Geometry and chaos on Riemann and Finsler manifolds

arXiv:chao-dyn/9709023v1 18 Sep 1997 Geometry and Chaos on Riemann and Finsler Manifolds Maria DI BARI1∗ Piero CIPRIANI1,2† 1 Dipartimento di Fisica ”E. Amaldi”, Università ”Roma Tre”, Via della Vasca Navale, 84 – 00146 ROMA, Italia 2 I.N.F.M. - sezione di ROMA. Submitted to: Planetary and Space Science Send proofs to: Piero CIPRIANI (address above) Send offprint request to: Piero CIPRIANI (address above) ∗ † e-mail: e-mail: [email protected] [email protected] Abstract In this paper we discuss some general aspects of the so-called geometrodynamical approach (GDA) to Chaos and present some results obtained within this framework. We firstly derive a naı̈ve and yet general geometrization procedure, and then specialize the discussion to the descriptions of motion within the frameworks of two among the most representative implementations of the approach, namely the Jacobi and the Finsler geometrodynamics. In order to support the claim that the GDA isn’t simply a mere re-transcription of the usual dynamics, but instead can give various hints on the understanding of the qualitative behaviour of dynamical systems (DS’s), we then compare, from a formal point of view, the tools used within the usual framework of Hamiltonian dynamics to detect the presence of Chaos with the corresponding ones used within the GDA, i.e., the tangent dynamics and the geodesic deviation equations, respectively, pointing out their general inequivalence. Moreover, to go ahead the mathematical analysis and to highlight both the peculiarities of the methods and the analogies between them, we work out two concrete applications to the study of very different, yet typical in distinct contexts, dynamical systems. The first is the well-known Hénon-Heiles Hamiltonian, which allows to exploit how the Finsler GDA is well suited not only for testing the dynamical behaviour of systems with few degrees of freedom, but even to get deeper insights on the sources of instability. We show the effectiveness of the GDA, both in recovering fully satisfactory agreement with the most well established outcomes and also in helping the understanding of the sources of Chaos. Then, in order to point out the general applicability of the method, we present the results obtained from the geometrical description of a General Relativistic DS, whose peculiarity is well known, as its very nature has been debated since long time, namely the Bianchi IX (BIX) cosmological model. Using the Finsler GDA, we obtain results with a built-in invariance, which give evidence to the non chaotic behaviour of this system, excluding any global exponential instability in the evolution of the geodesic deviation. 2 Introduction. The investigations on the occurrence of regular and chaotic behaviour in N dimensional dynamical systems are performed with a variety of methods and mathematical tools. Recently, this ensemble widened with the inclusion of the geometrodynamical approach (GDA), e.g., [Pettini 1993, Cipriani 1993]. As other new approaches, this tool has been suggested and applied to the study of stability properties of general dynamical systems, in the hope to bring the phenomenological analysis of their possibly chaotic behaviour back to an inquiry directed towards an explanation, at least qualitative, of the mechanisms responsible for the onset of Chaos. Within the framework of the geometrical picture, this explanation was sought through a possible link between a change in the curvature properties of the underlying manifold and a modification of the qualitative dynamical behaviour of the system. Within the Hamiltonian approach, the ingredients needed to make Chaos lie basically on the presence of stretching and folding of dynamical trajectories; i.e., in the existence of a strong dependence on initial conditions, which, together with a bound on the extension of the phase space, yields to a substantial unpredictability on the long time evolution of a system. Usually, the strong dependence is detected looking at the occurrence of an exponentially fast increase of the separation between initially arbitrarily close trajectories. To have true Chaos, this last property must be however supplemented by the compactness of the ambient space where dynamical trajectories live, this simply in order to discard trivial exponential growths due to the unboundedness of the volume at disposal of the DS. Stated otherwise, the folding is necessary in order to have a dynamics actually able to mix the trajectories, making practically impossible, after a finite interval of time, to discriminate between trajectories which were very nearby each other at the initial time. When the space isn’t compact, even in presence of strong dependence on initial conditions, it could be possible, in some instances (though not always), to distinguish among different trajectories originating within a small distance and then evolved subject to exponential instability. In the geometric description of dynamics, the recipe to find Chaos is essentially the same, with some generally speaking minor differences, which nevertheless prove sometimes to be very relevant for the understanding of the qualitative behaviour of the system. When the geometrization procedure is accomplished, the study of dynamical trajectories is brought back to the analysis of a geodesic flow on a suitable manifold M. In order to define Chaos, M should be compact, and geodesics on it have to deviate exponentially fast. As we will show below, now the point representative of the state of the system will move along geodesics with unit speed and the dynamics itself determines the natural way to measure distances on M (or, rather, on T{q} M). Though there exist many geometrizations, [Cipriani 1993, Di Bari 1996], in the following we will exploit mostly two of them, namely, the Jacobi and Finsler ones, essentially equivalent for most DS’s with many degrees of freedom, but whose differences result very enlightening on the sources of instability in the case of few dimensional systems. The GDA can be adopted to cope with very general DS’s, e.g., [Di Bari et al. 1997]; however, in the present paper, we will confine to discuss only that (still very large) subset of systems for which a comparison between the usual Hamiltonian approach and the two geometrical ones here discussed can be carried out without troubles. So, here and in the sequel, we will concentrate on N -dimensional conservative dynamical system. We further suppose, only to simplify the notation, that a choice of coordinates, x = {xi }, exists such that the equations of motion can be put in the newtonian form: ẍi + U,i = 0 i = (1, . . . , N ) , (1) 3 where U = U(x) is the potential, we denoted by a dot the derivative with respect to newtonian time and indicated the partial derivative with respect to one of the coordinates as U,i = ∂U . ∂xi The analysis of the stability of the dynamics given by eq.(1) is usually performed studying the evolution of two initially close trajectories, starting at t = 0 from two very nearby points P0 and P0 , i.e., with very similar initial conditions P0 = [x(0) , ẋ(0)] ; P0 = [x(0) + ξ(0) , ẋ(0) + ξ̇(0)] , where the disturbance vector, D = {DA }, (A = 1, . . . , 2N ), is defined through D = (ξ, ξ̇) ⇐⇒ {DA } = (D1 , . . . , DN , DN +1 , . . . , D2N ) = (ξ 1 , . . . , ξ N , ξ˙1 , . . . , ξ˙N ) , (2) i.e., Di = ξ i ; DN +i = ξ˙i , (i = 1, . . . , N ). This vector represents the uncertainty in the initial conditions, and connect the two points in the space of states of the system. With the choice of coordinates assumed above, it gives also the separation in phase space. The evolution of the small perturbation vector is governed by the tangent dynamics equations, [Benettin et al. 1976, Benettin et al. 1980], which read ξ¨i + U,ij ξ j = 0 i = (1, . . . , N ) , (3) where clearly U,ij = ∂2U , ∂xi ∂xj and the summation convention in the products is adopted. A quantitative indicator of the presence of Chaos in the dynamics is extracted from the solutions of eq.(3), measuring the possible exponential growth of the distance between trajectories, assumed to be represented by the norm, D(t) = kD(t)k, of the separation vector in phase space. The (maximal) Lyapunov characteristic number (LCN) is indeed defined as the asymptotic quantity, [Benettin et al. 1976]:   1 D(t) , (4) ln LCN = lim lim t→∞ D(0)→0 t D(0) where a suitable norm of a 2N -dimensional vector in phase space should have been defined, with the introduction of a (2N × 2N ) metric ΥAB : kD(t)k2 = ΥAB DA DB Up to this point, the choice of the metric is largely arbitrary, as the dynamics does not give any prescription on it, not to mention that, moreover, the phase space do not possess a truly metric structure. This indefiniteness is usually exploited in its full freedom, imposing an Euclidean metric to the full 2N -dimensional space, distinguishing also coordinates and momenta spaces, i.e., putting   D2 (t) = kD(t)k2 = ηij ξ i ξ j + ξ˙i ξ̇ j . (5) A DS is said to be chaotic in a region W of its phase space if W is compact, orbits starting inside W remains into W, and almost all orbits starting in its interior are characterized by a positive value of the maximal LCN, for almost any choice of the initial perturbation D(0). 4 Dynamics and Geometry Although the geometrization procedure can be carried out elegantly and in a full generality, substantially starting from different formulations of the least action principle, [Synge and Schild 1978, Cipriani 1993, Pettini 1993], we will show how a metric is naturally introduced looking at the dynamics itself. Let’s suppose that we are dealing with a DS as that given by eq.(1) and we perform a change of the time variable from the newtonian time, t, to a new variable, s, defined through a relation which in general depends, on time, coordinates and velocities: ds = f (t, xi , ẋi )dt . (6) As a result, the equations of motion (1) become 1 df dxi 1 d2 xi + + 2 U,i = 0 . 2 ds f ds ds f (7) Depending on the choice of the function f , it can happens that the new equations of motion can still be derived from a variational principle, δA = 0, where Z P2 Z P2 f (t, xi , ẋi )dt ds = A= P1 (8) P1 and the integral is taken between fixed endpoints: P1 = P1 (t1 ) = P1 (s1 ) and P2 = P2 (t2 ) = P2 (s2 ). Provided that f satisfies suitable conditions, A can be thought as the lenght of a geodesic joining P1 and P2 and satisfying equation (7); in this case, f also defines the lenght of the vector ẋ. When f depends explicitly on time t, in order to complete the procedure, it is necessary to include also the time as a coordinate. To this goal, we introduce a further time parameter w, and equation (6) becomes     dx/dw dt dx dt = f t, x, dw . (9) ds = f t, x, dt dt/dw dw If we put x0 = t and indicate with a prime the derivation with respect to the new time, w, equation (6) reads ds = F (xα , x′α )dw α = 0, 1, . . . , N , (10) where F (xα , x′α ) = f (xα , x′α )x′0 . (11) The number of equations of motion is now (N + 1). Indeed, from the variational principle it follows   d ∂F (xα , x′α ) ∂F (xα , x′α ) − = 0 β = (0, 1, . . . , N ) , (12) dw ∂x′β ∂xβ and the additional equation, i.e., that with β = 0, gives the conservation of energy. So when the time reparametrization involves an explicit dependence on the old time t, the geometrization procedure must be carried out using the function of (2N + 2) variables F (xα , x′α ), instead of the original one f (t, x, ẋ), depending on only (2N + 1) variables. In order to set a self consistent geometrodynamics, the function1 F (xα , x′α ) must fulfil the following conditions: 1 Or either the function f (x, ẋ), when the transformation do not involves explicitly the time t. 5 a) F (xα , x′α ) must be sign definite, e.g. positive F (xα , x′α ) > 0 ∀ x′α 6= 0 , (13) with F (xα , x′α ) = 0 if and only if x′α = 0 ∀ α = (0, 1, . . . , N ). b) F (xα , x′α ) must be a positively homogeneous function of first degree in the x′α , i.e. F (xα , kx′α ) = kF (xα , x′α ) , ∀k>0; (14) in such a way F dw is apparently invariant for reparametrization:     α α α dx α dx F x , dw1 ≡ F x , dw2 . dw1 dw2 (15) c) Finally, det ∂ 2 F 2 (xα , x′α ) 6= 0 . ∂x′β ∂x′γ (16) From the definition, eq.(11), the extended function F turns out to satisfy always condition b) above, while this is in general not true for the function f . So, if a built-in invariance for reparametrization, eq.(15), is sought, even if the trasformation t → s does not explicitly depends on t, it is sometimes however necessary to introduce the function F . If the conditions a), b) and c) are fulfilled, it is possible to exploit the transormation obtained, in order to derive a metric g on the manifold M. It can be shown that the quantity def gαβ (xα , x′α ) = 1 ∂ 2 F 2 (xα , x′α ) , 2 ∂x′β ∂x′γ (17) meets all the requirements demanded to a metric. By the properties of F , it easily follows, [Rund 1959, Di Bari 1996], that the line element of eq.(17) can be written as ds2 = F 2 (xα , x′α )dw2 = gβγ (xα , x′α )dxβ dxγ ; (18) and the equations of motion turn out to be the geodesic equations on (M,g), which read: η β d2 xα α dx dx + Γ =0, βη ds2 ds ds (19) where ρ ′ρ Γα βη (x , x )   ∂gδη ∂gηβ 1 δα ∂gδβ + − = g 2 ∂xη ∂xβ ∂xδ (20) are the connections of the metric, which, in the general case, depend also on the tangent vectors x′α . If we further add to the conditions a), b) and c) the following requirement: ∂ 2 F 2 (xα , x′α ) β γ v v >0 ∂x′β ∂x′γ ∀ v α 6= 0 , (21) then we will deal with a positive definite metric. As the dynamics determines the natural metric by itself, we see why it is not necessary to impose an a priori metric, as has been done to measure the magnitude, D(t), of the perturbation vector obtained solving eq.(3) along the trajectories, solutions of eq.(1). Within the framework of any geometrization, the norm of any vector of the tangent space, v ∈ Tq M, to the manifold on a point q ∈ M, simply reads: def v 2 = kvk2 = gαβ (xα , x′α )v α v β ; (22) 6 and we remark that in the most general case up to now discussed, the metric g as well the norm of the vector v, depend not only on the point xα of the manifold, but also on the tangent vector x′α . The metric gαβ (xα , x′α ) is a Finsler metric and its geometrical features are widely developed (see [Rund 1959] and references therein). In [Di Bari 1996] and in [Di Bari et al. 1997], the general mathematical formalism is derived and are also presented some applications to several DS’s. At this point we observe that, in the case the metric do not actually depends on the tangent vectors, i.e., when F 2 (xα , x′α ) = gαβ (xη ) dxα dxβ , dw dw (23) then the manifold reduces to a Riemannian one. We summarize our discussion emphasizing how, under very mild assumptions, the geodesic equations can be recovered simply re-interpreting the equations of motion of a DS, writing down them in terms of a new time parameter, and looking for a suitable transformation which make the new parameter one action of the system. We now specialize our presentation to the two metrics mentioned in the Introduction, which can be obtained from the general discussion just presented, by particular choices of the function F . The first one is the well known Jacobi metric, which is a very special case in that it is defined on a Riemannian manifold; whereas the other dynamical manifold we will explore is a pure Finsler metric, as described above, obtained, as we will see, by a very familiar function f . Though simpler, the Jacobi metric has shown its reliability in describing the main qualitative and quantitative properties of typical many degrees of freedom DS’s (see [Cipriani and Di Bari 1997a] and references therein for a more detailed account) and has been recently applied with partial success also to the investigations of dynamical behaviour of few dimensional systems. Indeed our attempt towards a generalization of the riemannian approach has been motivated just because of the limitations encountered by the Jacobi GDA in coping with two degrees of freedom systems or non standard ones, as those of General Relativitistic origin, which are characterized by a very peculiar kinetic energy form. In addition, it applies only to conservative systems2 , and it is then not suitable for the study of lagrangian systems with gyroscopic terms, as those arising either when a non inertial reference frame is chosen, e.g., the restricted three body problem, [Di Bari 1996, Di Bari et al. 1997], or when electromagnetic interactions are taken into account. The GDA in the framework of Finsler manifolds turns out to be much more general than the riemannian one, not only since it allows to describe a wider class of DS’s, but also because it gives more reliable results in all the situations the Riemannian approach get in troubles, as for few dimensional DS’s, at the border between regularity and Chaos, when the quasi integrable features of the dynamics, together with the small number of degrees of freedom, causes the results obtained within the riemannian approach to become noisy and less decipherable. Some of the results obtained for two and three degrees of freedom systems will be presented in the next sections, whereas a more detailed account, for both low and high dimensional DS’s, will appear elsewhere, [Cipriani and Di Bari 1997b, Di Bari and Cipriani 1997b] (see also [Cipriani and Di Bari 1997a]). Some preliminary results have been accounted for in [Di Bari et al. 1997, Di Bari and Cipriani 1997a]. 2 Being however relatively straightforward the inclusion of explicitly time dependent potentials. 7 Riemann Manifold: Jacobi Metric Exploiting the least action principle for holonomic conservative systems, in the formulation given by Maupertuis, the geometrization procedure is accomplished, leading to the well known Jacobi metric, e.g., [Arnold 1980]. The relation between the geodesic parameter and the newtonian time is given by the transformation’s law p √ dsJ = E − U(x) 2T dt , (24) 1 where E is the total energy, U(x) and T = aij ẋi ẋj are the potential and kinetic energies, respectively. 2 It is easy to see that the function fJ (x, ẋ) p √ fJ (x, ẋ) = E − U(x) 2T (25) satisfies all the conditions a), b) and c), it is not necessary to introduce the function F , and then the Jacobi metric is N dimensional and reads gij = [E − U(x)] aij i, j = (1, . . . , N ) . (26) Finsler Metric If we want to be able to cope with the most general, possibly peculiar, DS, we are naturally led to the introduction of the Finsler metric. While its derivation can be also obtained in a customary fashion, [Rund 1959], we will keep the spirit of the general presentation above: starting from the Hamilton leastaction principle, a new evolution parameter is defined through the Lagrangian of the system, L dsF = L(t, x, ẋ)dt . (27) In the formalism of equations (6) and following, we have now that the transformation law could in principle involve also the time t, fF = L(t, x, ẋ) . In addition to the possible explicit time dependence, the requirement to fulfil condition b), eq.(14), enforces to adopt the more general transformation law, introducing a new parameter w and defining the function of (2N + 2) arguments,   x′ α ′α F (x , x ) = L t, x, ′0 x′0 x α = (0, 1, . . . , N ) . (28) If we are dealing with a conservative system whose lagrangian function is 1 L = aij ẋi ẋj − U(xi ) , (29) 2 then the function F reads 1 (30) F = ′0 aij x′i x′j − U(xi )x′0 . 2x It is an easy, though tedious, task to derive the metric, [Di Bari 1996], which is now (N + 1)-dimensional, and depends, as remarked previously, also on the velocities, x′α : g00 = 1 ∂2F 2 = 3T 2 + U 2 , 2 ∂ 2 (x′0 )2 g0a = x′b 1 ∂2F 2 = −2T a , ab 2 ∂x′0 ∂x′a x′0 gab =
−2 1 ∂2F 2 + aab (T − U) . = aac abd x′c x′d x′0 ′a ′b 2 ∂x ∂x (31) 8 Instability and Geodesic Deviation Equation Within the framework of the GDA, the tool used to investigate the stability of the geodesic flow is given by the Jacobi–Levi-Civita (JLC) equations for the geodesic spread, [Synge 1926], see also [Pettini 1993, Cipriani 1993, Di Bari 1996], which involve the curvature properties of the manifold through the generalization of the ”Riemann” curvature tensor, Rα βγδ . We refer to the bibliography cited above, or also to [Cipriani and Di Bari 1997a], for more details. Briefly, if we want to follow the evolution of two initially close geodesic, whose separation at the time s = 0 was described by the vector of the tangent space z α , α = (0, 1, . . . , N ) we have to find the solutions of the JLC equations, which, in local coordinates, read   dxβ γ dxδ ∇ ∇z α = −Rα βγδ z = −Hα γ z γ , α = (0, 1, . . . , N ) , (32) ds ds ds ds where ∇/ds is the total (covariant) derivative along a geodesic, ∇v α def dv α ′β γ = + Γα ; βγ x v ds ds (33) and we introduced the so-called stability tensor H, [Cipriani 1993, Cipriani and Di Bari 1997a]. If we consider the most general case, we have an (N + 1)-dimensional Finsler manifold, and the curvature tensor also depends on both coordinates and velocity components along the geodesic, [Rund 1959],   ∗α   ∗α η η ∂Γ∗α ∂Γβδ ∂Γ∗α ∂Γβγ βγ ∂G βδ ∂G ∗η ∗α ∗η − + Γ∗α (34) − − Rα βγδ (xη , x′η ) = ηδ Γβγ − Γηγ Γβδ , ∂xδ ∂x′η ∂x′δ ∂xγ ∂x′η ∂x′γ where 1 α ′β ′δ Γ x x , 2 βδ   ∂Gη ∂Gη ∂Gη = Γγδβ − g αγ Cβαη ′δ + Cδαη ′β − Cδβη ′α ∂x ∂x ∂x Gα = (35) Γ∗γ δβ (36) and Cαβδ = 1 ∂gαβ . 2 ∂x′δ (37) If we restrict to the Riemannian case g = g(xα ), the symbols Cαβγ identically vanish, so that the Finsler α α connections Γ∗α βδ reduce to the usual Christoffel symbols Γβδ and the generalized curvature tensor R βγδ becomes the Riemann tensor. If, moreover, we consider a conservative system, there is no need, using the Jacobi metric, to introduce the additional parameter, and all the indices take only the values from 1 to N (and will be used latin indices). As we are interested in the global asymptotic behaviour of closeby geodesics and we don’t worry about the detailed evolution of all the components of the perturbation, in analogy with what has been done within the hamiltonian framework, we define an instability exponent, which is a quantitative indicator able to detect the possible exponential growth, in terms of the geodesic s-time, of the magnitude of the perturbation to the given geodesic:   z(s) 1 . ln δI = lim lim s→∞ z(0)→0 s z(0) (38) The magnitude of the perturbation z(s) is now defined in terms of the natural norm on the manifold def z 2 (s) ≡ kz(s)k2 = gαν z α (s)z ν (s) , (39) 9 where we have understood the dependence of the metric on s, through its arguments, xα = xα (s) and x′α = x′α (s). As we have seen above, the equations of motion of the newtonian mechanics are completely equivalent to the geodesic equations, which give rise exactly to the same trajectories, once rephrased the new time s in terms of the old one t, exploiting the defining realtionship amongst them, ds = f dt. Then we are left with a question: the equations which determine the evolution of the perturbations within the two framework, namely the tangent dynamics equations and the JLC ones, are also completely equivalent each other? Stated otherwise, the issue of the stability of motion will receive the same answers, irrespective of the tools used to investigate it? The answer to this question is in general negative; i.e., while the equations which describe a single trajectory coincide, once rephrased in terms of the same evolutionary parameter, the variational equations do not! Doubtless, in all those cases in which the issue about the occurrence of Chaos received an unambiguous answer, we expect that the differences among the distinct approaches do not reflect in qualitative disagreement. But, as remarked in the Introduction (see also [Cipriani and Di Bari 1997a]), we claim that the search for the deepest insights on the very origin of the onset of Chaos must be performed just at the border between regular and unstable behaviours. From there the most intringuing questions can arise, there the most paradoxical, and even contradictory, situations can occur and in this intermingled layer between Order and Chaos we have to look in order to improve our understanding of the phenomena, and even, perhaps, the very definitions we are trying to fit in the larger and larger spectrum of behaviours we are observing. So, we expect actually the same answers from all the approaches when the DS3 we are studying is either definitely regular or surely chaotic. Nevertheless, we cannot exclude that in the intermediate cases, either for a system whose global parameters assume values still in a region in which the instability is not fully developed, or because we are exploring a region where the structure of the phase space is densely covered by intermingled subsets of regular and chaotic character, the various tools can lead to different answers. This is exactly what happens in one single case presented below, whose explanation is rich of very interesting hints on the sources of Chaos in few dimensional DS’s and is discussed in details in [Cipriani and Di Bari 1997b]. Before to present the results of the comparison between the approaches, a short list of remarks seems to be advisable. • Among the undeniable merits the proposed approach possess, one can be fully appreciated mainly within gauge theories, as in the case of General Relativistic dynamical systems, which, incidentally, do not involve only the studies on cosmological models, but also the analysis of the motions of particles in strong gravitational fields, as accretion disk’s particles around neutron stars or black holes, the orbital motion of tightly bound binary stars, or even the GR corrections to the motion of small bodies in the solar system. The GDA is invariant by construction with respect to any arbitrary change of coordinates and time. • This property is not shared by the usual tangent dynamics equations, which are definitely not invariant after a time rescaling. This property can in the general case affects only quantitatively the calculations of the LCN’s, but in some particular situations can even change the qualitative outcomes, even when the rescaling is very smooth. 3 Or the region of its phase space we are investigating. 10 • To see why the usually adopted equations for small perturbations are not the most natural ones, to which to allot some privileged role, it suffices to look at the more or less equivalent geometrization leading to the Eisenhart metric, [Eisenhart 1929, Pettini 1993]. In this case, indeed, one can recover a form of the JLC equations which is completely equivalent to the tangent dynamics ones, through an affine parametrization of the geodesics. But, from the derivation itself, it is clear that this choice is only one among many possible. • The Eisenhart geometrization can be obtained also from a slight generalization of the procedure outlined above, only for N degrees of freedom conservative holonomic DS’s, starting from the (N + 1)-dimensional configuration space-time, and introducing a further parameter, sE , just in order to achieve an affine relation between the latter and the newtonian time, t ∼ x0 . The manifold then turns out to have (N + 2) dimensions and the line element reads ds2E = aij dxi dxj − 2U(x) (dx0 )2 + 2dx0 dxN +1 , (40) where xN +1 is the additional coordinate, that can be chosen as N +1 x 2 (t) = A t + B − Zt L dt , (41) 0 with A and B constant. As remarked above, with this choice the parametrization turns out to be affine: ds2E = A2 dt2 . • Nevertheless, the possibility to frame the tangent dynamics equations, i.e., one of the most used tools to investigate the chaotic behaviour of DS’s, into a geometric picture, is in itself a neat indication of the relevance the GDA can hold and a spur to address to it the attention deserved. • We finally emphasize the wider applicability of the Finsler approach, which is not limited to holonomic systems, and whose reliability will turn out clearly in the applications we sketch below. We now present some results (for a complete account on them, see [Cipriani and Di Bari 1997b] and [Di Bari and Cipriani 1997b], respectively) obtained from the application of both the GDA’s and the tangent dynamics equations to the two few dimensional dynamical systems mentioned before. In order to have a more understandable check about the equivalence of equations, we get rid (only here!) of the covariant form of the JLC equations, exploiting the definition of total covariant derivative, eq.(33), and of the curvature tensor, eq.(34), and expressing all the derivatives in terms of the newtonian time t. The equations to be compared with the tangent dynamics equations, eqs.(3), result much more involved when expressed in a form which is not covariant, and in terms of a parameter which is not the natural one. It is clear nevertheless that we couldn’t complete the steps in the opposite direction: while the change of the time parameter can be performed in the backwards direction, it not possible to start from eqs.(3) and to give them a covariant form. When the two steps have been done, and assuming for sake of simplicity that the kinetic energy form is euclidean (aij = ηij ), the JLC equations, (32), in the Jacobi and Finsler metrics read, respectively,      ∂ ∂ 1 dU 1 dU U,i + ż p p ; (42) (U,j z j + ẋp ż p ) + ẋi z p p z̈ i + U,ij z j = − E−U ∂x E − U dt ∂ ẋ E − U dt 11 and z̈ i + U,ij z j = −2ż t U,i + z̈ t ẋi , (43) where ∂ ż t dU (1 + U) + 2z i i z̈ = 4 L dt ∂x t  1 dU L dt  ∂ + 2ż ∂ ẋi i  1 dU L dt  . (44) The equations for the disturbances in the Finsler case have been splitted for α = (1, . . . , N ) and α = 0, and we remark that the latter appear at the r.h.s. of equation (43), i.e., the acceleration of the 0th component of the perturbation, influences all the other spatial components. From a direct comparison of the equations above, (42)÷(44), a striking difference with respect to the simpler eqs.(3) is apparent. Nevertheless, as emphasized above, we expect that they give the same results when used to detect instability in situations in which the behaviour is either markedly regular or strongly chaotic. And this is actually what happen. Numerical applications In this section we present some of the results obtained by numerical integration of the dynamics, obtained either from equations of motion (1) or from the geodesic equations, (7) along with the simultaneous numerical integration of the equations for the perturbations, eqs.(3) and (42)÷(44) for two DS’s, namely the Hénon–Heiles Hamiltonian and the Bianchi IX model. Hénon Heiles dynamics This simple dynamical system has raised to the role of a testing ground for any proposed method to investigate the chaotic dynamics of few dimensional DS’s in general, [Benettin et al. 1976], and in particular it has become a paradigmatic example of the process of onset (or, perhaps better, the diffusion) of Chaos along with the variation of a parameter of the system, [Hénon and Heiles 1964], in this case the energy itself playing this part. As it is well known, the Hénon-Heiles (HH) potential represents a two dimensional system introduced more than three decades ago, [Hénon and Heiles 1964], in order to explain some features of the motion of stars in a galactic potential. Since then, as said above, after the first surprising results obtained there and in subsequent studies, its relevance as an astronomical models has been entirely overshadowed by its intriguing features as DS. Among its interesting properties, because the relevance it holds in the present context, it emerges the rather abrupt transition from a quasi completely integrable dynamics at low en< 0.1, to a relevantly chaotic behaviour above this threshold, 0.1 < E ≤ 0.166, up to the escape ergy, E ∼ ∼ energy E = 1/6. The origin of this sudden modification of the qualitative behaviour of trajectories has been investigated in details; amongst them we recall the Toda’s attempt, [Toda 1974], towards a synthetic criterion to detect the onset of instability. While unsuccessful, [Benettin et al. 1977], that work spurred a lot of efforts devoted to an understanding more than phenomenological on the dynamical instability, [Cerruti-Sola and Pettini 1996, Cipriani and Di Bari 1997b]. It is also interesting to note that below the stochasticity threshold the classical perturbation theory gives excellent agreement with the outcomes of numerical experiments, [Lichtenberg and Lieberman 1983, §1.4a], whereas above that level the numerical phase portrait of the system turns out more and more different from that obtained analitically. This sudden breaking down of another quasi conserved quantity can help to shed light also on the behaviour 12 of higher dimensional DS’s. The Hénon-Heiles Hamiltonian reads H= ẋ2 ẏ 2 + + U(x, y) , 2 2 (45) with potential U(x, y) =
1 1 2 x + y 2 + x2 y − y 3 . 2 3 (46) We report in figure 1 the well known, e.g.,[Benettin et al. 1976], behaviour of LCN’s versus time (in a log-log scale) at the intermediate energy E = 0.125, for which there is a coexistence between regular and chaotic zones, for a representative set of initial conditions, (i.c). As we can see, the signature of Chaos is neatly detected, looking at the markedly different behaviours of the LCN’s. After some transients, and we remember that LCN’s are indeed asymptotic indicators, the definitely regular orbits, labeled by the first two digits, 1 and 2, show an unambiguous behaviour indicating the absence of any exponential instability: LCN (t) ∼ ln t −1 −→t , t t→∞ for i.c. 1 and 2 . (47) Analogously, the three surely chaotic orbits, emanating from i.c. 4÷6, show a clear-cut indication of a common exponential rate of growth of the euclidean norm of the deviation vector D. At variance, the initial condition labeled as 3 manifests a behaviour which is much more noisy than the other orbits, though it seems to give a nevertheless convincing indication of stable behaviour. We tested the peculiarity of this and associated orbits, i.e., those emanating from the same island on the Poincaré surface of section, x = 0, (PSS), looking at all the possible sources of the fluctuations, and even implementing a version of the renormalization procedure slightly modified with respect to the universally adopted one, proposed by [Benettin et al. 1976], which occasionally proved to be more convincing, see [Cipriani and Di Bari 1997b]. Moreover, while all the other initial conditions tested do not show any perceptible dependence with respect to a change on (the orientation of) the initial perturbation, D(0), the behaviour of the LCN for orbit 3 is sensitive, in the absolute value as well in the overall behaviour, to this choice, as shown by the curve labeled as 3′ in fig.1, obtained starting from the same i.c., but selecting a very bad orientation of the initial perturbation. With this choice the noise is quite relevant, neither is sensibly reduced performing a longer integration run, as can be seen looking at the small panel in the same figure. On the basis of these outcomes, we claim that, though almost certainly regular according to the accepted canonical definitions, the orbits emanating from the island(s) on the ẏ axis in the PSS, possess some characteristics which make them qualitatively slightly distinct with respect to the ones belonging, for example, to the main island on the y positive semi-axis or also to the orbits starting from inside the banana shaped region on the y < 0 side. As a matter of fact, we can roughly guess from figure 1 that the behaviour of the perturbation for this set of i.c., is not simply polynomial in time, but shows some intermittent phases of instability, probably of stretched exponential nature. This initially regarded as annoying behaviour has been instead revealed very fruitful, as it has persuaded to look more carefully to the proposed geometrical origin of instability and allowed us to find some convincing explanations for those peculiarities, which turn out to be very enlightening for a deeper understanding of the sources of Chaos, [Cipriani and Di Bari 1997b]. As heralded above, we do not expect to find any relevant difference in analyzing the qualitative properties 13 of the dynamics of the Hénon and Heiles system, when it displays a well defined nature. And actually we can see from figure 2 how the instability exponents defined in eq.(38) for the Jacobi metric give, in such situations, substantially the same answers. In fig.2 we have plotted the instability exponents, δIJ , (indicated simply as δJ in the figure) obtained by the numerical integrations of the geodesic equations for this metric, along with the corresponding geodesic deviation equations, in terms of the parameter √ sJ defined through the relation dsJ = 2T dt, in turn integrated simultaneously to the previous ones, in order to stop the run at the same value of the newtonian time t reached with the tangent dynamics equations. A part from a trivial numerical multiplicative factor, related to the non affine reparametrization, because of which the Jacobi parameter is always smaller (for the HH system at this energy) than the newtonian time, and consequently the instability exponent turns out to be greater than the LCN by the inverse factor4 , t √ Z √ E −−−→ 0.089 t ∼ t/10 ; sJ (t) = 2 T (t′ )dt′ = 2hT it t ∼ = √ t− 2 E=0.125 0 we find out a completely satisfactory agreement for the truly chaotic orbits, whereas, for the quietly regular ones, the qualitative agreement is equally convincing, though the behaviour of the δIJ is more noisy that that of the corresponding LCN’s. But the most striking difference concerns the answer given by the Jacobi GDA to the issue of the regularity of orbits of the (upper) island in the PSS: according to fig.2, orbit with i.c. labeled as 3 turns out to possess a strong dependence on initial conditions! In order to discard the possibility of an artifact due to a too short integration time, in the small frame in figure 2 we plot the long term behaviour of the δIJ , which highlights once more the firm decision of Jacobi geodesic deviation equations to depict as chaotic that orbit. While it is not difficult to understand to source of noise in the δIJ of regular orbits, 1 and 2, it turns out a definitely not easy task to single out the sources of such a surprising behaviour of the third one. The answers to these questions reside partially on the singular behaviour of the conformal factor fJ (x, ẋ) and consequently of the Jacobi metric itself, when the kinetic energy vanishes. Such an occurrence can be ruled out in the case of many degrees of freedom systems, [Cipriani and Di Bari 1997a], but cannot be excluded in the case of few dimensional DS’s, as in some cases the orbit can touch the zero velocity surface (or curve, when N = 2), which is the boundary, for standard DS’s5 , of the region allowed to motion. These singularities suffice to explain the oscillatory behaviour of instability exponents of the two regular orbits, but alone cannot help to grasp the origin of such a spiteful performance of the third one. Actually its very peculiar behaviour has been for us a very strong push to go beyond the first attempts to relate curvature and stability, and provided very deep hints on the relationhips between geometry and chaos, about which a detailed account can be found in [Cipriani and Di Bari 1997b]. Incidentally, we note that in the very interesting paper by [Cerruti-Sola and Pettini 1996], where the qualitative character of orbits has been investigated by means of an equation derived by the JLC ones, the behaviour of this (or similar) orbit has not been analyzed, whereas, here, we recover (see below) also a complete quantitative agreement 4 In the following formula the estimate on the average value of the kinetic energy is obtained exploiting the virial theorem in an approximated form. 5 As we will see below, this is not the case for General Relativistic DS’s, for which the zero kinetic energy surface do not coincide with the surface of zero velocities, as the kinetic energy form is not positive definite. Moreover, for standard (conservative) DS’s, the surface of zero kinetic energy is the (N − 1)-dimensional manifold {q, p} : {U (q) = E ; p = 0}, whereas, for non positive definite kinetic energy forms, the T = 0 manifold has in general dimension (2N − 2): {q, p} : {U (q) = E ; aij pi pj = 0}. 14 (once exploited the relationship among sJ or sF and t) between the tangent dynamics equations and the exact JLC equations, in the entire PSS x = 0. In [Cipriani and Di Bari 1997b], we moreover discover a geometric indicator of instability able to go beyond the Toda criterion, in that it can distinguish among chaotic and regular orbits, at the same energy. The singularity of the Jacobi metric mentioned above do not occur within the Finsler GDA, as the relationship between the geodesic parameter and the newtonian time is governed by the function F (xα , x′α ) defined above, which is derived from the lagrangian of the system, L(x, ẋ, t). It is well known, e.g. [Goldstein 1980], that, given a DS, the lagrangian function is defined up to an additive gauge function, m(x, t), which must only fulfil the condition to be the total derivative with respect to time of any differentiable function M (x, t) defined on configuration space. In the numerical study of the geodesic flow derived from the HH potential, we actually exploited this freedom in order to obtain an average equality among newtonian time and Finsler parameter; i.e., we chosen L̄(x, ẋ, t) = L(x, ẋ, t) + m(x, t) and consequently sF (t) = Zt L̄[x(t′ ), ẋ(t′ ), t′ ] dt′ = 0 Zt 0 L[x(t′ ), ẋ(t′ ), t′ ] dt′ + M [x(t), t] − M [x(0), 0] in such a way that hL̄it ∼ = 1, i.e., on the long run, sF (t) ∼ = t. This choice obviously didn’t modified the results, but only allowed a direct quantitative comparison between the digits of the Finsler instability exponents and the LCN’s. In figure 3 we show the sF -behaviour of the δIF’s (they also simply indicated as δF in the figure), for the same set of i.c. used to compute the LCN’s and the δJ ’s. The complete, quantitatively detailed, agreement with the results of the tangent dynamics equations is strikingly evident for all the i.c. analyzed, except again for the orbit with label 3. But, at variance with what occurred in the Jacobi description, now the results achieved within the GDA show a much less noise than the LCN’s! And the answer about the issue of instability of these initial conditions is strikingly definite. Moreover, we tested the behaviour of the δF ’s varying the initial orientation of the perturbation, being not able to distinguish between various choices, already at relatively small values of sF . Nevertheless, we didn’t content ourselves with the nice performances of the Finsler GDA and investigated both the reasons for a so satisfactory result and the causes at the origin of the seemingly deceptive outcomes of the Jacobi description. As remarked above, this at first sight misleading result of the Jacobi approach, has provided very useful insights on the analysis on the sources of instability of DS’s and of its relationships with the curvature properties of the manifold. A full account of the results achieved is presented in [Cipriani and Di Bari 1997b]. Bianchi IX dynamics In order to point out once more the power of the GDA and, at the same time, the cautions needed in order to avoid a too literal extrapolation to realistic DS’s of the results borrowed from the abstract Ergodic Theory, e.g.,[Chitre 1972, Szydlowski and Krawiec 1993], we then performed a numerical study of the dynamics of a General Relativistic three degrees of freedom dynamical system, which share with the Hénon-Heiles hamiltonian the strange destiny of becoming ever and ever more interesting as a paradigmatic DS within the theoretical context to which belongs (Celestial Mechanics and General Relativity, respectively), than as realistic model of the physical phenomenon it was intended to describe originally. The interest of this DS relies on the lively debate it raised among the cosmologists, since the last three 15 decades, e.g.,[Belinskii et al. 1970], and more recently even between people working on Ergodic and Dynamical Systems Theories, [Khalatnikov et al. 1985] and [Contopoulos et al. 1995, Hobill et al. 1994]. Indeed, amongst the mess of studies performed until recently, of both analytical and numerical character, about half of them were reporting convincing evidences on the regular features of the BIX dynamics; whereas the others claimed the existence of strong indications about the chaotic properties of the model. In this very last period, it seems to be in fashion the belief that this DS is indeed chaotic, [Cornish and Levin 1997], also because some amongst the previous results indicating perhaps even its integrability have been corrected, [Contopoulos et al. 1995]. It had been already suggested that these disagreements depend on the different time parameter adopted to describe the dynamics, [Hobill et al. 1991]. Actually, in a General Relativistic context, the gauge freedom in the selection of time and coordinates can be exploited with much more liberties, as the results should be invariant with respect to the choice made. The dynamics of the so-called Bianchi IX cosmological model6 , can be studied with a variety of approaches, although the most suitable from a DS viewpoint is actually the hamiltonian one. With an appropriate choice of variables, the BIX Hamiltonian reads H= 1 1 1 2 β̇ + β̇ 2 − α̇2 + U(α, β+ , β− ) , 2 + 2 − 2 (48) with potential U= 1 4α e 8   2  i h  √  √ 1 −8β+ 4 −2β+ , − e cosh 2 3β− + e4β+ cosh 4 3β− − 1 e 3 3 3 (49) where β+ , β− and α are functions of the scale factors a,b,c of the Universe, and measure the overall volume and the anisotropy of the 3-dimensional hypersurface τ = const.; the dot indicates differentiation with respect to the time parameter τ , defined in terms of the cosmological proper time t through the transformation dτ = dt/abc. The Hamiltonian (48) in nevertheless peculiar, in that the General Relativity imposes a constraint not encountered usually in classical DS’s: Einstein equations request that H be a null hamiltonian, i.e., H ≡ 0. The possibly chaotic properties of this relativistic dynamical system could be a signal about the occurrence of some kind of statistical behaviour of the primordial Universe, able to explain some observational evidences without the introduction of exotic theories, as the Inflation. Nevertheless, as remarked above, in the last years, it has assumed importance as a dynamical system in itself, because of the strikingly conflicting answers about its chaotic behaviour. It has become clear that the discrepancies could arise from different choices of the time parameter, and even some analytical results obtained by a discretization of the flow through the introduction of different maps, can hide some inaccurate approximantions, which turns out to be responsible of the misleading conclusions about the stochastic character of the dynamics. We refer to [Hobill et al. 1994] for an exhaustive review of the existing treatments, and to [Di Bari and Cipriani 1997a] and [Di Bari and Cipriani 1997b], respectively, for preliminary and detailed results. General Relativity requires covariance and so it is essential to analyze the Bianchi IX dynamics with methods whose results are independent of the choice of the coordinates, and, in particular, of the time variable. For this reason, we decided to implement the GDA to the study of its possibly chaotic behaviour. At the point we should make a comment: as it is well known, in order to properly speak about Chaos, it 6 See, e.g.,[Landau and Liftsits 1980, §§115÷118] and [Hobill et al. 1994] and references therein for a description of its original motivations and a discussion on the issues still open. 16 is necessary that the dynamics possess a strong dependence on intial conditions and moreover that the ambient space where the dynamics is decribed (the phase space within the Hamiltonian treatment) must be compact, i.e., the stretching caused by the exponential instability, must be accompanied by the folding due to the bounds imposed to the distance between any two point in this space by compactness. Looking at the Hamiltonian (48), we see at once that the second condition mentioned above is not fulfilled by the BIX dynamical system; so, even if we could be able to prove the occurrence of an exponential instability, it should not be rigorous to speak about Chaos, though it can be argued that when the dynamics shows an exponential instability characterized by a given time-scale τi , if the trajectory of the system remains trapped in an unstable region of phase space (or something else) for an interval of time Ttr much longer than the instability time, Ttr ≫ τi , then the effects on the predictability of the evolution would result substantially the same as if it were a truly chaotic dynamics. We will see below, however, that in this case we don’t need to deepen the reliability of this argumentation, as the BIX dynamics do not shows any exponential instability! The second preliminary observation concerns the possible geometrizations of this DS. It was already recognized that a geometrical description of the BIX dynamics could help to get rid of any gauge dependent approach, [Chitre 1972, Szydlowski and Krawiec 1993]. Nevertheless, these authors didn’t paid enough attention to the peculiar features imposed by the General Relativity to this particular DS. Indeed, the Hamiltonian (48), though in its own way, describes however a conservative dynamical system. On this light, in order to give a geometrical description of the dynamics, those authors exploited the Jacobi metric, using the line element defined above, i.e., the invariant Jacobi parameter dsJ = √ √ E − U 2T dτ , (50) where U is the potential defined in (49) and the kinetic energy T turns out from (48): T = 1 1 1 2 β̇ + β̇ 2 − α̇2 . 2 + 2 − 2 (51) It is evident why this treatment cannot be pursued further: it is true that the Jacobi parameter is invariant, but this make even more serious its singularities, repeatedly occurring along the evolution of the system, when the trajectory cross the zero kinetic energy surface ST =0 , which, because of the Hamiltonian constraint, coincides also with the surface where the potential U vanishes. As heralded above, this DS gives an example of non coincidence between the zero kinetic energy and zero velocities surfaces. Now, actually, the former no longer constitutes the external boundary of the region allowed to motion in configuration space, but it is instead inside that region, so it can be crossed by the actual trajectory infinitely many times during the evolution, and with whatever incidence angle. The comments made in the footnote 5 about the different dimensionalities of the manifolds of vanishing kinetic energy for standard and pecular DS’s help to intuitively grasp the very different chances of such an occurrence in the two situations. Incidentally, we observe that the interpretation of the results obtained within this (though unreliable) approach are nevertheless easily understandable: as the kinetic and potential energies along a trajectory equal each other, except for the sign, and as the potential is appreciably different from zero only near the walls, [Di Bari 1996, Di Bari et al. 1997], we see at once that the Jacobi description amounts essentially to reduce the continuous flow to a discrete bouncing map, neglecting the ever increasing amount of time elapsed between successive bounces. This leads to an overestimate of the instability rate, which is discussed in details in [Di Bari and Cipriani 1997b]. So, while the Jacobi GDA gives very satisfactory performances when applied to standard DS’s with many 17 degrees of freedom (see also [Cipriani and Di Bari 1997a]), it breaks down when applied to peculiar few dimensional spaces. These difficulties of the geometrical method have raised some criticisms against the overall approach, e.g.,[Burd and Tavakol 1993], and discouraged for some time any further attempt. But, in these cases again, the GDA on Finsler spaces reveals all its potential, allowing a geometrical, invariant by construction, transcription of the dynamics even of strongly singular DS’s. Indeed, in figure 4 we plot the instability exponents δIF , computed using the Finsler metric, for a small number of typical orbits, chosen from a nevertheless representative set of initial conditions, leading to distinct overall behaviours (for details see [Di Bari and Cipriani 1997b]). It is evident at once that we can discard the possibility of the occurrence of any global exponential instability. The presence of some light peacks, which become ever and ever smaller going towards the singularity7 , can be detected looking carefully at the sF behaviour of the instability exponents. These positive peacks, which tend for a while to stop the almost uniform decrease of δIF , occurs every time there is a transition from one Kasner epoch to another one, or within the minisuperspace picture, [Hobill et al. 1994], to the bounces of the trajectory against the potential walls. They reflect the occasional occurrence of an instability phase whose duration is however very small with respect to the ever increasing interval between two successive phases. On this light, this asymptotic behaviour of the instability exponents could partially support some recent claims about the occurrence in the BIX dynamics of a sort of chaotic scattering, as suggested in [Contopoulos et al. 1995] and [Cornish and Levin 1997]. For a more detailed analysis of this and other issues, we refer to [Di Bari and Cipriani 1997b], where it is also investigated the relationships existing between the dynamical features of th BIX model and the curvature properties of the associated Finsler manifold. Conclusions In this paper we presented an alternative derivation of the procedure which leads to the transcription of the dynamical properties of general lagrangian systems in geometrical terms. Using this elementary derivation, we shown the complete equivalence of first order variational principles, leading to the equations of motion and to geodesic equations, within the two frameworks, respectively. Then we addressed to the issue of the possible equivalence of second order variational equations, which convey the informations related to the equations describing the behaviour of the disturbances, i.e., the perturbation vectors in phase space, and the geodesic deviation vectors on the tangent space of the manifold, whose evolution is determined, respectively, by the usual tangent dynamics and the Jacobi– Levi-Civita equations. We compared the explicit expression of the former with those resulting by two specific geometrical transcriptions, and, once recognized the non coincidence, we tested their however possible equivalence through the application to two different but paradigmatic DS’s. We found a completely satisfactory agreement between the results obtained in the usual hamiltonian setting and those coming from the GDA. Moreover, the geometrical tools for the investigation of the stability properties of the dynamics show nevertheless some minor differences which, when fully exploited, bring very enlightening suggestions on the relationships between the curvature properties of the manifold and the possible onset of Chaos. A full account of the results of the numerical experiments along with the implications of these inves7 We recall that the BIX model represents a closed universe, with a Big Bang, followed by an anisotropic expansion which becomes more and more isotropic towards the maximum, and then a collapse to a Big Crunch, which tends to anisotropize again for |τ | → ∞. 18 tigations will be presented in [Cipriani and Di Bari 1997b], devoted to the Hénon-Heiles Hamiltonian, and in [Di Bari and Cipriani 1997b], for the Bianchi IX dynamical system. They, along with the existing results, obtained by other researchers, on both many degrees of freedom systems, [Pettini 1993], and few dimensional ones, [Cerruti-Sola and Pettini 1996], and by the authors, [Cipriani 1993, Di Bari 1996] and [Di Bari et al. 1997, Cipriani and Di Bari 1997a], strongly support the claims on the reliability of the GDA, and nevertheless point out the need of a careful check of the conditions needed in order to apply safely the geometrical transcription of the dynamics, to fully exploit the power of a tool which can give very deep insights on the origin of instability in DS’s, and also interesting hints to single out better criteria and/or indicators of chaotic behaviour, in particular in all those situations, and there are many of such a kind, where the boundary between Order and Chaos is so nuanced. 19 References [Arnold 1980] Arnold V.I. (1980) Mathematical Methods of Classical Mechanics Springer-Verlag. [Belinskii et al. 1970] Belinskii V. A., Khalatnikov I. M. and Lifshitz E. M. (1970) Adv.Phys., 19 , 525. [Benettin et al. 1976] Benettin G., Galgani L. and Strelcyn J. M. (1976) Phys.Rev. A, 14, 2338 [Benettin et al. 1977] Benettin G., Brambilla R. and Galgani L. (1977) Physica, 87A, 381 [Benettin et al. 1980] Benettin G., Galgani L., Giorgilli A. and Strelcyn J. M. (1980) Meccanica, 15, 9; ibidem page 20. [Burd and Tavakol 1993] Burd A. and Tavakol R. (1993) Phys. Rev. D, 47, 5336. [Cerruti-Sola and Pettini 1996] Cerruti-Sola M. and Pettini M. (1996) Phys Rev. E, 53, 179. [Chitre 1972] Chitre D. M. (1972) Ph.D. Thesis, Univ. of Maryland. [Cipriani 1993] Cipriani P. (1993) Ph.D.Thesis, Univ. of Rome “La Sapienza”, (in italian). [Cipriani and Di Bari 1997a] Cipriani P. and Di Bari M. (1997) Planetary and Space Science, (this issue), xxx. [Cipriani and Di Bari 1997b] Cipriani P. and Di Bari M. (1997) submitted. [Contopoulos et al. 1995] Contopoulos G., Grammaticos B. and Ramani A. (1995) J. Phys. A, 28, 5313. [Cornish and Levin 1997] Cornish N. J. and Levin J. (1997) Phys. Rev. Lett, 78, 998. [Di Bari 1996] Di Bari M. (1996) Ph.D.Thesis, Univ. of Rome “La Sapienza”, (in italian). [Di Bari and Cipriani 1997a] Di Bari M. and Cipriani P. (1997) in Proc. of 12th Italian Conference on General Relativity and Gravitational Physics, World Scientific, xxx. [Di Bari and Cipriani 1997b] Di Bari M. and Cipriani P. (1997) submitted. [Di Bari et al. 1997] Di Bari M., Boccaletti D., Cipriani P. and Pucacco G. (1997) Phys. Rev. E, 55, 6448. [Eisenhart 1929] Eisenhart L. P. (1929) Ann. of Math. 30, 591. [Goldstein 1980] Goldstein H. (1980) Classical Mechanics (2nd edition), AddisonWesley. 20 [Hénon and Heiles 1964] Hénon M. and Heiles C. (1964) Astron. Journ., 69, 73. [Hobill et al. 1994] Hobill D., Burd A. and Coley A. (1994) Deterministic Chaos in General Relativity, Plenum. [Hobill et al. 1991] Hobill D. W., Bernstein D., Welge M. and Simkins D. (1991) Class. Quantum Grav., 8, 1155. [Khalatnikov et al. 1985] Khalatnikov I.M., Lifshitz E.M., Khanin K.M., Shchur L.N. and Sinai Ya.G. (1985) Journ. Stat. Phys., 38, 97. [Landau and Liftsits 1980] Landau L.D. and Lifsits E.M. (1980) Classical Field Theory, 3rd edition, Pergamon. [Lichtenberg and Lieberman 1983] Lichtenberg A.J. and Lieberman M.A. (1983) Regular and Stochastic Motion, Springer-Verlag. [Pettini 1993] Pettini M. (1993) Phys. Rev. E, 47, 828. [Rund 1959] Rund H. (1959) The differential Geometry of Finsler Spaces, Springer-Verlag. [Synge 1926] Synge J.L. (1926) Phil. Trans. A, 226, 31. [Synge and Schild 1978] Synge J.L. and Schild A. (1978) Tensor Calculus Dover (reprint of 1949 edition). [Szydlowski and Krawiec 1993] Szydlowski M. and Krawiec A. (1993) Phys. Rev. D, 47, 5323. [Toda 1974] Toda M. (1974) Phys. Lett., 48A, 335. 21 FIGURE CAPTIONS Figure 1: LCN’s for the Hénon–Heiles Hamiltonian at E = 0.125. The initial conditions corresponding to each labeled curve are indicated in the figure. The curves labeled as 3 and 3′ refer to the behaviour of the LCN computed along the same orbit but with two different choices of the initial perturbation. The small frame shows the behaviour of these curves for an interval of time ten times longer. In this panel, the curve labeled as 3 has been shifted upwards, to avoid ovelapping. Figure 2: Instability exponents δJ , calculated in the Jacobi geometry for the HH system. The initial conditions are chosen and labeled as in figure 1. The differences in the vertical and horizontal scales with respect to those of figure 1 are simply due to the relation between sJ and t (see text). The inserted frame shows the long time behaviour (up to t = 105 ) of the instability exponent of the orbit with initial conditions labeled as 3; note that the vertical scale has been rescaled in order to highlight the smaller and smaller fluctuations . Figure 3: Instability exponents δF , calculated in the framework of Finsler gemetrodynamics. The initial conditions are chosen and labeled as in figure 1 . Figure 4: Short (upper panel) and long (lower panel) time behaviours of the instability exponents, δIF , computed in the Finsler framework for the Bianchi IX dynamical system. The initial conditions corresponding to each curve have been chosen to be representative of the overall set of typical behaviours of this system. It is evident the signature of a regular character of the dynamics, and also the stretching of time intervals between two successive peaks . 22 -1 10 4,5,6 10 -3 10 -4 3 3’ -2 10 10 -5 LCN 10 4 10 5 1 -3 10 3 2 -4 10 E=0.125 1 : y=0.2 vy=0.02 2 : y=0.33 vy=0.14 3,3’: y=0.015 vy=0.25 4 : y=0.2 vy=0.14 5 : y=-0.15 vy=0.02 6 : y=0.25 vy=0.3 3’ -5 10 3 10 4 t 10 0 10 4,5,6 1 0 0 3 -1 10 3 1 0 -1 δJ 1 0 -2 10 2 1 0 3 1 0 4 1,2 -3 10 E=0.125 1: y=0.2 vy=0.02 2: y=0.33 vy=0.14 3: y=0.015 vy=0.25 4: y=0.2 vy=0.14 5: y=-0.15 vy=0.02 6: y=0.25 vy=0.3 -4 10 2 10 3 sJ 10 -1 10 4,5,6 -2 10 δF 1 -3 10 2 3 -4 10 E=0.125 1: y=0.2 vy=0.02 2: y=0.33 vy=0.14 3: y=0.015 vy=0.25 4: y=0.2 vy=0.14 5: y=-0.15 vy=0.02 6: y=0.25 vy=0.3 -5 10 3 10 4 sF 10 0 10 -1 10 δI F -2 10 -3 10 0 10 1 10 sF 2 10 3 10 -3 10 δI F -4 10 -5 10 3 10 4 10 sF 5 10