbe o i e a ions o op imal e olu ion ime equi ed and he ini ial dis ance in o de o conside poin s as in ini esimally close in he phase space. We do no ha e many answe s o he ma e s men ioned in he las pa ag aph. Howe e , we ha e ound some sugges ions, which ha e esul ed om some simula ion expe imen s made wi h speci ic dynamical sys ems in which he heo e ical alue o Lyapuno ’s maximum exponen is known, in his s udy, we ha e ga he ed some o he mos gene al sugges ions made in he esea ch ma e ial ha we ha e e ised. The ini ial sepa a ion equi ed be ween wo poin s so ha hese can be conside ed as ini ial condi ions o wo di e en ajec o ies in he econs uc ed phase space ends o be ela ed o wha is called o bi al pe iod (Wol e al., 1985; Theile , 1986). This is he ime, which a sys em akes o co e an o bi . I is ecommended ha he ini ial sepa a ion be ween wo poin s should be a leas one o bi al pe iod. The di icul y o applying his ecommenda ion lies in ha he shape o he dynamical sys em gene a ing he da a mus be known since i is on sys em i sel ha he calcula ion o he o bi al pe iod is based. In he cases o which he shape o he dynamical sys em is unknown, we can ollow he ecommenda ions gi en by Hilbo n (1994) and Theile (1986). They main ain ha he ini ial sepa a ion equi ed be ween wo poin s so ha hey can be conside ed as ini ial condi ions o wo di e en ajec o ies, mus be g ea e han wha is known as au o-co ela ion ime (-) which is gi en by he exp ession (6). 1 - (6) ln(1/p) In Eq. (6) p is he au oco ela ion coe icien o lag 1. When p app oaches 1, Eq. (6) changes and becomes Eq. (7). 1 - (7) As a as he numbe o ini ial condi ions equi ed is conce ned, we ha e ound se e al ecommenda ions. Sano 44 A.M.L. JIMINEZ e al. o. 8i o o 4 o Io 20 30 4O so FIGURE 2 E olu ion o he dis ances be ween wo se ies gene a ed om he logis ic map o a 4 and do 0.001. and Sawada (1985) es ablish a lowe limi which will depend on he dimension o he econs uc ed space; o hem, he numbe o ini ial condi ions equi ed o he es ima ion o Lyapuno ’s maximum exponen mus be highe o equal o he dimension o he econs uc ed space (N >-- de). Hilbo n (1994) ecommends om 30 o 40 ini ial condi ions dis ibu ed o e he a ac o . O he au ho s es ablish a dependence on he dimension o he econs uc ed phase space and on he ini ial ole ance in o de o conside poin s as in ini esimally close. The exp ession (8), by Dimmig and Mi schke (1993), p o ides he numbe o ini ial condi ions equi ed o cha ac e iz- ing he a ac o acco ding o Lyapuno ’s maximum exponen N dmax/ (8) whe e d, is he dimension o he econs uc ed space and alma is he maximum ini ial dis ance equi ed o conside wo ajec o ies as close. Wol e al. (1985) ecommend om l0 de o 30 de Fo each o he se ies o dis ances we calcula e he local Lyapuno exponen (Ado). The a e age Lyapuno exponen () is gi en by (9) whe e N is he numbe o se ies o dis ances conside ed. Ano he aspec o ake in o accoun is he numbe o i e a ions T, op imal e olu ion ime o leng h o he se ies o dis ance ha a e sui able o calcula ing he Lyapuno exponen . In he case o chao ic sys ems, we know ha con e gence o heo e ic alues depends, among o he s a iables, on he use o an e olu ion ime (T) ha is no oo la ge. Thus, he exponen b ings he sensi i i y o he ini ial condi ions and no he con e gence p oduced by he bounda y o he a ac o alues o a egion o he phases space and o he ini ial dis ance (do) o conside wo neighbo ing ajec o ies. We know ha he e is di e gence in chao ic dynamics bu a he same ime, due o he olding mechanism, he alues go h ough poin s ha a e in ini esimally close o p e ious alues. In Fig. 2 we ha e shown he e olu ion o he dis ances be ween wo se ies gene a ed om he logis ic map, o a 4 and o he ini ial condi ions y01 0.1 and y02 0.101. A la ge T alue may p oduce an unde es ima ion o he Lyapuno exponen . Th ee c i e ia ha e been p oposed o ix he numbe o i e a ions o e olu ion imes (T): (a) To es ablish, a p io i, a ixed e olu ion ime (i.e. T 10), (b) To es ablish a inal dis ance ha can be when he a ac o diame e o a pe cen age is eached. In his case, he ime o e olu ion would be a iable and (c) To iden i y in he g aphic In d be o e T in he scale egion. Finally, he dis ance dmax, o he limi ed sys ems ha in e es us, canno be oo la ge. Due o he ac ha he alues Yi a e cons ained in size, he ini ial dis ances canno be la ge han he di e ence be ween he maximum alue, Ymix, and he minimum alue, Ymin. Mo eo e , he e a e p ac ical limi s when de e mining he ini ial dis ance o he ini e p ecision o he da a. The numbe o decimals is an in e io limi o he ini ial dis ance. Fo example, i he da a is egis e ed wi h h ee decimals, i would be senseless o ques ion a di e ence lesse han 0.001. Ano he e ec o he ini e p ecision is ha we can encoun e epea ed da a. To summa ize, he li e a u e abou non-linea dynamics only o e s a ew ecommenda ions, which some imes a e di e gen , abou he c i e ia o be used in o de o selec he op imal calculus pa ame e s in he es ima ion o Lyapuno exponen s by di ec me hods. These ew ecommenda ions a e ci cumsc ibed o he analysis o chao ic sys ems. We ha e ound no ecommenda ion o he es ima ion o A s a ing om he ime se ies o classic sys ems. The eason o his is he in e es in dis inguishing a iabili y due o a chao ic beha io o de e minis dynamic sys ems o a iabili y caused by whi e noise o linea s ochas ic p ocesses, and less in he iden i ica ion o non- linea e ms om he analysis o ime se ies. In his s udy, we ha e cen e ed in he dependence o he Lyapuno exponen , ob ained by means o di ec es ima ion, o he ini ial dis ance and he ime e olu ion. We ha e used gene a ed se ies o chao ic sys ems and gene a ed se ies o classic sys ems wi h a ying complex- i y. To gene a e he se ies we ha e used he logis ic map (10). Y +l ay (1 Y ) (10) We know ha he logis ic equa ion is a s uc u ally uns able sys em. Tha is, i s beha io depends on he alue o he pa ame e a. In Table I we ha e speci ied he alues o a used in his esea ch, i s beha io and he diame e o he co esponding a ac o (Hilbo n, 1994; S oga z, 1994). METHOD In his sec ion, we desc ibe he dependence o he Lyapuno exponen bo h on he ini ial dis ance and he LYAPUNOV EXPONENT IN SHORT TIME SERIES 45 TABLE Value o a Beha io Diame e o a ac o Lyapuno exponen 3.2 Limi cycle (p 2) 3.52 Limi cycle (p 2) 3.55 Limi cycle (p 8) 3.58 Chaos (low ex en ) 4 Chaos (high ex en ) 0.2910.51,0.81] -0.91 0.5110.37,0.88] -0.19 0.53[0.36,0.89] -0.1 0.56[0.34,0.90] 0.1 1[0,1] 0.69 e olu ion ime in se ies gene a ed by he logis ic map. We w o e a p og am in he Ma hema ica p og amming language ( . 2.1 o Windows) and we gene a ed se ies wi h close ini ial alues. The p oximi y c i e ia used was he one ecommended by Sano and Sawada (1985) acco ding o which wo gene a ed se ies o a dynamic sys em a e conside ed o be close i he ini ial dis ance be ween hem (do) is be ween 1 and 5% o he a ac o diame e . Wi h he p og am made o gene a e he da a and gi en ha sensi i i y o insensi i i y o ini ial condi ions is a cha ac e is ic o a g oup o ajec o ies wi h close ini ial condi ions, o he alues a 3.2, 3.52, 3.55, 3.58, 4} We ini ially gene a ed 50 se ies o N-- 100 da a wi h ini ial condi ions (Yo) whose dis ance was 1% o he a ac o diame e (see Table I). F om he 50 se ies gene a ed we ob ained: 49 se ies o dis ances wi h do 1%, 48 wi h do 2%, 47 wi h do 3%, 46 wi h do 4% and 45 wi h do 5%. The ini ial alue o he i s se ies gene a ed (Yo) coincided wi h he minimum alue o he a ac o ampli ude (see Table I) wi h he excep ion o he se ies gene a ed o a 4, whe e we s a ed om y0-- 0.1. Fo his las alue, we used h ee addi ional dis ances among he ini ial condi ions 0.1, 0.25 and 0.5% and 50 dis ance se ies we e gene a ed. The Lyapuno exponen o 1 <-- T --< 99 and he ini ial dis ance (do) we e ob ained by means o he exp ession A=ln In (11) Yio YjO -o We calcula ed he a e age Lyapuno exponen (X) as es ima o o de A o he coe icien s ob ained wi h exp ession (11) o each do and T. We g aphically ep esen ed X as a unc ion o index T o each alue o do in o de o s udy he incidence o hese pa ame e s. RESULTS We o ganized he esul s in wo di e en sec ions. In he i s pa , we show hose esul s co esponding o he T T -,2 e)e 3.2, do=5 iles 0,04 FIGURE 3 Beha io o X acco ding o T when inc easing he do o a 3.2. 46 A.M.L. JIMNEZ e al. FIGURE 4 Beha io o X when inc easing do o 10 and 20%. alues o a whose beha io con e ges on a classic a ac o . In he second pa , we o e he esul s o he alues o a wi h a chao ic beha io . We ha e p e e ed o show he g aphics co esponding o he e olu ion o X acing T ins ead o he alue ables because we see hem as mo e illus a i e and easie o in e p e . Classic A ac o a 3.2 In Fig. 3, we ep esen ed he a e age Lyapuno exponen (X) as a unc ion o he e olu ion ime (T) o he se o dis ances ha we e conside ed. In he di e en g aphics o Fig. 3, we ha e d awn a b oken line h ough he alue o he heo e ic Lyapuno exponen . The con e gence o his alue can be seen wi h he inc ease o he e olu ion ime T. The con e gence o m is independen om he ini ial dis ance (do) conside ed. In all cases he e is an oscilla o y dec ease o X. Fo he se do {1%,2%,3%} he con e gence s ops in T 14, ob aining he alue o X closes o he heo e ic when T 13. -,z, - ba=3.52, d=4%, Bias=O.01, T,,=17 ,2 0,0 -,2 T c)==3.SZ = =0.00 ’.=7 d) 3.52, d 4%, T e) 3.52, d= 5, Bias 0.05, T,, 17 FIGURE 5 Beha io o X acco ding o he e olu ion ime (T) o a 3.52. LYAPUNOV EXPONENT IN SHORT TIME SERIES 47 T b) 3,55, do 2%, 8as O.01, T c) 3,55, d 3%, Odas <0.0I T d) SS, d,=,m, nab 002 Bias 0,01, T.-7 FIGURE 6 Beha io o X acco ding o he e olu ion ime (T) o a 3.55. Fo he se do {4%, 5% he con e gence s ops in T 16. The closes mean Lyapuno exponen (A) is ob ained o T 15 and T 13 ( he bias is 0.04). To see i inc easing he ini ial dis ance would dec ease he bias (X-A), we analyzed in 16 and 15 se ies o dis ances he beha io o A acco ding o he e olu ion ime (T) when inc easing he ini ial dis ance (do) be ween he ajec o ies o 10 and 20% o he a ac o diame e . We ha e p esen ed he esul s in Fig. 4. I can be obse ed (g aphic (a)), how he alue o he bias and he shape o he con e gence a e simila o hose ob ained o he ini ial dis ances analyzed p e iously. On he con a y, when he alue o he ini ial dis ance inc eases o 20% he bias inc eases d ama ically o 0.51. a 3.52 Figu e 5 shows he beha io o A acco ding o he e olu ion ime o he uni o ini ial dis ances conside ed: do {1%,2%,3%,4%,5%} As wi h he se ies ep esen ed in Fig. 3, o a 3.52 he mean Lyapuno exponen declines oscilla o y when T inc eases. Mo eo e , some di e ences can be obse ed in ela ion o do. In he g aphics o Fig. 5, we ha e d awn a con inuous line. This is pe pendicula o he axis o he in e sec ion o he alue o T, which p o ides he bes es ima ion o A. As is cus oma y, a b oken line ep esen s he heo e ic alue o he Lyapuno exponen . We can see (g aphics a, b and c in Fig. 5) ha X ends o he heo e ic alue when he ini ial dis ances a e 1-3%. When inc easing he ini ial dis ance o 4 and 5% (g aphics d and e in Fig. 5), he limi o X is no he heo e ic alue bu a la ge one. Fo he alues o dis ances and e olu ion imes conside ed, he di ec es ima ion p o ides alues ha a e biased posi i ely o A. a 3.55 Figu e 6 shows he beha io o X acco ding o T o he dis ances conside ed. In he g aphics in Fig. 6, we ha e d awn an o dina e line A--0 ( he g ay line) wi h he aim o assessing possible quali a i e e o s when iden i ying he subjacen a ac o o he da a gene a ing sys em. The To (op imal) is he alues o T o which he bias is less. In g aphs (a) and ( ) in Fig. 6, we can see an ini ial pe iod wi h g ea a iabili y in which A oscilla es be ween 48 A.M.L. JIMINEZ e al. T 3.56, dn 1% slow and p ac ically s abilizes i sel oscilla ing be ween -0.04 and -0.02 when he ini ial dis ance is 1%, as can be seen in Fig. 7. Fo he es o he do wi h he e olu ion ime, oscilla ion s abilizes be ween 0.03 and 0.001. In any case, we can deduce om he Fig. 6, ha using sho and e en e olu ion imes ( i s sec ion o he dispe sion diag ams) can be mo e p oblema ic han using longe e olu ion imes as, al hough he bias inc eases wi h T, he e a e no quali a i e e o s no ed. FIGURE 7 Oscilla ions o X in he in e al: T [60, 100]. he zone o chao ic beha io (A > 0) o he e en T alues, and he zone o ecu en beha io (X > 0) o he odd alues. A e his i s span, when T g ows o he alue o A ha is highe han he heo e ical alue (X -0.1), he di ec es ima ions o con e ge slowly and in an oscilla ing way. The inc ease in he ini ial dis ance p ocess b ings he bounda y o he con e gence p ocess o X o 0. In he in e al 5 -< T --< 11 we can ind he alues ha p o ide he bes es ima ions o . A common alue o To ha p o ides eliable es ima ions (bias -< 0.02) o he se o dis ances unde conside a ion is To 7. F om his ini ial pe iod, bo h he decline o ,( o e en alues o T, as well as he g ow h o odd alues, is e y Chao ic Beha io a 3.58 Amongs he many alues o a whose beha io is chao ic in he in e al [3.58, 4], we used p ecisely he ex emes, which co espond o he s ange a ac o o he smalles and la ges diame e s, espec i ely. Figu e 8 shows he esul s ela i e o he o m o he dependence o X as opposed o he e olu ion ime. In he se o g aphs in Fig. 8, we can dis inguish h ee sec ions: in he i s sec ion o app oxima ely 1 <_ T <-- 4, we can see a sha p inc ease o X owa ds he heo e ical alue. T 0 20 40 0 80 00 b)a=3.S& do=2% To={4,8}.Bla$=O .58. =3 %, T,., 14, 8) I 0.01 o )a=3.58, do=4%, To= [4. 8), Bi= e)==3.,.’,8, do=5%. T.,={4.S}. IIZI .c0.01 do T 7,00 8,00 FIGURE 8 Beha io o X acco ding o he e olu ion ime (T) when inc easing he ini ial dis ance o a 3.58. LYAPUNOV EXPONENT IN SHORT TIME SERIES 49 ,10 ,05 ,05 .04 .02 .01, -.01 -,0 -,0 T c) 3.58 cJo -.3% ,05 ,03 01o -.02 -,03 -,04 " -,0 d)a 3.5 do 4% ,04 ,02 ,01, ,00 -,01 -,0. -,0 -,04 ,0 ::N FIGURE 9 Beha io o X o T > 8 o a 3.58. We can conside he second sec ion as s abiliza ion, in which A oscilla es a ound a cons an alue e y nea o he heo e ical alue o he e en alues o T. This second span becomes p og essi ely sho e as he ini ial dis ances g ow. Thus, o do--1%, he alues o X emain p ac ically cons an un il T 25, whils o do 5%, his sec ion inishes in T 8. Du ing his ime o s abiliza ion, as he ini ial dis ance inc eases, e o is mo e likely in iden i ying he ype o a ac o p esen in he da a, especially o he odd alues o T. The hi d sec ion, which is clea ly iden i iable in he igu es, is he longes . In his sec ion, A oscilla es wi h a ia ions ha a e p ac ically cons an in size, a ound 0, independen o he ini ial dis ance. The inc ease in e olu ion ime o highe han op imal T unde es ima es he alue o X I seems ha wi h T, he a e age exponen X would be cen e ed on 0 o all he ini ial dis ances. We ha e widened his sec ion speci ically in o de o obse e whe he he beha io in his a ea is eally independen o he ini ial dis ance o no . The esul s a e shown in Fig. 9. Taking he alues o X o T > 8, we can see ha , wi h he ini ial dis ances do inc eases he possibili y o quali a i e e o when using longe e olu ion imes. In g aph (a) in Fig. 9, p ac ically 100% o he A alues a e g ea e han 0. He e, al hough he bias is sha p, ne e heless he quali a i e conclusion ega ding he na u e o he a ac o con ained in he se ies would be co ec . G aphs (b)-(d) in Fig. 9 shows how he dis ibu ion o alues a ound A--0 in e s wi h he inc ease in he ini ial dis ance and he numbe o nega i e exponen s inc eases p og essi ely. The beha io o X in ela ion o T is simila o ha obse ed o he alues o a wi h ecu en beha io a in e als o 2, 4 and 8. In g aph ( ) in Fig. 8 we can see how, o he alues o T ha cons i u e he i s and second sec ion o he e olu ion o X o he se o ini ial dis ances, he bias is independen o hese. a--4 G aphs (a) and (h) in Fig. 10 show he dependence o X wi h espec o he e olu ion ime o he se o do ha was applied. In g aphs (b)-(g) (see Fig. 10) we can app ecia e a e y apid ini ial inc ease which is mo e o less lineal, owa ds he heo e ical alue o X. This g ow h is in e up ed ab up ly in A alues below he heo e ical alue. Fo g owing alues o he se o ini ial dis ances do--= 1%, 2%, 3%, 4%, 5% }, he maximum X alues eached a e: 0.59, 0.58, 0.55, 0.55 and 0.53. I appea s ha we can con i m a di ec ela ionship be ween he bias