nlin.CD 2019-08-19 0

Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chao…

Authors: ** Ch. Skokos, J. Bountis, T. C. Bountis (※ 실제 논문 저자는 Ch. Skokos

Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method
Geometrical prop erties of lo cal dynamics in Hamiltonian systems: the Generalized Alignmen t Index (G A LI) metho d Ch. Sk ok os a , b , ∗ , T.C. Boun tis a , Ch. An tonop oulo s a a Dep artment of Mathematics, Division of Applie d Analysis and Center for R ese ar ch and Applic ations of Nonline ar Systems (CRANS), University of P atr as, GR-26500 Patr as, Gr e e c e b Astr onomie et Syst` emes Dynamiques, IMCCE, Observatoir e de Paris, 77 avenue Denfert–R o cher e au, F- 750 14, Paris, F r a nc e Abstract W e inv estigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolutio n of v olume elements formed by unit d eviatio n v ectors ab out their orbits. The b eha vior of these v olumes is strongly influenced b y the regu- lar or c h aoti c n ature of the motion, the num b er of deviation vec tors, their linear (in)dep endence and the spectrum of Ly ap u no v exp onents. The differen t time evo lu- tion of these v olum es can b e used to iden tify rapidly and efficien tly th e nature of the dynamics, leading to the introd u ction of qu an tities that clearly distinguish b et ween c h aotic b eha vior and quasip erio dic motion on N -dimensional tori. More sp ecifically w e in tr o du ce the Generalized Alignment I n dex of ord er k (GALI k ) as the volume of a ge neralized parallelepip ed, whose edges are k initially linearly indep enden t un it deviation vect ors from the stud ied orbit w hose magnitude is normalized to unity at ev ery time step. W e sh o w analyticall y and v erify numerically on particular examples of N degree of fr eedom Hamiltonian systems that, for c haotic orb its, GALI k tends exp onen tially to zero with e xp onent s that in v olve the v alues of sev eral Lyapuno v exp onen ts. In the case of regular orbits, GALI k fluctuates around non–zero v alues for 2 ≤ k ≤ N and go es to zero for N < k ≤ 2 N follo wing p o w er laws that dep end on the dimension of the torus and th e n um b er m of deviation v ectors initially tan- gen t to the torus : ∝ t − 2( k − N )+ m if 0 ≤ m < k − N , and ∝ t − ( k − N ) if m ≥ k − N . The GALI k is a generalizat ion of the Smaller Alig nment Index (SALI) (GALI 2 ∝ SALI). Ho w ev er, GALI k pro vides significantly more detailed information on the lo- cal dynamics, allo ws f or a faster and clearer d istinction b et ween order and c h aos than SALI and wo rks ev en in cases where the SALI metho d is inconclusive. Key wor ds: Hamiltonian systems, Chaos detection metho ds, Chaotic motion P ACS: 05.45.-a, 05.45.Jn, 05.45.Ac Preprint su bmitted to Ph y sica D 19 August 2019 1 In t ro duction Determining the c haotic or regular nature of orbits in c onservative dynamical systems is a fundamen tal issue of nonlinear science. The difficult y with con- serv ativ e systems, of course, is tha t regular and c haotic orbits are distributed throughout phase space in very complicated w ays, in con trast with dissipa- tiv e systems, where all orbits ev en tually fall on regular or c haotic attractors. Ov er the y ears, sev eral metho ds distinguishing regular from c haot ic motion in conserv ativ e systems ha v e b een prop osed and applied, with v arying degrees of suc cess. These methods can b e div ided in tw o ma jor categories: Some are based on the study of the ev olutio n of small deviation ve ctors fro m a given orbit, while others rely on the analysis of the particular orbit itself. The most commonly emplo y ed metho d for distinguishin g b etw een order and c haos, whic h b elongs to the category related to the study of deviation v ectors, is the ev aluation of the maximal Lyapuno v Characteristic Exp onen t (LCE) σ 1 ; if σ 1 > 0 the orbit is c haotic. The theory of Ly apuno v expo nents was a pplied to c har a cterize chaotic orbits b y Oseledec [1], while the connection b et w een Ly apunov exp onen ts and exp onen tial div ergence of nearb y orbits w as g iv en in [2,3]. Benettin et al. [4 ] studied the problem of the computation of a ll LCEs theoretically a nd prop osed in [5] an algorithm for their num erical computation. In particular, σ 1 is computed as the limit for t → ∞ of the quan tit y L 1 ( t ) = 1 t ln k ~ w ( t ) k k ~ w (0) k , i.e. σ 1 = lim t →∞ L 1 ( t ) , (1) where ~ w (0), ~ w ( t ) are deviation vec tors from a giv en orbit, at times t = 0 a nd t > 0 r esp ectiv ely . It has b een sho wn tha t the ab o ve limit is finite, indep enden t of t he c ho ice of the metric for the phase space and conv erges to σ 1 for almost all initial v ectors ~ w (0) [1,4,5]. Similarly , all other LCEs , σ 2 , σ 3 etc. are computed as the limits for t → ∞ of some appropriate quan tities, L 2 ( t ), L 3 ( t ) etc . (se e [5] for more details). W e note that throughout the presen t pap er, whenev er w e need to compute the v alues of the maximal LCE o r of sev eral LCEs w e apply resp ectiv ely the algorithms prop osed by Benettin et al. [2,5]. Since 1980 , new metho ds hav e b een in t ro duced for the effectiv e computation of LCEs (e. g. [6], see also [7] and references therein). The true p ow er of these tec hniques is rev ealed in the study o f mu lti–dimensional sy stems, when only a small ∗ Corresp onding author. Email addr esses: hskokos @imcce.fr (Ch. Skok os), bou ntis@math.up atras.gr (T.C. Bount is), antonop @math.upatr as.gr (Ch. An tonop oulos). URLs: htt p://www.imc ce.fr/ ˜ hskokos (Ch. Sk ok os), http://w ww.math.upat ras.gr/ ˜ bountis (T.C. Boun tis), http://w ww.math.upat ras.gr/ ˜ antonop (Ch . An tonop oulos). 2 n umber of LCE are of intere st. In such cases, these me tho ds are s ignifican t ly more efficien t than the metho d of [5], whic h computes the whole sp ectrum of LCEs. On the other hand, they are less or equally efficien t when compared with the metho d of [2] for the computation of the maximal LCE, whose v alue is sufficien t f or the determination of the regular or c haotic nat ure of an orbit. Among o ther c haoticity detectors, b elonging to the same category with the ev aluation of the maximal LCE, are the fast Ly apunov indicator (F LI) and its v arian t s [8 ,9,10,11,12], the mean exp onen tial growth of nearb y orbits (MEGNO) [13,14], the smaller alignmen t ind ex (SALI) [1 5,16,17], the relativ e Ly apuno v indicator (RLI) [18], as well as metho ds based on the study of p ow er sp ec- tra of deviation v ectors [19], as w ell as spectra of quan tities r elat ed to these v ectors [20,21,22]. In the category of metho ds ba sed on the analysis of a time series constructed b y the co or dinates of the orbit under study , one may list the frequency map analysis of Lask ar [23,24,25,26,2 7 ,28], the metho d o f the lo w frequency pow er (LFP) [29,30], the ‘0–1’ test [31], as w ell as some other more recen tly in tro duced tec hniques [32,33]. In the presen t pap er, w e generalize and improv e considerably t he SALI metho d men tioned ab ov e by in tro ducing the Generalized ALignmen t Index (G ALI). This index retains the adv an t ages of the SALI – i.e. its simplicit y and effi- ciency in distinguishing b et w een regular and c haotic motion – but, in addi- tion, is faster than t he SALI, displays p o w er law deca ys that dep end on torus dimensionalit y and can also be applied successfully to cases where the SALI is inconclusiv e, lik e in the case of c haotic orbits whose t wo largest Ly apunov exp o nen ts are equal or almost equal. F or the computation of the GALI w e use information fr o m the ev o lution of mor e than two deviation v ectors from the reference orbit, while SALI’s com- putation requires the ev o lution of o nly tw o suc h v ectors. In particular, GALI k is prop ortional to ‘v olume’ elemen ts fo rmed by k initially linearly indep enden t unit de viation v ectors whose magnitude is normalized to unit y at ev ery time step. If the orbit is c haotic, GALI k go es to zero exp onen tia lly fast by the law GALI k ( t ) ∝ e − [( σ 1 − σ 2 )+( σ 1 − σ 3 )+ ··· +( σ 1 − σ k )] t . (2) If, on the other hand, the orbit lies in a n N –dimensional torus, GALI k displa ys the follo wing b eha viors: Eithe r GALI k ( t ) ≈ constan t f or 2 ≤ k ≤ N , (3) or, if N < k ≤ 2 N , it deca ys with diffe r ent p ower laws , dep ending on the n umber m of deviation v ectors whic h initially lie in the tangen t space of the 3 torus, i. e. : GALI k ( t ) ∝      1 t 2( k − N ) − m if N < k ≤ 2 N and 0 ≤ m < k − N 1 t k − N if N < k ≤ 2 N and m ≥ k − N (4) So, the GALI allows us to study mor e efficien tly the ge ometric al prop erties of the dynamics in the neighborho o d of an orbit, esp ecially in higher dimensions, where it allows for a m uc h faster determination of its chaotic nature, o ve rcom- ing the limitations of the SALI metho d. In the case of regular motion, GALI k is either a constan t, or deca ys b y p o wer laws that dep end on the dimen sional- it y of the sub space in which the orbit lies, w hic h can prov e use ful e .g., if our orbits a r e in a ‘stic ky’ region, o r if our system happ ens to p ossess few er or more than N indep enden t in tegrals of the mot ion (i.e. is partially in tegra ble or super- in tegrable res p ectiv ely). This pap er is organized as follow s: In section 2, w e recall the definition of the SALI describing also its b eha vior for regular a nd c haotic orbits o f Hamil- tonian flo ws and symplectic maps. In section 3 , w e in tro duce the GALI k for k deviation v ectors, explaining in detail its nume rical computation, while in section 4 w e study theoretically the b eha vior of the new index for c haotic and regular orbits. Section 5 pres en ts applications of the GALI k approac h to v arious Hamiltonian systems of differen t n um b ers of degrees of freedom, con- cen trating on its part icular adv antages. Finally , in section 6, we summarize the results and presen t our conclusions, while the app endices are devoted re- sp ectiv ely to the definition of the w edge pro duct and the explanation of the explicit connection b et w een GALI 2 and SALI. 2 The SALI The SALI metho d w as introduced in [15] and ha s b een applied succe ssfully to detect regular and chaotic motion in Ha milto nian flo ws as we ll as s ymplectic maps [34,16,35,36,17,37,38,39,40,4 1,42,43,44]. It is an index t ha t tends ex- p onen tially to zero in the case of c haotic orbits, while it fluctuates aro und non–zero v alues for regular tra jectories of Hamiltonian systems and 2 N – dimensional sy mplectic maps with N > 1. In the case of 2–dimensional (2D) maps, the SALI tends to zero b oth fo r regular and chaotic or bits but with v ery differen t time rates, whic h allow s us again to distinguish b et w een the t wo cases [15]: In particular the SALI te nds to ze ro follo wing an exp onen tia l la w for c haot ic o rbits and decays to zero following a p o w er la w for regular orbits. The ba sic idea b ehind the success of the SALI metho d (whic h ess en tially distinguishes it from the computation of LCEs) is the in tr o duction of one 4 additional deviation v ector with respect to a reference orbit. Indeed, by con- sidering the relation b etw een t wo deviation v ectors (ins tead of one deviation v ector and the reference orbit), o ne is able to circum v en t the difficulty of the slo w con v ergence of Ly apuno v exp onen ts to non–zero (or ze ro) v alues a s t → ∞ . In order to compute the SALI, therefore, one follow s sim ultaneously the time ev olution of a reference orbit along with t w o deviation v ectors with initial conditions ~ w 1 (0), ~ w 2 (0). Since we are only in terested in the directions of these t wo v ectors w e normalize t hem, at ev ery time step, ke eping the ir norm equal to 1, setting ˆ w i ( t ) = ~ w i ( t ) k ~ w i ( t ) k , i = 1 , 2 (5) where k · k denotes the Euclidean norm and the hat ( ∧ ) ov er a v ector denotes that it is of unit magnitude. The SALI is then defined as: SALI( t ) = min {k ˆ w 1 ( t ) + ˆ w 2 ( t ) k , k ˆ w 1 ( t ) − ˆ w 2 ( t ) k} , (6) whence it is eviden t that SALI( t ) ∈ [0 , √ 2]. SALI = 0 indicates that the t w o deviation v ectors ha v e b ecome aligned in the same direction (and are equal or opp osite to each other); in o ther words, they are linearly dep enden t . Let us observ e, at this p oint, that seeking the minim um of the tw o p ositiv e quan tities in (6) (whic h are b ounded abov e b y 2) is essen tially equiv alen t to ev aluating the pro duct P ( t ) = k ˆ w 1 ( t ) + ˆ w 2 ( t ) k · k ˆ w 1 ( t ) − ˆ w 2 ( t ) k , (7) at ev ery v alue of t . Indeed, if the minimum of these tw o quan tities is zero (as in the case o f a c haotic reference orbit, see below), so will b e the v a lue of P ( t ). On the ot her hand, if it is not zero, P ( t ) will b e prop ortional to t he constant ab out whic h this minim um oscillates (as in the case of regular motion, see b elo w). This suggests that, instead of computing the SALI( t ) from (6), one migh t as w ell ev aluate the ‘exte rior’ or ‘w edge’ pro duct of the t w o deviation v ectors ˆ w 1 ∧ ˆ w 2 for whic h it holds k ˆ w 1 ∧ ˆ w 2 k = k ˆ w 1 − ˆ w 2 k · k ˆ w 1 + ˆ w 2 k 2 , (8) and whic h represen ts the ‘area’ of the parallelogra m formed b y the tw o devia- tion v ectors. F or the definition of the w edge pro duct see App endix A and for a pro of of (8) see App endix B. Indeed, the ‘w edge’ pro duct can pro vide m uc h 5 more useful information, as it can b e generalized to represen t the ‘volume ’ of a para llelepip ed formed b y the v ectors ˆ w 1 , ˆ w 2 , . . . , ˆ w k , 2 ≤ k ≤ 2 N , regarded as deviations fro m a n orbit of an N –degree of freedom Hamiltonian system, or a 2 N –dimensional symplectic map. It is the main purp ose of this paper to study precisely suc h a generalization and rev eal considerably more qualitativ e and quantitativ e information ab out the lo cal and global dynamics of these systems. Before we pro ceed to describe this generalization, ho w ev er, let us first summarize what w e kno w ab out the prop erties of the SALI for the case o f tw o deviation ve ctors ˆ w 1 , ˆ w 2 : (1) In the case of c hao tic orbits, the deviation ve ctors ˆ w 1 , ˆ w 2 ev en tually be- come aligned in the direction of the maximal Ly apunov exp onen t, and SALI( t ) falls exp onen tially to zero. An analytical study of SALI’s b eha v- ior for chaotic orbits w a s carried out in [17] where it w as sho wn that SALI( t ) ∝ e − ( σ 1 − σ 2 ) t (9) σ 1 , σ 2 b eing the t wo larg est LCEs. (2) In the case of regular motion, on the other hand, the orbit lies on a torus and the vec tors ˆ w 1 , ˆ w 2 ev en tually fa ll on its tangent space, following a t − 1 time ev olution, ha ving in general differen t direc tions. In this case, the SALI oscillates ab out v alues that are differen t from zero (for more details see [16]). This b eha vior is due to the fact t ha t for regular orbits the norm of a deviation ve ctor increases lin early in time along the flo w. Th us, our normalization pro cedure bring s ab out a dec rease of the magnitude of the co ordinates p erp endicular to the torus at a rate prop o r t ional to t − 1 and so ˆ w 1 , ˆ w 2 ev en tually fall on the tangent space of the torus. Note that in the case of 2D maps the torus is actually an inv ariant curv e and its tangen t space is 1–dimensional. So, in this case , the tw o unit deviation v ec- tors ev entually b ecome linearly dep enden t and SALI b ecomes zero follo wing a p ow er la w. This is, of course, diffe ren t than the exp onen tial deca y of SALI for chaotic orbits and th us SALI can distinguish easily b etw een the t w o cases ev en in 2D maps [15]. Th us, although the b eha vior of SALI in 2D maps is clearly understo o d, the fact remains that SALI do es no t alw a ys hav e the same b eha vior for regular orbits, as it may oscillate a bo ut a constant or deca y to zero by a p ow er law, dep ending on the dimensionalit y of the tangen t space of the reference orbit. It might, therefore, b e in teresting to ask whether t his index can be gene ralized, so that differen t p o w er la ws ma y b e found to c har- acterize regular motio n in higher dimensions. It is one of the principal aims of this pap er to show that such a gen eralization is p ossible. Let us make one final remark concerning the b ehav ior of SALI for c hao t ic orbits: Lo oking a t equation (9), one might wonder what w ould happ en in the case of a chaotic o r bit whose t w o largest Lyapuno v expo nen ts σ 1 and 6 σ 2 are equal or a lmost equal. Although this may not b e common in generic Hamiltonian systems, suc h cases can b e found in the literature. In one suc h example [39], v ery close to a part icular unstable p erio dic orbit o f a 1 5 degree of freedom Hamiltonian system, the t w o largest Lyapuno v exp onen ts are nearly equal σ 1 − σ 2 ≈ 0 . 000 2 . Ev en thoug h, in that example, SALI still tends to zero at the rat e indicated b y (9), it is eviden t that the c haotic nature of an orbit cannot be rev ealed v ery fast by the SALI method. It is, therefore, clear that a more detailed analysis of the lo cal dynamics is needed to further explore the prop erties of sp ecific orbit s, remedy the draw bac ks and impro ve up on the adv an t a ges of the SALI. F or example, if w e could define an index that dep ends on sever al Ly apuno v exp onen ts, this might accelerate considerably the iden tification of c haotic motion. 3 Definition of t he GALI Let us conside r an autonomous Hamiltonian system of N degrees of freedom ha ving a Hamiltonian function H ( q 1 , q 2 , . . . , q N , p 1 , p 2 , . . . , p N ) = h = constan t (10) where q i and p i , i = 1 , 2 , . . . , N are the generalized co or dina t es a nd conjugate momen ta resp ectiv ely . An or bit of t his system is defined b y a v ector ~ x ( t ) = ( q 1 ( t ) , q 2 ( t ) , . . . , q N ( t ) , p 1 ( t ) , p 2 ( t ) , . . . , p N ( t )), with x i = q i , x i + N = p i , i = 1 , 2 , . . . , N . The time ev olution o f this orbit is gov erned by Hamilton equations of motion d~ x dt = ~ V ( ~ x ) = ∂ H ∂ ~ p , − ∂ H ∂ ~ q ! , (11) while the time ev o lution of an initial deviation v ector ~ w (0) = ( dx 1 (0) , . . . , dx 2 N (0)) from the ~ x ( t ) solution of (11) ob eys the v ariational equations d ~ w dt = M ( ~ x ( t )) ~ w , (12) where M = ∂ ~ V /∂ ~ x is the Jacobian m atrix of ~ V . The SALI is a quan tit y suitable for c hec king whether or not t w o normalized deviation vectors ˆ w 1 , ˆ w 2 (ha ving norm 1), eve n tually b ecome linearly dep en- den t, b y falling in the same direction. The linear dep endence of the tw o v ectors is equiv alen t to the v anishing of the ‘a rea’ of the para llelogram ha ving as edges 7 the tw o vec tors. Generalizing this idea w e now follow the ev olution of k de- viation ve ctors ˆ w 1 , ˆ w 2 , . . . , ˆ w k , with 2 ≤ k ≤ 2 N , and determine whether these ev en tually b ecome linearly dep enden t, b y c heck ing if the ‘v olume’ of the parallelepip ed hav ing these ve ctors as edges go es to zero. This v olume will b e computed as the norm of t he w edge pro duct of these v ectors (see App endix A for a definition of the w edge pro duct). All normalized deviation ve ctors ˆ w i , i = 1 , 2 , . . . , k , b elong to the 2 N –dimensional tangen t space of the Hamilto nia n flo w. Using as a ba sis of this space the usual set o f orthono r ma l v ectors ˆ e 1 = (1 , 0 , 0 , . . . , 0) , ˆ e 2 = (0 , 1 , 0 , . . . , 0) , . . . , ˆ e 2 N = (0 , 0 , 0 , . . . , 1) (13) an y deviation v ector ˆ w i can b e written as ˆ w i = 2 N X j =1 w ij ˆ e j , i = 1 , 2 , . . . , k (14) where w ij are real n um b ers satisfying 2 N X j =1 w 2 ij = 1 . (15) Th us, e quation (A.12) of App endix A give s           ˆ w 1 ˆ w 2 . . . ˆ w k           =           w 11 w 12 · · · w 1 2 N w 21 w 22 · · · w 2 2 N . . . . . . . . . w k 1 w k 2 · · · w k 2 N           ·           ˆ e 1 ˆ e 2 . . . ˆ e 2 N           . (16) Using then equation (A.13) the w edge pro duct of these k deviation v ectors tak es the form ˆ w 1 ∧ ˆ w 2 ∧ · · · ∧ ˆ w k = X 1 ≤ i 1 σ 2 , a leading order estimate of the deviation vec tor’s Euclidean norm (for t large enough), is given 11 b y k ~ w i ( t ) k ≈ | c i 1 | e σ 1 t . (30) Consequen tly , the matrix C in (A.12) of co efficien ts of k normalized deviation v ectors ˆ w i ( t ) = ~ w i ( t ) / k ~ w i ( t ) k , i = 1 , 2 , . . . , k with 2 ≤ k ≤ 2 N , using as basis of the v ector s pace the set { ˆ u 1 , ˆ u 2 , . . . , ˆ u 2 N } b ecomes C ( t ) = [ c ij ] =           s 1 c 1 2 | c 1 1 | e − ( σ 1 − σ 2 ) t c 1 3 | c 1 1 | e − ( σ 1 − σ 3 ) t · · · c 1 2 N | c 1 1 | e − ( σ 1 − σ 2 N ) t s 2 c 2 2 | c 2 1 | e − ( σ 1 − σ 2 ) t c 2 3 | c 2 1 | e − ( σ 1 − σ 3 ) t · · · c 2 2 N | c 2 1 | e − ( σ 1 − σ 2 N ) t . . . . . . . . . . . . s k c k 2 | c k 1 | e − ( σ 1 − σ 2 ) t c k 3 | c k 1 | e − ( σ 1 − σ 3 ) t · · · c k 2 N | c k 1 | e − ( σ 1 − σ 2 N ) t           , (31) with s i = sign( c i 1 ) and i = 1 , 2 , . . . , k , j = 1 , 2 , . . . , 2 N and so w e ha ve  ˆ w 1 ˆ w 2 . . . ˆ w k  T = C ·  ˆ u 1 ˆ u 2 . . . ˆ u 2 N  T (32) with ( T ) denoting t he transp ose of a matrix. The wedge pro duct o f t he k normalized deviation v ectors is then computed as in equation (17) by : ˆ w 1 ( t ) ∧ ˆ w 2 ( t ) ∧ · · · ∧ ˆ w k ( t ) = X 1 ≤ i 1 σ 2 so that the norm of eac h deviation v ector can b e w ell a pproximated by equation (30). If the first m Ly apunov exp onen ts, with 1 < m < k , are equal, o r v ery close to eac h other, i.e. σ 1 ≃ σ 2 ≃ · · · ≃ σ m equation (43) b ecomes GALI k ( t ) ∝ e − [( σ 1 − σ m +1 )+( σ 1 − σ m +2 )+ ··· +( σ 1 − σ k )] t , (44) whic h still describ es a n exp onen tial deca y . How ev er, for k ≤ m < N the GALI k do es not tend t o zero as there exists at least one determinant of the matrix C that do es not v anish. In this case, of c ourse, one should increase the num b er of dev iation v ectors un til an exponential decreas e of G ALI k is a chiev ed. The extreme situation that all σ i = 0 corresp onds to motion on quasip erio dic tori, where all orbits are regular and is describ ed b elo w. 4.2 The ev aluation of GALI for r e gular o rbits As is we ll–kno wn, regular orbits o f an N degree of freedom Hamiltonian sys- tem (1 0) t ypically lie on N –dimensional tori. If suc h tori are f o und around a stable p erio dic orbit, they can b e accurately describ ed by N formal in tegrals of motion in in v olution, so that the system w ould app ear lo cally integrable. This means that w e could perfor m a lo cal transformation to action–angle v ariables, considering as actions J 1 , J 2 , . . . , J N the v alues of the N formal integrals, so 15 that Hamilton’s equations of motion, lo cally attain the f o rm ˙ J i = 0 ˙ θ i = ω i ( J 1 , J 2 , . . . , J N ) i = 1 , 2 , . . . , N . (45) These can be easily integrated to giv e J i ( t ) = J i 0 θ i ( t ) = θ i 0 + ω i ( J 10 , J 20 , . . . , J N 0 ) t i = 1 , 2 , . . . , N , (46) where J i 0 , θ i 0 , i = 1 , 2 , . . . , N are the initial c onditions. By denoting as ξ i , η i , i = 1 , 2 , . . . , N small dev iations of J i and θ i resp ectiv ely , the v ariational equations of system (45), describing the ev olution of a deviation v ector are ˙ ξ i = 0 ˙ η i = P N j =1 ω ij · ξ j i = 1 , 2 , . . . , N , (47) where ω ij = ∂ ω i ∂ J j    ~ J 0 i, j = 1 , 2 , . . . , N , (48) and ~ J 0 = ( J 10 , J 20 , . . . , J N 0 ) = constant, r epresen ts the N –dimensional ve ctor of the initial actions. The solution of these equations is: ξ i ( t ) = ξ i (0) η i ( t ) = η i (0) + h P N j =1 ω ij ξ j (0) i t i = 1 , 2 , . . . , N . (49) F rom equations (49) we see that an initial deviation vec tor ~ w (0) with co o r di- nates ξ i (0), i = 1 , 2 , . . . , N in the action v ariables and η i (0), i = 1 , 2 , . . . , N in the ang les, i. e. ~ w (0) = ( ξ 1 (0) , ξ 2 (0) , . . . , ξ N (0) , η 1 (0) , η 2 (0) , . . . , η N (0)), ev olv es in time in suc h a wa y that its action co ordinates remain constan t, while its angle co ordinates increase linearly in time. This b ehavior implies an almost linear increase of the norm of the deviation ve ctor. T o see this, let us assume that initially this v ector ~ w (0) has unit magnitude, i. e. N X i =1 ξ i (0) 2 + N X i =1 η i (0) 2 = 1 (50) 16 whence the t ime ev olution of its norm is giv en b y k ~ w ( t ) k =      1 +    N X i =1   N X j =1 ω ij ξ j (0)   2    t 2 +   2 N X i =1   η i (0) N X j =1 ω ij ξ j (0)     t      1 / 2 , (51) while the normalized deviation ve ctor ˆ w ( t ) b ecomes: ˆ w ( t ) = 1 k ~ w ( t ) k   ξ 1 (0) , . . . , ξ N (0) , η 1 (0) +   N X j =1 ω 1 j ξ j (0)   t, . . . , η N (0) +   N X j =1 ω N j ξ j (0)   t   . (52) Since the no r m (51 ) of a deviation v ector, for t large enough, increases pr a cti- cally linearly with t, the normalized deviation vec tor (52) tends to fall on t he tangen t space of the torus, since its co ordinates p erp endicular to the torus (i. e. the co ordinates along the action directions ) v anish following a t − 1 rate. This b ehav ior has already b een shown n umerically in the case of a n in tegrable Hamiltonian of 2 degrees of freedom in [16]. Using as a basis of the 2 N –dimensional tangen t space o f the Hamiltonian flo w the 2 N un it v ectors { ˆ v 1 , ˆ v 2 , . . . , ˆ v 2 N } , such tha t the first N of them, ˆ v 1 , ˆ v 2 , . . . , ˆ v N corresp ond to the N action v ariables and the remaining ones, ˆ v N +1 , ˆ v N +2 , . . . , ˆ v 2 N to the N conj ug a te angle v ariables, any unit deviation v ector ˆ w i , i = 1 , 2 , . . . can b e written as ˆ w i ( t ) = 1 k ~ w ( t ) k   N X j =1 ξ i j (0) ˆ v j + N X j =1 η i j (0) + N X k =1 ω k j ξ i j (0) t ! ˆ v N + j   . (53) W e p oint out tha t the quantities ω ij , i, j = 1 , 2 . . . , N , in (48), dep end only on the particular reference orbit and not on the c hoice of the deviation v ector. W e also note that the basis ˆ u i , i = 1 , 2 , . . . , 2 N dep ends on the sp ecific torus on whic h the motion o ccurs and is related to the usual v ector basis ˆ e i , i = 1 , 2 , . . . , 2 N of e quation (13 ) , through a non–singular transformat ion, similar to the one of equation (35), having the form:  ˆ v 1 ˆ v 2 . . . ˆ v 2 N  T = T o ·  ˆ e 1 ˆ e 2 . . . ˆ e 2 N  T (54) with T o denoting the transformation matrix. The basis { ˆ e 1 , ˆ e 2 , . . . , ˆ e 2 N } is used to describe the ev o lution of a deviation ve ctor with resp ect to the orig ina l q i , p i i = 1 , 2 , . . . , N co ordinates of the Hamiltonian system (10), while the basis { ˆ v 1 , ˆ v 2 , . . . , ˆ v 2 N } is used to describ e the same ev olut io n, if w e consider the original system in action–a ngle v ariables, so that the equations of motion are the ones giv en b y (45). 17 A t this p oint w e mak e the following remark: If the initial deviation v ector already lies in the tangent s p ac e of the torus it will remain constan t for all time! Indeed, taking for the initial conditions o f this ve ctor ξ i (0) = 0 , i = 1 , 2 , . . . , N , (55) with N X i =1 η i (0) 2 = 1 , (56) w e conc lude from equation (49) that ξ i ( t ) = 0 , η i ( t ) = η i (0) . (57) i.e. the deviation vec tor remains unc hanged having it s norm alw ays eq ual to 1. In pa rticular, suc h a v ector has the form ˆ w ( t ) = (0 , 0 , . . . , 0 , η 1 (0) , η 2 (0) , . . . , η N (0)) . (58) Let us now study t he case of k , general, linearly indep enden t unit deviation v ectors { ˆ w 1 , ˆ w 2 , . . . , ˆ w k } with 2 ≤ k ≤ 2 N . Using as vec tor basis the set { ˆ v 1 , ˆ v 2 , . . . , ˆ v 2 N } we g et:  ˆ w 1 ˆ w 2 . . . ˆ w k  T = D ·  ˆ v 1 ˆ v 2 . . . ˆ v 2 N  T (59) If no deviation v ector is initially lo cated in t he t a ngen t space of the torus, matrix D has the f orm D = [ d ij ] = 1 Q k m =1 k ~ w m ( t ) k · ·           ξ 1 1 (0) · · · ξ 1 N (0) η 1 1 (0) + P N m =1 ω 1 m ξ 1 m (0) t · · · η 1 N (0) + P N m =1 ω N m ξ 1 m (0) t ξ 2 1 (0) · · · ξ 2 N (0) η 2 1 (0) + P N m =1 ω 1 m ξ 2 m (0) t · · · η 2 N (0) + P N m =1 ω N m ξ 2 m (0) t . . . . . . . . . . . . ξ k 1 (0) · · · ξ k N (0) η k 1 (0) + P N m =1 ω 1 m ξ k m (0) t · · · η k N (0) + P N m =1 ω N m ξ k m (0) t           , (60) where i = 1 , 2 , . . . , k a nd j = 1 , 2 , . . . , 2 N . R ecalling our earlier discussion (see (50)-(53)), we not e that the norm of v ector ~ w i ( t ) for long times , gro ws linearly with t as M i ( t ) = k ~ w i ( t ) k ∝ t. (61) 18 Defining then b y ξ 0 ,k i and η k i the k × 1 column matrices ξ 0 ,k i =  ξ 1 i (0) ξ 2 i (0) . . . ξ k i (0)  T , η k i =  η 1 i (0) η 2 i (0) . . . η k i (0)  T , (62) the matrix D of (60) assumes the m uch simpler form D ( t ) = 1 Q k i =1 M i ( t ) ·  ξ 0 ,k 1 . . . ξ 0 ,k N η k 1 + P N i =1 ω 1 i ξ 0 ,k i t . . . η k N + P N i =1 ω N i ξ 0 ,k i t  = = 1 Q k i =1 M i ( t ) · D 0 ,k ( t ) . (63 ) Supp ose now that w e hav e m linearly indep enden t deviation v ectors, with m ≤ k and m ≤ N , initially lo cated in the ta ng en t space of the torus and let them b e t he first m deviation v ectors in equation (59). This implies, in the ab o v e notatio n, that the ξ i v ectors in (63) now hav e the form ξ m,k i =  0 0 . . . 0 ξ m +1 i (0) ξ m +2 i (0) . . . ξ k i (0)  T (64) where the first sup erscript, m , refers to t he n umber of first comp onen ts b eing equal to zero. Th us, the matrix D of (63) in this case reads D ( t ) = 1 Q k − m i =1 M m + i ( t ) ·  ξ m,k 1 . . . ξ m,k N η k 1 + P N i =1 ω 1 i ξ m,k i t . . . η k N + P N i =1 ω N i ξ m,k i t  = = 1 Q k − m i =1 M m + i ( t ) · D m,k ( t ) , (65) where the first superscript of D m,k ( t ) in equations (63) and (65) has an analo- gous meaning as in the ξ m,k i . W e note that f or k = m we define Q 0 i =1 M m + i ( t ) = 1. Using again equation (A.13), we write the w edge pro duct of the k normalized deviation v ectors as ˆ w 1 ( t ) ∧ ˆ w 2 ( t ) ∧ · · · ∧ ˆ w k ( t ) = X 1 ≤ i 1 0 tangent initial deviation ve ctors Finally , let us consider the b eha vior of GALI k for the sp ecial case where m initial dev iation v ectors, with m ≤ k and m ≤ N , are lo cated in the tangent space o f the to rus. In this case, matrix D , whose elemen ts app ear in the definition of S ′ k , has the form giv en b y (65). Th us, all determinants app earing in the definition of S ′ k ha v e as a common factor the quan tit y 1 / Q k − m i =1 M m + i ( t ), whic h de creases to zero follow ing a p o w er law 1 Q k − m i =1 M m + i ( t ) ∝ 1 t k − m . (77) Pro ceeding in exactly the same manner as in the m = 0 case a b ov e, w e deduce that, in the case of 2 ≤ k ≤ N the fastest gro wing k × k determinan ts resulting from the matrix D m,k are o f the for m:     η k i 1 η k i 2 · · · η k i m ω i m +1 n 1 ξ 0 ,k n 1 t ω i m +2 n 2 ξ 0 ,k n 2 t · · · ω i k n k − m ξ 0 ,k n k − m t     ∝ t k − m , (78) with i l ∈ { 1 , 2 , . . . , N } , l = 1 , 2 , . . . , k with i l 6 = i j for l 6 = j , and n l ∈ { 1 , 2 , . . . , N } , l = 1 , 2 , . . . , k − m with n l 6 = n j , for l 6 = j . Hence, w e conclude that the b ehavior of S ′ k , and consequen tly of GALI k is defined b y the behavior of determinan ts havin g the form of (78) which, when com bined with (77) implies that GALI k ( t ) ≈ constan t f or 2 ≤ k ≤ N . (79) The case of N < k ≤ 2 N deviation v ectors, how ev er, with m > 0 initially tangen t v ectors, yields a considerably differen t result. F ollow ing en tirely anal- ogous arguments as in the m = 0 case, w e find that, if m < k − N , S ′ k and GALI k ev olve prop ortionally to t 2 N − k /t k − m = 1 /t 2( k − N ) − m . On the other hand, if m ≥ k − N , one can show that the fastest growing determinan t is prop ortional to t N − m . In this case, S ′ k and GALI k ev olve in time fo llo wing a quite differen t pow er law : t N − m /t k − m = 1 /t k − N . Summarizing the results of t his section, w e see that GALI k for r egula r motio n remains essen tially constan t when k ≤ N , while it tends to zero fo r k > N 23 follo wing a p ow er la w whic h dep ends on the n um b er m ( m ≤ N and m ≤ k ) of deviation v ectors initially ta ngen t to the torus. In conclusion, w e ha v e sho wn that: GALI k ( t ) ∝              constan t if 2 ≤ k ≤ N 1 t 2( k − N ) − m if N < k ≤ 2 N and 0 ≤ m < k − N 1 t k − N if N < k ≤ 2 N and m ≥ k − N . (80) 5 Numerical v erification and applications In order to a pply the GALI metho d to Hamiltonian systems a nd verify t he theoretically predicted b ehavior of the previous sections, w e shall use tw o simple examples with 2 (2D ) a nd 3 (3 D) degrees of f reedom: the well–kn o wn 2D H ´ enon–Heiles system [48], describ ed by the Hamiltonian H 2 = 1 2 ( p 2 x + p 2 y ) + 1 2 ( x 2 + y 2 ) + x 2 y − 1 3 y 3 , (81) and the 3D Hamiltonian system: H 3 = 3 X i =1 ω i 2 ( q 2 i + p 2 i ) + q 2 1 q 2 + q 2 1 q 3 , (82) studied in [49,5]. W e k eep the parameters of t he tw o systems fixed at the energies H 2 = 0 . 125 and H 3 = 0 . 09, with ω 1 = 1, ω 2 = √ 2 and ω 3 = √ 3. In order to illustrate the b ehav ior of GALI k , for differen t v alues of k , w e shall consider some represen tativ e cases of c ha otic and regular orbits of the t w o systems . Additionally , w e shall study the higher–dimensional example of a 15D Hamil- tonian, describing a c hain of 15 particles with quadratic and quartic nearest neigh b or inte raction, kno wn as the famous F ermi–P asta–Ulam (FPU) mo del [50] H 15 = 1 2 15 X i =1 p 2 i + 15 X i =1  1 2 ( q i +1 − q i ) 2 + 1 4 β ( q i +1 − q i ) 4  (83) where q i is the displacemen t of the i th particle from its equilibrium p oin t and p i is the conjugate momen tum. This is a mo del w e hav e recen tly analyze d in [39] a nd w e s hall use here the same v alues of t he energy H 15 = 26 . 68777 and β = 1 . 0 4 as in that study . 24 1 2 3 4 5 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 L 1 ( t ) L 2 ( t ) L 1 ( t ) - L 2 ( t ) ( a ) L i ( t ) l o g t 0 2 0 0 4 0 0 6 0 0 8 0 0 - 1 6 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 | | w ( t ) | | - 1 ( b ) sl o p e = - 1 / l n ( 1 0 ) sl o p e = - 2 1 / l n ( 1 0 ) sl o p e = - 4 1 / l n ( 1 0 ) G A L I 4 G A L I 3 G A L I 2 l o g ( G A L I s ) t Fig. 1. (a) The evo lution of L 1 ( t ) (solid cur ve), L 2 ( t ) (dashed cur v e) and L 1 ( t ) − L 2 ( t ) (dotted curve) for a chao tic orb it with initial cond itions x = 0, y = − 0 . 25, p x = 0 . 42, p y = 0 of the 2D system (81). (b) The evolutio n of GALI 2 , GALI 3 and GALI 4 of the same orb it. Th e plotted lines corresp ond to f unctions pr op ortional to e − σ 1 t (solid line), e − 2 σ 1 t (dashed line) and e − 4 σ 1 t (dotted line) for σ 1 = 0 . 047. Note that the t –axis is lin ear. The ev olution of the norm of th e deviation v ector ~ w ( t ) (with k ~ w (0) k = 1) used for the computation of L 1 ( t ), is also plotted in (b) (gray curv e). 5.1 A 2 D Hamiltonian system Let us consider first a c ha o tic orbit of the 2D Hamiltonian (81), with initial conditions x = 0, y = − 0 . 25, p x = 0 . 42, p y = 0. In figure 1(a) w e see the time ev olution of L 1 ( t ) of this orbit. The computation is car r ied out un til L 1 ( t ) stops ha ving large fluctuatio ns a nd approac hes a p ositiv e v alue (indicating the chaotic nature of the orbit), whic h could b e considered a s a go o d appro x- imation of the maximal LCE, σ 1 . Actually , for t ≈ 10 5 , we find σ 1 ≈ 0 . 047 . W e recall that 2D Hamiltonian systems hav e only one p ositiv e LCE σ 1 , since the second largest is σ 2 = 0. It also holds that σ 3 = − σ 2 and σ 4 = − σ 1 and th us fo rm ula (43), whic h describes the time ev olution of GALI k for c hao t ic orbits, giv es GALI 2 ( t ) ∝ e − σ 1 t , GALI 3 ( t ) ∝ e − 2 σ 1 t , GALI 4 ( t ) ∝ e − 4 σ 1 t . (84) In figure 1(b) w e plot GALI k , k = 2 , 3 , 4 for the same c haotic or bit as a f unc- tion of time t . W e plot t in linear scale so that, if ( 84) is v alid, the slop e of GALI 2 , GALI 3 and GALI 4 should approximately b e − σ 1 / ln 10, − 2 σ 1 / ln 10 25 and − 4 σ 1 / ln 10 resp ectiv ely . F rom figure 1 (b) we see that lines ha ving pre- cisely these slop es, for σ 1 = 0 . 0 4 7, approx imate quite accurately t he computed v alues of the GALIs. The biggest deviation b et w een the theoretical curv e and n umerical data a pp ears in the case of G ALI 4 where the theoretical prediction underestimates the deca ying r ate of the index, but ev en in this case the differ- ence do es not app ear to o significan t. Note, ho w ev er, t he imp orta n t difference in the times it takes to decide ab out the c haotic nature of the or bit : W aiting for the maximal LCE to con v erge in figure 1(a), one needs more than 1 0 4 time units, while, as w e see in figure 1(b), the GALI k ’s pro vide this information in less than 400 time units! W e also note that , plotting in this example the ev olution of the quan tity k ~ w ( t ) k − 1 (with k ~ w (0) k = 1), whic h is used to determine L 1 ( t ) in (1 ) and is practically iden t ified with t he F ast Ly apunov Indicator (FLI), w e o btain in fig ure 1(b) a graph similar to that of GALI 2 ( t ). This is not surprising, as both k ~ w ( t ) k − 1 and GALI 2 ( t ) tend exp onen tia lly to zero followin g a deca y prop ortional to e − σ 1 t (see equations (30) and (84) ) . F rom t he results of figure 1(b) w e see that the different plotted quan tit ies reac h the limit of computer’s accuracy (10 − 16 ) a t differen t times and in par t icular GALI 2 at t ≈ 800, GALI 3 at t ≈ 400 , GALI 4 at t ≈ 150 and k ~ w ( t ) k − 1 at t ≈ 720 . The CPU time nee ded for computing the ev olutio n of the indices up to these times were: 0.220 sec for k ~ w ( t ) k − 1 , 0 .2 95 sec for GALI 2 , 0 .1 65 sec for GALI 3 and 0.070 sec for GALI 4 resp ectiv ely . Th us, in this case also, it is clear that the higher order GALI k (with k > 2) can iden tify the chaotic nature of an orbit faster tha n the metho ds of the maximal LCE, the FLI or the SALI (equiv alen t to GALI 2 , see b elo w). It is interes ting to remark at this p oin t (as mentioned in section 4.1), that the accuracy of the exp onen tial laws (84) is due to the fact that the lo cal Ly apunov expo nents ce ase to fluctuate significan tly ab out their limit v a lues, after a relatively short time interv al. T o see this, w e ha v e plotted in fig ur e 1(a), t he t w o nonnegative lo cal Ly a punov exp onen ts L 1 ( t ), L 2 ( t ), as w ell as their difference. Note that L 1 ( t ) − L 2 ( t ) b egins to b e w ell approximated b y σ 1 − σ 2 = σ 1 already for times t o f order 10 2 units. A similar b eha vior o f suc h L 1 ( t ) − L i ( t ), i = 2 , 3 , . . . , 2 N differences are observ ed fo r the other Hamiltonians w e studie d in this pap er having 3 or more degrees of freedom. As explained in detail in App endix B, GALI 2 practically coincides with SALI in the case o f c haotic o r bit s. This b ecomes eviden t from figure 2 where we plot the absolute difference b et w een GALI 2 and SALI o f the c hao t ic orbit of figure 1 as a function of time t . The tw o indices practically coincide a f ter ab out t ≈ 300 units, since their difference is at the limit of computer’s accuracy (10 − 16 ), although their actual v alues are of order 10 − 5 (see figure 1(b)). Let us now study the b eha vior o f GALI k for a regular orbit of the 2D Hamil- 26 0 1 0 0 2 0 0 3 0 0 - 1 6 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 l o g | S A L I - G A L I 2 | t Fig. 2. The absolute difference b etw een GALI 2 and SALI of the c h aotic orbit of figure 1 as a function of time t . tonian (81). F rom (80) it follows that in the case of a Hamiltonian system with N = 2 degrees of f reedom GALI 2 will alwa ys remain differen t fr o m zero, while GALI 3 and GALI 4 should decay to zero following a p ow er law, whose exp o nen t depends on the num b er m of deviation v ectors t ha t are initially tan- gen t t o the torus on whic h the orbit lies. Now, for a regular orbit of the 2D Hamiltonian (81) a nd a random choice of initial deviation ve ctors, w e exp ect the GALI indices to b eha v e as GALI 2 ( t ) ∝ constan t , GALI 3 ( t ) ∝ 1 t 2 , GALI 4 ( t ) ∝ 1 t 4 . (85) A simple qualitativ e w ay of studying the dynamics of a Hamilto nia n system is b y plotting the succe ssiv e in tersections o f the orbits with a P oincar´ e Surface of Section (PSS) [45]. In 2D Hamiltonians, the PSS is a t w o dimensional plane and the p oin ts of a regular orbit (whic h lie on a torus) fall on a smo oth closed curv e. This prop erty a llo ws us to choose initial deviation v ectors tangen t to a torus in the case of system (81). In particular, w e consider the regular orbit with initial conditions x = 0 , y = 0 , p x = 0 . 5, p y = 0. In fig ure 3 , we plot the in tersection p oin ts o f this orbit with the PSS defined b y x = 0 (panel (a)) and y = 0 (pa nel (b)). F r o m the morphology of the t w o closed c urv es of figure 3, it is e asily se en t ha t deviation ve ctors ˆ e 1 = (1 , 0 , 0 , 0) and ˆ e 4 = (0 , 0 , 0 , 1) are tangen t to t he to rus. In Figure 4, we plot the time ev olution of SALI, GALI 2 , GALI 3 and GALI 4 for the regular orbit of figure 3, for v arious c hoices of initial dev iation v ectors. 27 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 ( a ) p y y - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 . 0 0 . 1 - 0 . 3 - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 ( b ) p x x Fig. 3. The Poinca r ´ e Surface of S ectio n (PS S ) d efined by (a) x = 0 and (b) y = 0 of t he regular orbit with initial conditions x = 0, y = 0, p x = 0 . 5, p y = 0 for the H ´ enon–Heiles system (81). In figure 4(a ) the initial deviation v ectors are r a ndomly c hosen so that none of them is tang ent to the torus. In this case SALI and GALI 2 fluctuate around non–zero v alues, while GALI 3 and GALI 4 tend to zero followin g the theo- retically predicted p ow er la ws, see (85). In figure 4(b) w e presen t results for the indices when w e ha ve m = 1 initial deviation v ector tangen t to the torus (in particular v ector ˆ e 1 ). In this c ase the indices ev olv e as predicted b y (80), i. e. SALI and GALI 2 remain practically constant, while G ALI 3 ∝ 1 /t and GALI 4 ∝ 1 /t 3 . Fina lly , in figure 4(c) w e hav e plotted our results using m = 2 initial deviation v ectors tangent to the torus (v ectors ˆ e 1 and ˆ e 4 ). Again the predictions of (80) are seen to b e v alid since GALI 3 ∝ 1 /t and G ALI 4 ∝ 1 /t 2 . The differen t behavior of SALI (or GALI 2 ) fo r regular and c haotic orbits has already b een successfully used for discriminating b et w een regions of order and c haos in v arious dynamical systems [1 7 ,36,40,41,42,43,44]. F or example, by in tegra ting orbits whose initial conditions lie on a grid, and b y a ttributing to each grid p oin t a color according to the v alue of SALI at the end of a giv en in tegra t io n time, one can obtain clear a nd informative pictures of the dynamics in the full phase space of sev eral Hamiltonian systems of phys ical significance [17,36,43]. Figures 1(b) and 4 clearly illustrat e that GALI 3 and GALI 4 tend to ze ro b o th for regular and c haotic orbits, but with v ery differen t time rates. W e may use this differenc e to distinguis h b et we en c haot ic and regular motio n following a differen t approach than SALI or GALI 2 . Let us illustrate this b y considering 28 1 2 3 4 5 - 1 6 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 sl o p e = - 2 sl o p e = - 4 ( a ) S A L I G A L I 4 G A L I 3 G A L I 2 l o g ( G A L I s ) l o g t 1 2 3 4 5 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 sl o p e = - 1 sl o p e = - 3 ( b ) S A L I G A L I 4 G A L I 3 G A L I 2 l o g ( G A L I s ) l o g t 1 2 3 4 5 - 1 0 - 8 - 6 - 4 - 2 0 sl o p e = - 1 sl o p e = - 2 ( c ) S A L I G A L I 4 G A L I 3 G A L I 2 l o g ( G A L I s ) l o g t Fig. 4. Time ev olution of S ALI (g ra y curves), GALI 2 , GALI 3 and GALI 4 for the regular orb it of figur e 3 in log–log scale for different v alues of th e n um b er m of deviation vec tors initially tangent to the torus: (a) m = 0, (b) m = 1 and (c) m = 2. W e note that in panel (a) the cur v es of S ALI and GALI 2 are very close to eac h other and thus cann ot b e distinguished. In ev ery panel, dashed lines corresp ond ing to particular p o w er la ws are also plotted. the computation of GALI 4 : F rom (84 ) and (85), we expect GALI 4 ∝ e − 4 σ 1 t for c haotic orbits and G ALI 4 ∝ 1 /t 4 for regular ones. T hese time rates imply that, in general, the time needed for the index to b ecome zero is m uc h larger for r egula r orbits. Th us, instead of simply registering the v alue of the index at the end of a giv en time interv al (as w e do with SALI or GALI 2 ), let us record the time, t th , needed for G ALI 4 to reac h a v ery small threshold, e. g. 10 − 12 , and color eac h grid p oin t according to the v alue of t th . 29 Fig. 5. Regions of d ifferen t v alues of the time t th needed for GALI 4 to b ecome less than 10 − 12 on the PSS defined by x = 0 of the 2D H ´ enon–Heiles Hamiltonian (81). The outcome of this pro cedure for t he 2D H´ enon–Heiles system (81) is pre- sen ted in figure 5. Each orbit is inte grated up to t = 500 units and if the v alue of GALI 4 , at the end of the integration is larger than 10 − 12 the corresponding grid p oin t is colored b y the ligh t gray color used for t th ≥ 400. Th us w e can clearly distinguish in this figure among v ar ious ‘degrees’ o f c haotic b eha vior in regions colored blac k or dark gra y – corresp o nding to small v a lues of t th – and regions of regular motion colored ligh t gra y , corresp onding to large v alues of t th . At the b order b et w een them w e find p oints ha ving intermediate v alues of t th whic h b elong to t he so–called ‘stic ky’ c ha otic regions. Thu s, this approac h yields a v ery detailed chart of the dynamics, where even tin y islands of sta- bilit y can b e iden tified inside the large chaotic sea. W e note that f o r ev ery initial condition the same set of initial deviation ve ctors was used, ensuring the same initial v alue of GALI 4 for all orbits and j ustifying the dynamical in terpretatio n of the color scale of figure 5. 5.2 A 3 D Hamiltonian system Let us now study the b eha vior o f the GALIs in the case of the 3D Hamiltonian (82). F ollow ing [49 ,5 ] the initial conditions of the o rbits of this system are defined b y a ssigning a r bitr ary v alues to the p ositions q 1 , q 2 , q 3 , as w ell as the so–called ‘harmonic energies’ E 1 , E 2 , E 3 related to the momen ta thro ugh p i = s 2 E i ω i , i = 1 , 2 , 3 . (86) 30 1 2 3 4 5 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 L 1 ( t ) L 2 ( t ) ( a ) L 1 ( t ) , L 2 ( t ) l o g t 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 - 1 6 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 ( b ) s l o p e = - ( 1 - 2 ) / l n ( 1 0 ) s l o p e = - ( 2 1 - 2 ) / l n ( 1 0 ) s l o p e = - ( 3 1 - 2 ) / l n ( 1 0 ) s l o p e = - 4 1 / l n ( 1 0 ) s l o p e = - 6 1 / l n ( 1 0 ) G A L I 6 G A L I 5 G A L I 3 G A L I 4 G A L I 2 l o g ( G A L I s ) t Fig. 6. (a) The ev olution of L 1 ( t ), L 2 ( t ) for the chaoti c orbit with in itial condition q 1 = q 2 = q 3 = 0, E 1 = E 2 = E 3 = 0 . 03 of the 3D system (82). (b) T he ev olution of GALI k with k = 2 , . . . , 6 of the same orbit. The plotted lines corresp ond to functions prop ortional to e − ( σ 1 − σ 2 ) t , e − (2 σ 1 − σ 2 ) t , e − (3 σ 1 − σ 2 ) t , e − 4 σ 1 t and e − 6 σ 1 t for σ 1 = 0 . 03, σ 2 = 0 . 008. Note that the t –axis is linear. Chaotic or bits o f 3D Ha milto nian systems generally ha v e tw o p ositiv e Ly a- puno v exp onen ts, σ 1 and σ 2 , while σ 3 = 0. So, for appro ximating the behavior of GALIs according to (43), b oth σ 1 and σ 2 are needed. In particular, (43) giv es GALI 2 ( t ) ∝ e − ( σ 1 − σ 2 ) t , GALI 3 ( t ) ∝ e − (2 σ 1 − σ 2 ) t , GALI 4 ( t ) ∝ e − (3 σ 1 − σ 2 ) t , GALI 5 ( t ) ∝ e − 4 σ 1 t , GALI 6 ( t ) ∝ e − 6 σ 1 t . (87) Let us consider the chaotic or bit with initia l conditio ns q 1 = q 2 = q 3 = 0, E 1 = E 2 = E 3 = 0 . 03 of t he 3D system ( 82). W e compute σ 1 , σ 2 for this o r bit as the long time limits of the Lyapuno v exponent quantities L 1 ( t ), L 2 ( t ), applying the tech nique prop osed by Benettin e t al. [5]. The results are presen ted in figure 6(a). The computatio n is carried out un t il L 1 ( t ) and L 2 ( t ) stop ha ving la r g e fluctuatio ns and approa ch some p ositiv e v alues (since the orbit is c ha o tic), whic h could b e considered a s go o d appro ximations of their limits σ 1 , σ 2 . Actually for t ≈ 10 5 w e ha ve σ 1 ≈ 0 . 0 3 and σ 2 ≈ 0 . 008. Using these v alues as go o d approxim ations of σ 1 , σ 2 w e see in figure 6(b) that the slop es of all G ALIs are well repro duced b y (87). Next, w e consider the case o f regular orbits in o ur 3D Ha miltonian system. In the general case, where no initial deviation v ector is tangent to t he torus 31 0 1 2 3 4 5 6 - 5 - 4 - 3 - 2 - 1 0 ( a ) l o g ( L 1 ( t ) ) l o g t 1 2 3 4 5 - 1 6 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 ( b ) sl o p e = - 2 sl o p e = - 4 sl o p e = - 6 G A L I 6 G A L I 5 G A L I 4 G A L I 3 G A L I 2 l o g ( G A L I s ) l o g t Fig. 7. (a) The ev olution of L 1 ( t ) for th e regular orbit with initial condition q 1 = q 2 = q 3 = 0, E 1 = 0 . 005, E 2 = 0 . 085, E 3 = 0 of the 3D system (82). (b) The ev olution of GALI k with k = 2 , . . . , 6 of the same orbit. The p lotte d lines corresp ond to functions prop ortional to 1 t 2 , 1 t 4 and 1 t 6 . where the regular orbit lies, the GALIs should b eha v e as: GALI 2 ( t ) ∝ constan t , GALI 3 ( t ) ∝ constan t , GALI 4 ( t ) ∝ 1 t 2 , GALI 5 ( t ) ∝ 1 t 4 , GALI 6 ( t ) ∝ 1 t 6 . (88) according to (80). In order to v erify expression (8 8) w e shall follow a sp ecific regular orbit of t he 3D system (82) w ith initial conditions q 1 = q 2 = q 3 = 0, E 1 = 0 . 005, E 2 = 0 . 08 5, E 3 = 0. The regular nature of the orbit is rev ealed b y the slo w con v ergence of its L 1 ( t ) to zero, implying that σ 1 = 0, se e figure 7(a). In figure 7( b) , w e plot the v alues of all GALIs of t his or bit with resp ect to time t . F rom these results we se e that the differen t b ehav iors of GALIs are v ery w ell approx imated b y form ula (88). F rom the results o f figures 6 and 7, therefore, w e conclude that in the case of 3D Hamiltonian sy stems no t only G ALI 2 , but also GALI 3 has differen t b ehav - ior for r egular and c ha otic o rbits. In particular GALI 3 tends exp onen tially to zero for c haotic orbits (ev en faster than GALI 2 or SALI), while it fluc tuates around non–zero v alues for regular orbits. Hence, the natural question arises whether GALI 3 can b e used instead of SALI for t he faster detection of c haotic and regular motion in 3D Hamilto nians a nd, by extension, whether GALI k , with k > 3, should b e preferred for systems with N > 3. The ob vious compu- tational dra wback, of course, is that the ev aluation of G ALI k requires that w e n umerically follo w the ev olution of more than 2 deviation vec tors. 32 First of all, let us p oin t out that the computation of SALI, applying (6), is sligh t ly faster tha n GALI 2 , for whic h one needs to ev aluate sev eral 2 × 2 determinan ts. F or example, for orbits of t he 3D Hamiltonian (82) the CPU time needed for the computation o f SALI fo r a fixed time inte rv al t , w as ab out 97% of the CPU time needed for the computation of G ALI 2 for t he same t ime in terv al. Although this difference in not significant, w e prefer to compute SALI instead of GALI 2 and compare its efficiency with the computation o f GALI 3 . It is obvious that the computatio n o f GALI 3 for a giv en time in terv al t needs more CPU time than SALI, since w e follow the ev o lution of three deviation v ectors instead of t wo. This is particularly true for regular orbits as the index do es not become zero and its ev olution has to be follo wed for the whole pre- scrib ed time in terv al. In the case of chaotic orbits, ho w ev er, the situation is differen t. Let us consider, for example, the c haotic orbit of figure 6. The usual tec hnique to c haracterize an orbit as ch aotic is to che c k, after some time inter- v al, if its SALI has b ecome less than a v ery small threshold v alue, e . g. 10 − 8 . F or this particular orbit, this threshold v alue w as reach ed for t ≈ 760. Adopt- ing the same threshold to c haracterize an orbit a s c hao t ic, w e find that GALI 3 b ecomes less than 1 0 − 8 after t ≈ 335, requiring only as muc h as 65 % o f the CPU time needed for SALI to reac h the same threshold! So, using GALI 3 instead o f SALI, w e g ain considerably in CPU time fo r c haotic orbits, while we lose fo r regular orbits. Thus , the efficiency of using GALI 3 for discriminating b et wee n chaos and order in a 3D system dep ends on the p ercen tage of phase space o ccupied b y c haotic orbits (if all or bits a r e r egu- lar GALI 3 requires more CPU time tha n SALI). More crucially , how ev er, it dep ends on the c hoice of the fina l time, up to whic h eac h orbit is in tegrated. As an example, let us in tegra te, up to t = 1000 time units, all o rbits whose initial conditions lie on a dense grid in the subspace q 3 = p 3 = 0, p 2 ≥ 0 of a 4–dimensional PSS, with q 1 = 0 of the 3D system (82) , a ttributing to eac h gr id p oin t a color according to the v alue of GALI 3 at the end of the in tegration. If G ALI 3 of an orbit b ecomes less than 10 − 8 for t < 1000 the ev olutio n of the orbit is stopp ed, its G ALI 3 v alue is registered and the or bit is c haracterized as c hao t ic. The o ut come of this exp erimen t is pr esen ted in figure 8. W e find that 77% of the orbits of figure 8 are c haracterized a s c haotic, hav ing GALI 3 < 10 − 8 . In order to ha v e the same p ercen tage of orbits identified as c haot ic using SALI (i. e. ha ving SALI < 10 − 8 ) the same exp erimen t has to b e carried out for t = 2 000 units, requiring 53% more CPU time. D ue to the high p ercen tage of c haotic o rbits, in this case, ev en when the SALI is computed for t = 1000 the corresponding CPU time is 1 2% higher than the one needed fo r the computation of figure 8, while only 55% of the orbits are identified a s c haotic. Th us it b ecomes eviden t that a carefully designed application of GALI 3 – or GALI k for that matter – can significan tly diminish the computational time needed for a reliable discrimination b et wee n regions 33 Fig. 8. Regions of different v alues of the GALI 3 on th e su bspace q 3 = p 3 = 0, p 2 ≥ 0 of the 4–dimensional PSS q 1 = 0 of the 3D sys tem (82) at t = 1000. of order and c haos in Hamiltonian systems with N > 2 degrees o f freedom. 5.3 A m ulti–dimensional Hamiltonian system Let us finally turn to a m uch higher–dimensional Hamiltonian system ha v- ing 15 degrees of freedom, i. e. t he one show n in (83). With fixed b oundary conditions q 0 ( t ) = q 16 ( t ) = 0 , ∀ t, (89) it is know n that there exists, for all energies , H 15 = E , a simple p erio dic o rbit, satisfying [51,39] q 2 i ( t ) = 0 , q 2 i − 1 ( t ) = − q 2 i +1 ( t ) = q ( t ) , i = 1 , 2 , . . . , 7 , (90) where q ( t ) = q ( t + T ) ob eys a simple nonlinear equation admitting Jacobi elliptic function solutions. F or the parameter v alues H 15 = 26 . 68777 and β = 1 . 04 used in a n earlier study [39], w e kno w that this orbit is unstable and has a sizable c haotic region around it. As initial conditions for (90) w e tak e q (0) = 1 . 322 and p i (0) = 0 , i = 1 , 2 , . . . , 15 . (91) First, w e consider a c haot ic orbit which is lo cated close to this p erio dic so- lution, b y taking as initial conditions q 1 (0) = q (0), q 3 (0) = q 7 (0) = q 11 (0) = 34 1 2 3 4 5 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5 0 . 4 0 ( a ) L 1 ( t ) L 2 ( t ) L 3 ( t ) L 4 ( t ) L i ( t ) l o g t 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 - 1 6 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 sl o p e = - ( 1 - 2 ) / l n ( 1 0 ) sl o p e = - ( 2 1 - 2 - 3 ) / l n ( 1 0 ) sl o p e = - ( 3 1 - 2 - 3 - 4 ) / l n ( 1 0 ) ( b ) G A L I 2 G A L I 4 G A L I 3 l o g ( G A L I s ) t Fig. 9. (a) The evol ution of L 1 ( t ), L 2 ( t ), L 3 ( t ) and L 4 ( t ) for a c haotic orbit of the 15D system (83). (b) The ev olution of GALI 2 , GALI 3 and GALI 4 for the same orbit. The p lotte d lines corresp ond to functions p rop ortional to e − ( σ 1 − σ 2 ) t , e − (2 σ 1 − σ 2 − σ 3 ) t and e − (3 σ 1 − σ 2 − σ 3 − σ 4 ) t , for σ 1 = 0 . 132 , σ 2 = 0 . 117, σ 3 = 0 . 104, σ 4 = 0 . 093. Note that the t –axis is linear. − q (0) + 10 − 7 , q 5 (0) = q 9 (0) = q 15 (0) = q (0) − 10 − 7 , q 2 i = 0 for i = 1 , 2 , . . . , 7 and p i (0) = 0 for i = 1 , 2 , . . . , 14 , p 15 (0) = 0 . 00323. The chaotic nature of this orbit is reve aled b y the fact that it s maximal LCE is p ositive (see figure 9(a)). In fact, from the results of fig ure 9(a) we deduce reliable estimates of the system’s four larg est Ly apuno v expo nen ts: σ 1 ≈ 0 . 132 , σ 2 ≈ 0 . 117, σ 3 ≈ 0 . 104 and σ 4 ≈ 0 . 093 . Th us, w e hav e a case where sev eral L CEs hav e p ositiv e v alues, the la r gest tw o o f them b eing very close to eac h other. The b ehav ior of the GALIs is again quite accurately appro ximated b y the theoretically predicted exp o nen tial laws (43). This b ecomes eviden t b y the res ults presen ted in figure 9(b), where w e plot the time ev olution of G ALI 2 , GALI 3 and GALI 4 as w ell as the exponential la ws that theoretically desc rib e the ev olution of these indices . In this case, GALI 2 do es deca y to zero relatively slo wly since σ 1 and σ 2 ha v e similar v alues and hence, using GALI 3 , GALI 4 or a G ALI of higher order, one can determine the c hao tic natur e o f the orbit muc h faster. It is w or t h mentioning that (43) describ es m uch mo r e accurately the ev olution of GALI k when the orbit we wish to study is v ery close to the unstable p erio dic solution (90) itself. This is due to the fact that in t hat case, the LCEs are directly related to the eigen v alues of the mono drom y matrix asso ciated with the v ariational equations of t his unstable p erio dic orbit, see equation (25). In fact, for our c hoice of parameters, this matrix has t w o equal pairs of real eigen v alues with magnitude greater than one, while all other eigenv alues lie on the unit circle in the complex plane. As a c onsequence , the o r bit has t w o nearly iden tical p ositive Ly apuno v exp onen ts (as well as their t w o negative 35 1 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 L 1 ( t ) L 2 ( t ) L 3 ( t ) L 4 ( t ) ( a ) L i ( t ) l o g t 0 5 0 1 0 0 1 5 0 2 0 0 - 1 4 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 G A L I 2 G A L I 3 G A L I 4 sl o p e = - ( 1 - 2 ) / l n ( 1 0 ) sl o p e = - ( 2 1 - 2 - 3 ) / l n ( 1 0 ) sl o p e = - ( 3 1 - 2 - 3 - 4 ) / l n ( 1 0 ) ( b ) l o g ( G A L I s ) t Fig. 10. (a) T he evo lution of L 1 ( t ), L 2 ( t ), L 3 ( t ) and L 4 ( t ) for an orbit wh ic h is very close to the un stable p erio dic orb it (91) of th e 15D system (83). (b) T he evol u- tion of GALI 2 , GALI 3 and GALI 4 of the same orbit. The plotted lin es corresp ond to fu nctions prop ortional to e − ( σ 1 − σ 2 ) t , e − (2 σ 1 − σ 2 − σ 3 ) t and e − (3 σ 1 − σ 2 − σ 3 − σ 4 ) t , for σ 1 = 0 . 3885, σ 2 = 0 . 3883, σ 3 = 0, σ 4 = 0. Note that the t –axis is linear. coun terparts), while all other exp onen ts are zero. This is sho wn in figure 10 (a), where w e plot the ev olution o f the L i ( t ) for i = 1 , 2 , 3 , 4, whose limits for t → ∞ are the 4 largest Ly apuno v exp onen ts. F rom these results w e deduce σ 1 ≈ 0 . 3885 , σ 2 ≈ 0 . 3883 , while the decrease of L 3 ( t ) and L 3 ( t ) to zero indicate that σ 3 = σ 4 = 0. In figure 10(b) w e no w o bserv e that G ALI 2 remains practically constan t fo r this particular time in terv al (actually it dec reases to zero extremely slow ly follo wing the exp onen tia l law e − ( σ 1 − σ 2 ) t = e − 0 . 0002 t ). On the o ther hand, GALI 3 and GALI 4 deca y exp onen tially to zero following the la ws, GALI 3 ∝ e − (2 σ 1 − σ 2 − σ 3 ) t , G ALI 3 ∝ e − (3 σ 1 − σ 2 − σ 3 − σ 4 ) t , giv en b y equation (43). 6 Discussion and conclusions In this paper w e ha v e in tro duced and applied the G eneralized Alignmen t In- dices o f or der k (GALI k ) as a to ol for study ing lo cal and global dynamics in conserv ativ e dynamical systems, such as Hamiltonian systems of N degrees of freedom, or 2 N – dimensional symplectic maps. W e hav e sho wn that these indices can b e succes sfully emp lo y ed not only to distinguish individual o rbits as chaotic or regular, but also to efficien tly c hart large domains of phase space, c hara cterizing the dynamics in the v arious regions by differen t b eha viors of the indices ra nging from regular (GALI k s ar e constan t o r deca y b y w ell–defined p o wer laws) to c haotic (G ALI k s expo nen tially go to zero). 36 A different approach than simply calculating the maximal Lyapuno v exp onen t is to compute t he so–called Smaller Alignment Index (SALI), following the ev olution o f two initially differen t deviation v ectors. This approach has b een used b y sev eral authors and has pr ov ed quite success ful, as it c an determine the nature of the dynamics mo r e rapidly , reliably and efficien tly than t he maximal LCE. In the presen t pa per, motiv ated b y the observ ation that the SALI is in fact prop ort io nal to the ‘area’ of a para llelogram, hav ing as edges the t wo normalized deviation v ectors, w e hav e generalized SALI by defining a quan tity called GALI k , represen ting the ‘v olume’ of a parallelepip ed having a s edges k > 2 initially linearly inde p enden t unit de viation v ectors. In practice, GALI k is computed as the ‘norm’ of the ‘ex terior’ or w edge pro duct of the k normalized deviation v ectors. F or the n umerical ev aluation of GALI k , we need to compute t he reference orbit w e are in terested in from the fully no nlinear equations of the system, as w ell as follow the time evolution of k deviation v ectors, solving the (linear) v ariational equations ab out the or bit. How many suc h vec tors should w e take? Since the phase space o f the dynamical system is 2 N –dimensional, k should b e less than or equal to 2 N , otherwise G ALI k will b e equal to zero already from the start. Ho wev er, ev en though w e may choose our deviation v ectors initially linearly indep enden t, t hey may b e c om e dep endent as time ev olves , in whic h case the phase space ‘v olume’ represen ted b y GALI k will v anish! This is precisely what happ ens for all k > 2 if our reference orbit is chaotic, and also if it is regular and k > N , but at v ery differen t time ra t es. In particular, w e show ed analytically and v erified n umerically in a num b er of examples of Hamiltonian systems that for c haotic orbits GALI k tends ex- p onen tially to zero following a rate whic h dep ends on the v alues of sev era l Ly apunov exp onen t s (see equation (43)). On the other hand, in the case of regular o r bits, GALI k with 2 ≤ k ≤ N fluctuates around non–zero v alues, while, for N < k ≤ 2 N , it tend s to ze ro follo wing a p ow er la w (see equation (80)). The exp onen t of the p o w er law dep ends on the v alues of k and N , a s we ll as on the num b er m of deviation v ectors that may hav e b een chosen initially tangen t to t he to rus on which the or bit lies. Clearly , these differen t beha viors of the GALI k can b e exploited for the rapid and accurate determination of the ch aotic v ersus regula r nature of a give n orbit, or of an ensem ble of orbits. V arying the n um b er of deviation v ectors (and bringing more LCEs into pla y), w e can, in f a ct, ac hiev e high rates of iden tification o f c haotic regions, in a computationally a dv antageous w ay . Sec- ondly , regular motion can b e iden tified by the index b eing nearly constan t for small k , w hile, when k exceeds the dimension of the orbits’ s ubspace, GALI k deca ys by w ell–defined p ow er laws . This ma y help us iden tif y , for example, cases where the motion o ccurs on c an tori of dimens ion d < N (see e.g. [45]) and the orbits b ecome ‘stic ky’ on island chains, b efore turning truly c haotic 37 and exp onen tial dec a y tak es o v er. W e ha v e also studied on sp ecific Hamiltonians with N > 2 the c omputational efficiency of t he GALI k . One migh t susp ect, of course, that the b est choice w ould b e GALI N since this is the index that exhibits the most differen t b ehav- ior for regular and c haotic orbits. On the other hand, it is clear that follo wing a great num b er of de viation v ectors requires considerably more computation time. It turns out, how ev er, that, if chaos o ccupies a ‘large’ po rtion of phase space, a w ell–tailored application of GALI k , with 2 < k ≤ N , can significantly diminish the CPU time required for t he detailed ‘c harting’ of phase space, compared with SALI ( k = 2), as we demonstrated on sp ecific examples in section 5.2 (see figure 8). Although the results presen ted in this pap er we re obta ined for N degree of freedom Hamilto nian systems, it is easy to see that t hey also apply to 2 N – dimensional symplectic maps. So , equations (43) and (80) whic h de scrib e the b eha vior of GALI k , with 2 ≤ k ≤ 2 N , f o r c haotic and regular orbits resp ec- tiv ely are exp ected to hold in that case also. One remark is in order, ho w ev er: In the case of N = 1, i. e. for 2D maps, the first condition of equation (80) cannot b e fulfille d. Th us, for regular orbits of 2D maps, an y 2 initia lly inde- p enden t deviation v ectors w ill b ecome aligned in the direction tangen t to the corresp onding in v arian t curv e and GALI 2 will tend to ze ro followin g a pow er la w of the form GALI 2 ∝ 1 /t 2 . This b ehavior is already kno wn in the literature [15]. Ac kno wledgemen ts This w o r k w as partially supp orted by the Europ ean So cial F und (ESF), Op er- ational Program fo r Educational and V o cational T raining I I ( EPEAE K I I) and particularly the Programs HERAKLEITOS, pro viding a Ph. D. sch olarship for the t hird author ( C. A.) and the Progra m PYTHA G ORAS I I, partially sup- p orting the first author (Ch. S.). Ch. S. w as also supp orted b y the Mar ie Curie In tra –Europ ean F ello wship No MEIF–CT–2006–0256 78. The second author (T. B.) w ishes to ex press his gratitude to t he b eautiful Cen tro Internacional de Ciencias of the Univ ersidad Autonoma de Mexico for its exce llen t hospital- it y during his visit in January – F ebruary 2 0 06, when some of this w ork w as completed. In particular, T. B. wan ts to t ha nk the main researc hers of this Cen ter, Dr. Christof Jung and Thomas Seligman for numerous conv ersations on the stability of m ulti–dimensional Hamiltonian systems . Finally , w e w ould lik e to thank the referees fo r ve ry useful comme n ts whic h help ed us impro v e the clarit y of the pap er. 38 A W edge pro duct F ollow ing an intro duction to the theory of w edge pro ducts as presen ted in textb o oks suc h as [52], let us consider an M –dimensional v ector space V ov er the field o f real num b ers R . The exterior algebra of V is denoted by Λ( V ) and its mu ltiplication, known as the w edge pro duct or the exterior pro duct, is written as ∧ . The w edge product is asso ciativ e: ( ~ u ∧ ~ v ) ∧ ~ w = ~ u ∧ ( ~ v ∧ ~ w ) (A.1) for ~ u, ~ v , ~ w ∈ V and bilinear ( c 1 ~ u + c 2 ~ v ) ∧ ~ w = c 1 ( ~ u ∧ ~ w ) + c 2 ( ~ v ∧ ~ w ) , ~ w ∧ ( c 1 ~ u + c 2 ~ v ) = c 1 ( ~ w ∧ ~ u ) + c 2 ( ~ w ∧ ~ v ) (A.2) for ~ u, ~ v , ~ w ∈ V and c 1 , c 2 ∈ R . The w edge pro duct is also alternating on V ~ u ∧ ~ u = ~ 0 (A.3) for all v ectors ~ u ∈ V . Th us w e ha v e that ~ u ∧ ~ v = − ~ v ∧ ~ u (A.4) for all v ectors ~ u, ~ v ∈ V and ~ u 1 ∧ ~ u 2 ∧ · · · ∧ ~ u k = ~ 0 (A.5) whenev er ~ u 1 , ~ u 2 , . . . , ~ u k ∈ V are linearly dep enden t. Elemen ts of the form ~ u 1 ∧ ~ u 2 ∧ · · · ∧ ~ u k with ~ u 1 , ~ u 2 , . . . , ~ u k ∈ V are called k –v ectors. The subspace of Λ( V ) generated b y all k –v ectors is called the k – th exterior p o w er of V and denoted b y Λ k ( V ). The exterior algebra Λ( V ) can b e written as the direct sum of eac h of the k –th p o w ers o f V: Λ( V ) = M M k =0 Λ k ( V ) = Λ 0 ( V ) ⊕ Λ 1 ( V ) ⊕ Λ 1 ( V ) ⊕ · · · ⊕ Λ M ( V ) (A.6) where Λ 0 ( V ) = R and Λ 1 ( V ) = V . Let { ˆ e 1 , ˆ e 2 , . . . , ˆ e M } b e an orthonormal basis of V, i. e. ˆ e i , i = 1 , 2 , . . . , M are linearly indep enden t v ectors of unit magnitude and ˆ e i · ˆ e j = δ ij (A.7) 39 where ( · ) denotes the inner pro duct in V and δ ij =      1 f or i = j 0 f or i 6 = j . (A.8) It can b e easily seen t ha t the set { ˆ e i 1 ∧ ˆ e i 2 ∧ · · · ∧ ˆ e i k | 1 ≤ i 1 < i 2 < · · · < i k ≤ M } (A.9) is a basis of Λ k ( V ) since a n y w edge pro duct of the form ~ u 1 ∧ ~ u 2 ∧ · · · ∧ ~ u k can b e written as a linear com bina t ion of the k –v ectors of equation (A.9). This is true b ecause ev ery v ector ~ u i , i = 1 , 2 , . . . , k can b e written as a linear com binatio n of the basis v ectors ˆ e i , i = 1 , 2 , . . . , M and using the bilinearity of the w edge pro duct this can b e ex panded to a linear com bination of w edge pro ducts of those ba sis v ectors. An y w edge pro duct in whic h the same basis v ector app ears mo r e than once is zero, while any w edge pro duct in whic h the basis v ectors do not app ear in the pro per order can b e reordered, c hanging the sign whene v er t w o basis v ectors change places. The dimension of Λ k ( V ) is equal to the binomial co efficien t dimΛ k ( V ) =    M k    = M ! k !( M − k )! (A.10) and thus the dimension of Λ ( V ) is equal to the sum of the binomial co efficien ts dimΛ( V ) = M X k =0    M k    = 2 M . (A.11) The co efficien ts of a k –v ector ~ u 1 ∧ ~ u 2 ∧ · · · ∧ ~ u k are the minors of the matr ix that describes the v ectors ~ u i , i = 1 , 2 , . . . , k in terms of the basis ˆ e i , i = 1 , 2 , . . . , M . Let us write these relations in matrix form           ~ u 1 ~ u 2 . . . ~ u k           =           u 11 u 12 · · · u 1 M u 21 u 22 · · · u 2 M . . . . . . . . . u k 1 u k 2 · · · u k M           ·           ˆ e 1 ˆ e 2 . . . ˆ e M           = C ·           ˆ e 1 ˆ e 2 . . . ˆ e M           (A.12) C b eing the matrix o f the co efficien ts of vectors ~ u i , i = 1 , 2 , . . . , k with resp ect to the or thonormal basis ˆ e i , i = 1 , 2 , . . . , M and u ij , i = 1 , 2 , . . . , k , j = 40 1 , 2 , . . . , M b eing real n um b ers. Then the wedge pro duct ~ u 1 ∧ ~ u 2 ∧ · · · ∧ ~ u k is defined b y ~ u 1 ∧ ~ u 2 ∧ · · · ∧ ~ u k = X 1 ≤ i 1

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