A COMPARISON BETWEEN METHODS TO COMPUTE LYAPUNOV EXPONENTS G. TANCREDI, A.SA NCHEZ

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1 THE ASTRONOMICAL JOURNAL, 121:1171È1179, 21 February ( 21. The American Astronomical Society. All rights reserved. Printed in U.S.A. A COMPARISON BETWEEN METHODS TO COMPUTE LYAPUNOV EXPONENTS G. TANCREDI, A.SA NCHEZ Departamento de Astronom a, Facultad de Ciencias, Universidad de la Repu blica, Igua 4225, 114 Montevideo, Uruguay; gonzalo=ðsica.edu.uy, andrea=ðsica.edu.uy AND F. ROIG Instituto Astronoü mico e Geof sico, Universidade Sa8 o Paulo, Avenida Miguel Este fano 42, Sa8 o Paulo, Brasil; froig=iagusp.usp.br Received 2 January 25; accepted 2 October 31 ABSTRACT Lyapunov characteristic exponents measure the rate of exponential divergence between neighboring trajectories in the phase space. For a given autonomous dynamical system, the maximum Lyapunov characteristic exponent (hereafter LCE) is computed from the solution of the variational equations of the system. There are many dynamical systems in which the formulation and solution of the variational equations is a cumbersome task. In those cases an alternative procedure, Ðrst introduced by Benettin et al., is to replace the variational solution by computing two neighbor trajectories (the test particle and its shadow) and calculating the mutual distance. In this paper, we deal with a comparison between these two di erent techniques for the calculation of LCE: the variational method and the two-particle method. We point out a problem that can appear when the two-particle method is used, which can lead to a false estimation of a positive LCE. The explanation of this phenomenon can be analyzed in two di erent situations: (1) for relatively large initial separations the two-particle method is not a good approximation to the solution of the variational equations, and (2) for small initial separations the two-particle method have problems related to the machine precision, even when the separation can be many order of magnitudes larger than the machine precision. We show some examples of false estimates of the LCE that have already appeared in the literature using the two-particle method, and Ðnally we present some suggestions to be taken into account when this method has to be used. Key words: methods: numerical È solar system: general 1. INTRODUCTION where d(t) is the distance in the phase space between two trajectories (a reference trajectory and a shadow one), initially separated by a distance d. For the result to be mean- Lyapunov characteristic exponents are a widely used tool for the estimation of chaoticity in dynamical systems. They ingful, the exponential divergence should be measured in basically measure the rate of exponential divergence the immediate neighborhood of the reference trajectory; between neighboring trajectories in the phase space. For a thus the LCE should now be deðned as given autonomous dynamical system x5 \ f (x), the maximum Lyapunov characteristic exponent (hereafter c \ lim s(t). (3) LCE) is deðned as c \ lim s(t), with t?= t?= d? s(t) \ 1 o m(t) o Then, for practical calculations, a periodic renormalization ln, (1) of the distance d(t) is necessary in order to avoid the shadow t o m o to remain far away from the reference trajectory. The renormalization method introduced by Benettin et al. (1976) con- where m(t) is the solution of the variational equations of the system, m5 \ + fém and m \ m(). Then the system is said sists in approaching the shadow trajectory to the reference to be regular x (integrable) if c \ and chaotic if c[. The one along the separation vector down to the initial distance numbers s(t) are called Lyapunov characteristic indicators d. If this renormalization is made at a Ðxed time interval q, (LCI; Froeschle, Froeschle, & Lohinger 1993) and for practhen we can write tical calculations it is not possible to take t ] O. Thus, the evolution of s(t) is followed up to some time and we plot s \ 1 k kq ; k ln d(iq), (4) d log s(t) versus log t. If the curve has a negative constant i/ slope, the system is regular; if it presents an inñection of the and thus slope which comes close to and the plot converges to a c \ lim s. (5) certain value of s(t), the system is chaotic. k There are many dynamical systems in which the formulation and solution of the variational equations is a cumber- The renormalization is also necessary to avoid k?= d? some task. In those cases, an alternative procedure to saturation,ïï which is caused because a chaotic region is calculate c, Ðrst introduced by Benettin, Galgani, & normally bounded in the phase space, and thus the distance Strelcyn (1976), is to replace the deðnition in equation (1) by between the two trajectories cannot grow by more than the s(t) \ 1 average size of the chaotic region. It is worth noting that all d(t) ln, (2) these problems are avoided with the solution of the variational equations, since it is a linear solution, and so its t d 1171

2 1172 TANCREDI, SAŠ NCHEZ, & ROIG Vol. 121 behavior is independent of the initial condition and of the size of the chaotic region (for a discussion see, for example, Wisdom 1983). Both the solution of variational equations (hereafter variational method) and the simultaneous integration of two trajectories (hereafter two-particle method) have been extensively used in the literature to calculate the LCE. The aim of this paper is to point out a problem that can appear when the two-particle method is used, which can lead to a false estimation of a positive LCE ( 2). In 3we discuss the dependence of the false estimates of the LCEs with di erent initial parameters, and we present a conjecture to explain this phenomenon. We will show some examples from the literature where false estimates have already appeared ( 4). In the conclusion ( 5) we give some recommendations in case one is forced to use the two-particle method. 2. FALSE ESTIMATE OF LCEs We recall that the Lyapunov characteristic indicators (the LCIs) are Ðnite estimates of the Lyapunov characteristic exponents (the LCEs). We then use the term LCI when we refer to the result of a numerical computation and to LCE when we refer to the expected theoretical value. The main di erence between the variational and twoparticle methods is clear: the solution m(t) is simply the linearized distance d(t). Roughly speaking, both methods can be compared with the approximation of a function obtained by using the tangent at a point or the secant by two points, respectively. There is no a priori reason to think that the results obtained with each method could be di erent. At least they should be qualitatively similar, provided that we are far away from a singularity of the system, in which case the linear approach of the distance is not valid and the deðnition of LCE (eq. [1]) becomes meaningless. However, we can Ðnd many cases of quasi-regular or even regular (integrable) systems in which the results of two methods are not the same. As an example, let us consider the two body problem (a massless particle in a central gravitational Ðeld) and calculate the LCI using the two methods. For this calculation we used the integration algorithm Radau by Everhart (1985). We adapted the integrator to solve either the di erential equations for the particle and its shadow (two-particle method) or the di erential equations for the particle and the variational equations (variational method). For the two-particle method, if the particle has a position and a velocity vector p we create the shadow by, an homothesis of factor (1 ] b), i.e., p 1 \ (1 ] b)p. (6) FIG. 1.ÈLCI for the solution of the two-body problem calculated using the two-particle method and the variational method using the RADAU integrator. The initial separation was b \ d /p \ 1~8 and the renormali- zation time step was q \ 1 yr. techniques, with always the same qualitative result, though there are important di erences in the quantitative results among the set of variables and numerical algorithms used (see 3). Figure 2 presents curves log m versus log t for di erent numerical algorithms; the calculations were done for the same initial conditions, renormalization time step, and The initial distance d is then simply d \ b o p o. (7) The initial orbits for the following calculations in the two body problem correspond to the orbit of Ceres around the Sun (semimajor axis a \ 2.77 AU, eccentricity e \.76). As we can see in Figure 1, the variational method gives the actual result, but the two-particle method gives a positive LCI, which is clearly wrong. We made several experiments using di erent values of the distance, d, and of the renor- malization time step, q. We used also di erent sets of variables and coordinate systems and even di erent integration FIG. 2.ÈLCI for the solution of the two-body problem calculated using the variational equations with renormalization (nearly straight downward line), and the two-particle method with di erent numerical integration algorithms: Radau (top curve), the Keplerian algorithm of this paper (middle curve), SWIFT LYAP (bottom curve). For all cases d /p \ 1~8 and q \ 1 yr.

3 No. 2, 21 METHODS TO COMPUTE LYAPUNOV EXPONENTS 1173 initial separation. We compare results using the Radau integrator presented above, the subroutine to compute Lyapunov exponents with the two-body method that comes with the Swift package (SWIFT LYAP; Levison & Duncan 1994),1 and a very simple numerical algorithm based on the solution of KeplerÏs equation, hereinafter called Keplerian algorithm. This algorithm consists in the following steps: 1. Given the position and velocity of the particle, we create the shadow by an homothesis of factor (1 ] b), as explained above. 2. The initial mutual distance d between the particle and shadow is computed. 3. The position and velocity of particle and shadow are transformed into orbital elements a semimajor axis, e eccentricity, u argument of perihelion, M mean anomaly. 4. During a time step q, a, e and u remain constant, and the mean anomaly evolves as: M i`1 \ M i ] nq, (8) where n is the mean motion. 5. After solving KeplerÏs equation, the orbital elements are transformed back to position and velocity in Cartesian coordinates. The mutual distance at the end of the time step is then computed. 6. Afterward the renormalization procedure is applied, the shadow particle is brought back to a distance d along the separation vector in position and velocity. Return to point 3. This method is obviously much faster than the numerical solution of the gravitational di erential equation, letting us make many tests in short CPU time. Note that all numerical algorithms that use the twoparticle method give positive values of the LCI (see Fig. 2). The other alternative is to apply the renormalization procedure whenever the separation distance reaches a Ðxed value, rather than at Ðxed time steps. The results of applying this method are shown in Figure 3 for two di erent initial separations. The renormalization was done when the distance reached 1 times its original value. For similar initial separation we can compare the results with those of Figure 1 to conclude that, though we get a smaller positive LCI, we still obtain a false estimate Some T entative Explanations Holman & Murray (1996) presented a possible explanation for the overestimation of LCEs observed in quasiregular systems when the two-particle method is used. They argued that the false positive LCE is generated in the twoparticle method because after the i-th renormalization the mutual distance d between the particles is given approx- imately by i d i ^ d (1 ] aqn) exp (jq), (9) where d is the initial distance, q is the interval between successive renormalizations, a and n are constants associated with the initial power-law transient separation, and j is ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ 1 Available at FIG. 3.ÈLCI for the solution of the two-body problem calculated using the two-particle method with renormalization at Ðxed separation distance. The initial separation was b \ d /p \ 1~6 for the top curve and b \ d /p \ 1~8 for the lower one. The renormalization was done whenever the distance reached hundred times its original value. the maximum LCE. Combining this expression with the renormalization formula (eq. 4) they obtained ln 1 ] (aqn) s ^ j ]. (1) k q Then, they claimed that for those trajectories with j D or j > ln (1 ] aqn)/q (i.e., quasi-regular trajectories) there is an overestimation of the LCE that is larger for smaller values of q. This does not a ect the strong chaotic trajectories, since in that case j? ln (1 ] aqn)/q. Although this is in good agreement with the behavior observed in their experiments, it is possible to prove that their explanation is not valid. This is because a in equation (9) is not actually a constant as they assumed, but rather it depends on k. We will show this in the following. Consider that the modulus of the distance between the two trajectories evolves with time following the law d(t) \ d (1 ] j r tn) exp (j c t). (11) In this expression, j and j are constants associated with the rate of divergence r of the c regular and chaotic motion, respectively. If the trajectory is regular, then j \ and the distance will grow according to (1 ] j tn). In c other words, an initial error d in the reference trajectory r will propagate following a polynomial of order n (it is common to say that the error propagates linearly with time). On the other hand, if the trajectory is chaotic, j [. Thus, for very short times the polynomial divergence dominates, c but for longer times the exponential divergence becomes more important. It is clear that j is the maximum LCE. Now, suppose c that we make a renormalization at a Ðxed interval q and, in between the steps, the distance evolves

4 1174 TANCREDI, SAŠ NCHEZ, & ROIG Vol. 121 following the function d(t) deðned in equation (11). Then at t \ q the distance before renormalization is d \ d(q); at t \ 2q is d \ d d(2q)/d(q); consequently at 1 t \ (i ] 1)q would be 2 d[(i ] 1)q] d \ d i`1 d(iq) or in terms of equation (11), (12) [1 ] j qn(i ] 1)n] exp [j (i ] 1)q] d \ d r c. (13) i`1 (1 ] j qnin) exp (j iq) r c Using the binomial theorem, we can write this expression as d \ d (1 ] j qna ) exp (j q), (14) i`1 r i`1 c where a \ 1 i`1 1 ] j qnin ; n An r jb in~j. (15) j/1 It is simple to see that lim a \, since a D 1/i for large i?= i i values of i. Applying equation (14) to the renormalization formula (eq. [4]), we obtain s \ 1 k kq ; k ln [(1 ] jr qna ) exp (j q)] \ j ] S, (16) i c c k i/1 where S \ 1 k kq ; k ln (1 ] j qna ). (17) r i i/1 LetÏs show that lim S \. First, we have that for k?= k large values of i we can write ln (1 ] j r qna i ) D ln A 1 ] j r qn 1 ib ^ jr qn 1 i and thus for large values of k we obtain ] ÉÉÉ, (18) S D 1 k kq ; k j qn r ] ÉÉÉ \ ln (j r qnk), (19) i kq i/1 which clearly goes to zero for k ] O. Then, c \ lim s \ j. If k?= a is k assumed c to be constant ( \ a), equation (16) becomes i identical to equation (1) (the one assumed by Holman & Murray 1996); but we have just demonstrated that the assumption is not valid, a goes to zero for large k. It is worth noting that an expression i similar to equation (16) can be found when applying the variational method. Note that the modulus of the solution of the variational equations can be also represented by an expression similar to equation (11), i.e., m(t) \ m (1 ] j r tn) exp (j c t), (2) where j is associated with the maximum eigenvalue of the system and c j and n are related to the degeneracy of some eigenvalues. Then, r a periodic renormalization of m(t) combined with equation (4) (now substituting d by m), leads again to equation (16). Since in practise the renormalization procedure is carried out up to a Ðnite k, then a will always have a Ðnite nonzero value. In this case, the overestimation k of the LCE observed in quasi-regular systems could be justiðed in the framework of the explanation given by Holman & Murray (1996). However, since lim a \, this eventual overestimation should decrease (or k?= even k disappear) as soon as we extend our calculation to larger values of k, i.e., advancing the trajectories for a larger time span keeping the same q between renormalizations. Of course this is not the behavior observed in quasi-regular trajectories when using the two-particle method. In fact the false positive LCE is independent of the total time span of the calculation. On the other hand, if we assume for a moment that in practice ÏÏ a is almost constant for any k, as Holman & Murray (1996) k assumed, then we should also observe an overestimation of the LCE when applying the renormalization formula (eq. [4]) in the variational method, since the asymptotic behavior of s is the same for both methods (eq. [16]). Again, this is not what k is observed in the experiments. For example, in Figure 2 the almost downward straight line corresponds to a calculation where we apply the renormalization procedure to the variational method exactly in the same way as for the two-particle method. It is possible to verify that the variational method never produces false positive LCEs. Then, we conclude that: (1) the explanation given by Holman & Murray (1996) is not necessary correct since the assumption a \ const. is wrong, and (2) even assuming that in practise a could be e ectively constant, their explanation does not correctly account for the discrepancy observed between the two-particle and variational method, respectively. We stress the fact that, based on the explanation given by Holman & Murray (1996), the rescaling technique should also lead to a false LCE when the variational solution is used. In the next section, we will show that, in fact, the origin of the false estimates of LCEs in the two-particle method relies in the accumulation of round-o errors during the computation of the mutual distances between the two particles in the course of the successive renormalizations. These errors are not present in the variational solution since this does not involve the calculation of mutual distances. 3. DEPENDENCE OF LCI WITH d AND q Before formulating a conjecture to explain the false estimation of LCE in regular systems when using the twoparticle method, let us study the dependence of the LCI with the initial distance between the two trajectories d. We use the Keplerian algorithm for the following calculations. Figure 4 shows the LCI of a two-body problem calculated using the two-particle method with di erent values of d. Although all the LCIs give wrong estimates (they should be zero), we note that for very large and very small values of d the estimation is worse, and there seems to be an optimal ÏÏ value of d /p \ 1~8 for which the estimation is better. A more detailed analysis of the dependence of the LCIs with d is presented in Figure 5. The two-particle method is again applied to the two-body problem for a wide set of initial separation. The Ðnal value of s at t \ 17 yr is plotted against log (d /p ). The curves from top to bottom correspond to calculations in: (1) double precision with renormalization time step, q \ 1 yr; (2) double precision with q \ 1 yr; and (3) quadruple precision with q \ 1

5 No. 2, 21 METHODS TO COMPUTE LYAPUNOV EXPONENTS 1175 FIG. 4.ÈLCI for the solution of the two-body problem calculated using the two-particle Keplerian algorithm of this paper. Di erent curves correspond to di erent initial separations d. The renormalization time step was 1 yr in all cases. yr. [Note that for d /p \ 1~1,1~15, and 1~2 we just give upper limits to the LCEs, since the curves log s(t) versus log t do not level o.] Two distinct regimes can be distinguished: one for large values of d (the right-hand branch of the V-shaped curve in Fig. 5) and another one for small values of d (left branch); the change from one regime to the other depends on the precision of the calculation and slightly on the renormalization time step. For the right branch, larger values of s correspond to larger d. This is because for large values of d we are far away from the condition of neighboring trajectories. The distance, d, is no more equivalent to its linear approach m and the deðnition equation (2) is no longer applicable (recall that the deðnition of LCEs uses m not d!). In the deðnition of the LCE, the distances are measured on the surface tangent to the manifold, while in the two-particle method the tangent vector is approximated by a secant. This secant vector belongs to another space, which is itself tangent to a manifold di erent than the original one. This manifold deðnes a di erent dynamical problem, and then its associated LCE does not need to be equal to that of the original problem. Under this interpretation, it is not expected that the spurious LCIs have any dependence on the precision of the calculations nor on the renormalization time step, because it is just a geometrical problem. In fact, the convergence of the right-hand branches of the middle and bottom curves proves that the machine precision has no e ect on the spurious values. Also, we see that for very large values of d, (e.g., for d /p Z 1~3), a good agreement in the Ðnal value of the LCE is found for di erent renormalization steps (see Fig. 5 and the three upper curves of Fig. 6). These numerical results are in disagreement with the interpretation by Holman 7 Murray (1996), since they would expect a dependence with the renormalization step. As it is expected from the deðnition of the LCEs, the spurious values become closer to the true value (i.e., zero) as d ]. This tendency is at least true up to a certain d, where the curve s versus d has a turning point (see Fig. 5). We found that in our particular problem the turning point occurs for values of d many orders of magnitudes higher FIG. 5.ÈLCI for the solution of the two-body problem as a function of the initial separation d. The calculation was made with the Keplerian algorithm of this paper, using di erent renormalization time steps and di erent machine precision. For some calculations we give only an upper limit of the estimated LCI (dotted line). See text for explanation. FIG. 6.ÈLCI for the two-body problem calculated with the Keplerian algorithm for di erent values of d and q. The top three curves correspond to d /p \ 1~3 and the three bottom ones correspond to d /p \ 1~1.

6 1176 TANCREDI, SAŠ NCHEZ, & ROIG Vol. 121 than the machine precision. When using a renormalization step of 1 yr, the turning point occurs at d /p D 1~8 in double precision calculations (where the machine precision is D 1~16)2; while it occurs at d /p D 1~15 in quadruple precision calculations (where the precision is D 1~34). To understand the cause of this problem we will make the following experiments with the results of the Keplerian algorithm: let us project the separation vector connecting the particle and its shadow on the phase space of orbital elements a, e, u, M, where a is the semimajor axis, e the eccentricity, u the argument of perihelion, and M the mean anomaly. After a certain time interval, the separation vector is generally in the direction of the mean anomaly axis, and hence its projection dm on this axis after renormalization has in modulus almost the size of the initial separation. On ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ 2 We deðne machine precision as the quantity p for which (1 ] p) [ 1is consider to be equal to. the other hand, the projections in the other axis are many orders of magnitudes smaller. Between two renormalizations, a, e, and u su er no changes, while M evolves linearly with time (see the description of the algorithm in 2). Then, after a renormalization time q we can write da \ da, (21) de \ de, (22) du \ du, (23) while the separation dm is the sum of two components dm \ dm ] qdn \ dm [ 3 q da. (24) 2 a5@2 (For the particle we use the following standard units: semimajor axis a \ 1, mean motion n \ 1, period T \ 2n). In Figure 7 we present the evolution of da and dm with time for an initial distance d /p \ 1~1 and two renorma- FIG. 7.ÈSeparation vector d is projected onto the phase space of orbital elements a, e, u, M and the components da and dm are shown here as a function of time. (a) For a renormalization time step of 1 yr; (b) for a renormalization time step of 1 yr. da is expressed in units of 1~13 and dm in 1~9 radian. Calculations were made with the Keplerian algorithm for an initial separation d /p \ 1~1.

7 No. 2, 21 METHODS TO COMPUTE LYAPUNOV EXPONENTS 1177 lization steps q \ 1 and 1 yr. As mentioned above dm is of the order of d. Considering that a for the particle and the shadow are on the order of 1, the di erence da could not be smaller than the machine precision. Again we can see in Figure 7 that da is generally of that order, but it presents random Ñuctuations on signiðcant digits that, following equation (24), produce spurious contributions to dm. The random Ñuctuations are generated in the renormalization procedure because the shadow is approached to the particle along the separation vector (in Cartesian coordinates), and this separation tends to be aligned with the orbit so da ] (numerically means, to the machine precision). But, depending on the position on the orbit, the values of da could be much larger than the machine precision. The accumulation of these spurious values cause a false estimation of the LCE. For a smaller d, the contribution of da on the values of dm is larger, and hence larger LCIs are obtained. Since these spurious values are generated in the renormalization procedure, it is expected that the Ðnal values of the LCIs depend also on the renormalization time step. This is clear if we look at the left-hand branches of the top and middle curves in Figure 5, and also the three lower curves in Figure 6, where we can see a strong dependence of the spurious LCIs with the renormalization time step. Comparing the evolution of da and dm with time for q \ 1 and 1 yr (Fig. 7), we conclude that the shorter the renormalization time step the larger the number of spurious values of da and hence of dm. This is obvious since, for a given time span, there are more renormalizations for a shorter renormalization time step, and there are more chances for large spurious values to occur. For the numerical integrations using the gravitational di erential equations instead of the Keplerian algorithm, the problem is slightly di erent. In Figure 1 we present results for a single value of q and d. Similar plots as the ones presented in Figure 5 can be obtained in the integrations of the gravitational di erential equations, with a V - shaped dependence of the LCI with d. The explanation for the right-hand branch is the same as described above. In the left-hand branch the role played by da is now played by the separation of the velocity vector dv. The velocity vectors for the particle and the shadow tend to be aligned, i.e., o dv Š o ]. Since dr is on the order of d, the renormaliza- tion procedure introduces large Ñuctuations on dv. Consider that VŠ 2\k A2 r [ 1 ab ; (25) hence A 2V dv \ k [ 2 r2 dr [ da a2b. (26) Large spurious values of dv would imply large values of da and hence of dm, leading to values of dr? dr. (Note that da and dm are not calculated in the numerical integrations, but obviously they are implicitly a ecting dr and dv.) The accumulation of these spurious values cause the false estimate of the LCE. 4. EXAMPLES IN THE LITERATURE The two-particle method has been extensively used in many research Ðelds for the computation of the Lyapunov exponents. As we mentioned earlier, it is much easier to implement and in some cases it is the only alternative available. As an example we present a few cases where this method has been used in the study of the orbital dynamics of solar system bodies; e.g., Sussman & Wisdom (1988), Franklin, Lecar, & Soper (1989), Lecar, Franklin, & Murison (1992), Whipple & Shelus (1993), Franklin, Lecar, & Murison (1993), Saha & Tremaine (1993), Murison, Lecar, & Franklin (1994), Franklin (1994), Mu ller & Dvorak (1995), Tancredi (1995, 1998). These papers deal with many di erent problems, like the chaotic behavior of Pluto, main belt asteroids, NEAs, Jupiter family comets, and outer Jovian satellites. Holman & Murray (1996) already showed that the computations of the Lyapunov exponents of some asteroids in the outer belt made by Murison et al. (1994) with the twoparticle method were mostly wrong. Murison et al. (1994) results were correct only in the cases where the Lyapunov times were less than a few thousand years (the Lyapunov time is the inverse of the Lyapunov exponent). Their shadow orbits were started at a distance of 1~6 in phase space. In Figure 5 we observe that for this initial separation, spurious values of the Lyapunov time on the order of a few thousand years are obtained depending on the renormalization time. The two-particle method is not suitable to analyze the slow chaos observed in the asteroid main belt. Sussman & Wisdom (1988) used the two-particle method to compute the Lyapunov time of Pluto but they were very careful in the implementation. They used a high-precision computer (which allows them to use initial separations of the order of 1~2); but they also tested their results with the variational method. We repeat their calculations with the variational method and we obtain a Lyapunov time of the same order. Tancredi (1995) and Tancredi (1998) studied the dynamical evolution of Jupiter family comets and NEAs, respectively. He obtained Lyapunov times clustered in the range 5È15 yr and generally lower than 1 yr. As observed in Figure 5, for such high values of the LCI (1~2È1~3 yr~1), the two-particle method do not introduce errors for initial relative separation smaller than 1~4, as he had used. Whipple & Shelus (1993) and Saha & Tremaine (1993) studied the chaotic evolution of the outer Jovian satellites. The Ðrst authors centered on J-VtIII Pasiphae, while the later authors considered the four retrograde satellites (J- VIII Pasiphae, J-IX Sinope, J-XI Carme, J-XII Ananke). The integrations of Whipple & Shelus (1993) were not long enough to estimate a positive Lyapunov exponent. Figure 2c of Saha & Tremaine (1993) shows the variation of c(t) with time in a logarithmic scale. As stated by the authors, All four curves show some sign of levelling o, but only Sinope is clearly chaotic.ïï The integrations by Saha & Tremaine (1993) were performed with a symplectic algorithm, they used a initial separation of neighboring orbits of da/ a \ 1~4 and a renormalization time of 512 yr. They also mentioned, A better procedure would have been to integrate the tangent equations, but the integrator we use is not suited to this task.ïï We reproduce their calculations using the same initial conditions and two di erent algorithms: (1) the Radau integrator and the variational equations; (2) the Radau integrator and the two-particle method with the distances calculated in the elements phase space. With the variational method we obtain a positive LCI for Pasiphae and Sinope, but no levelling o for Carme and

8 1178 TANCREDI, SAŠ NCHEZ, & ROIG Vol. 121 Ananke (Fig. 8a). With the two-particle method, a similar LCE is obtained for Sinope; for Pasiphae we obtain a slightly higher LCE, and the curves for Carme and Ananke show signs of levelling o similar to the results obtained by Saha & Tremaine (1993) (compare their Fig. 2c and our Fig. 8b). One should also compare the slopes of the corresponding curves in Figures 8a and b and note that the slopes in the two-particle method are higher than in the variational method. The two-particle method gives false estimates of the Lyapunov exponent higher than the actual values. As stated by the authors the implementation of the variational method is not a straightforward task in a symplectic integrator, whereas it is much easier in a general integrator of ordinary di erential equations such as Radau or Bulirsh- Stoer. In recent years the simplectic integrator has been widely used within the solar system dynamicist community, in particular after the public release of the package Swift developed by Levison & Duncan (1994). This package includes two codes to compute the LCI: one with the twoparticle method that it is not recommended and the other implements the variational method but it is still under test.3 Recently, Murray & Holman (1999) measured the Lyapunov exponent of the system composed by the Sun ] the giant planets by comparing pairs of integrations with very small di erences in the initial conditions. They introduced a shift in the coordinate of Uranus of the order of 1~14 AU. By using this tiny separation they hoped to avoid the requirement of renormalization. Nevertheless, since they obtained a Lyapunov time of D 7 ] 16 yr in a D 2 ] 18 yr integration, following equation 11 the Ðnal separation should be of the order of.25 AU. Two criticism can be pointed out to this procedure to calculate LCIs: (1) with such tiny initial separations that are less than two order of magnitudes greater than the machine precision, the errors ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ 3 Available at in the integration algorithm becomes very relevant; in particular if they claim a relative error in energy of the order of 1~9; (2) though, one would expect that the separation at the end of the integration would mainly be along the angular variables, it may be large enough to question the validity of the assumption of sampling the same chaotic region, in particular on view of the quasi-resonance structure of the system. A detailed analysis of the results of Murray & Holman (1999) is out of the scope of this paper, we just point out the inconvenience to avoid renormalization. 5. CONCLUSIONS We have shown in this paper that in certain cases it is possible to obtain a spurious estimation of the LCEs when using the two-particle renormalization method of Benettin et al. (1976). This behavior can be explained by two major causes: (1) When the initial distance d /p is larger than a certain value (of the order of 1~8 in double precision in our integrations of heliocentric orbits) the deðnition of distance between two neighboring orbits over the manifold does not match the deðnition of linearized distance over the tangent space to the manifold. Then the deðnition of LCE as given by equation (2) is no longer valid. (2) On the other hand, if d /p is much smaller than this value, the renormalization procedure for the two particles is able to introduce large numeric errors in the calculation of the mutual distance; e.g., the value of some parameters can fall near to the machine precision and they can indirectly a ect the value the principal components of the mutual distance, leading to spurious large values. Since the errors are introduced in the renormalization procedure, it is natural to expect a dependence of the estimated LCEs with the number of renormalizations performed (i.e., with the renormalization time step) in the sense that the smaller the step the worse the estimation. However, the renormalization does not provoke the same e ect in the variational method; because, as we FIG. 8.ÈLCI for the outer Jovian satellites Pasiphae ( P ÏÏ), Sinope ( S ÏÏ), Carme ( C ÏÏ) and Ananke ( A ÏÏ) computed with (left) the Radau integrator and the variational equations; (right) the Radau integrator and the two-particle method with the distances calculated in the elements phase space.

9 No. 2, 21 METHODS TO COMPUTE LYAPUNOV EXPONENTS 1179 explain in 2.1, the exact solution given by the variational equations lead to a correct result even if a renormalization procedure is applied. We can conclude that the two-particle method is not recommended to calculate LCIs in those cases where the solution can fall in a region of regular or quasi-regular motion of the phase space. In these cases the variational method should be preferable. For a region of strong chaoticity, the LCIs calculated with the two-particle method give acceptable values. Nevertheless, it is recommended to make several test with di erent initial separations and renormalization time steps. To be conðdent on the estimate of the LCE, no variation on the Ðnal value of the LCI should be observed among the di erent runs. One of the authors (F. R.) whish to thanks the Sa8 o Paulo State Science Foundation (FAPESP) for partially supporting this work. Benettin, G., Galgani, L., & Strelcyn, J.-M. 1976, Phys. Rev. A, 14, 2338 Everhart, E. 1985, Dynamics of Comets: Their Origin and Evolution, ed. A. Carusi & G. B. Valsecchi (Dordrecht: Reidel), 185 Franklin, F. 1994, AJ, 17, 189 Franklin, F., Lecar, M., & Murison, M. 1993, AJ, 15, 2336 Franklin, F., Lecar, M., & Soper, P. 1989, Icarus, 79, 223 Froeschle, C., Froeschle, Ch., & Lohinger, E. 1993, Celest. Mech. Dyn. Astron., 56, 37 Holman, M., & Murray, N. 1996, AJ, 112, 1278 Lecar, M., Franklin, F., & Murison, M. 1992, AJ, 14, 123 REFERENCES Levison, H. F., & Duncan, M. J. 1994, Icarus, 18, 18 Mu ller, P., & Dvorak, R. 1995, A&A, 3, 289 Murison, M. A., Lecar, M., & Franklin, F. A. 1994, AJ, 18, 2323 Murray, N., & Holman, M. 1999, Science, 283, 1877 Saha, P., & Tremaine, S. 1993, Icarus, 16, 549 Sussman, J., & Wisdom, J. 1988, Science, 241, 433 Tancredi, G. 1995, A&A, 299, 288 ÈÈÈ. 1998, Celest. Mech. Dyn. Astron., 7, 181 Whipple, A. L., & Shelus, P. J. 1993, Icarus, 11, 265 Wisdom, J. 1983, Icarus, 56, 51