Covariant Lyapunov vectors for rigid disk systems
aa r X i v : . [ n li n . C D ] J un Covariant Lyapunov Vectors for Rigid Disk Systems
Hadrien Bosetti ∗ and Harald A. Posch † Computational Physics Group, Faculty of Physics,University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria (Dated: October 26, 2018)We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard disk system in a rectangular box with periodic boundary conditions. The systemis large enough to allow the formation of Lyapunov modes parallel to the x axis of the box. TheOseledec splitting into covariant subspaces of the tangent space is considered by computing the fullset of covariant perturbation vectors co-moving with the flow in tangent-space. These vectors areshown to be transversal, but generally not orthogonal to each other. Only the angle between co-variant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrummay become small, but the probability of this angle to vanish approaches zero. The stable andunstable manifolds are transverse to each other and the system is hyperbolic. I. INTRODUCTION
Lyapunov exponents measure the exponential growth, or decay, of infinitesimal phase space perturbations of achaotic dynamical system. For a D -dimensional phase space, there are D exponents, which, if ordered according tosize, λ i ≥ λ i +1 , are referred to as the Lyapunov spectrum. The classical algorithm for the computation is basedon the fact that almost all volume elements of dimension d ≤ D in tangent space (with the exception of elementsof measure zero) asymptotically evolve with an exponential rate which is equal to the sum of the first d Lyapunovexponents. Such a d -dimensional subspace may be spanned by d orthonormal vectors, which may be constructed bythe Gram-Schmidt procedure and, therefore, are referred to as Gram-Schmidt (GS) vectors. The GS-vectors are notcovariant, which means that at any point in phase space they are not mapped by the linearized dynamics into theGS vectors at the forward images of that point [1]. As a consequence, they are not invariant with respect to thetime-reversed dynamics. Due to the periodic re-orthonormalization of the GS vectors only the radial dynamics isexploited for the computation of the exponents, whereas the angular information is discarded.Although the angular dynamics is not a universal property and may depend, for example, on the choice of thecoordinate system [2], it would be advantageous for many applications, to span the subspaces mentioned above bycovariant vectors and to study also the angular dynamics of and between these vectors. It has the additional advantageto preserve the time-reversal symmetry for these tangent vectors, a property not displayed by the GS vectors. Recently,an efficient numerical procedure was developed by Ginelli et al. [1] for the computation of covariant Lyapunov vectors.Here we apply their algorithm to a two-dimensional system of rigid disks.The choice of hard elastic particles is motivated by the fact that their dynamics is comparatively simple, andtheir ergodic, structural and dynamical properties are well known and are thought to be typical of more realisticphysical systems [3]. Secondly, hard-particle systems in two and three dimensions serve as reference systems forthe most successful perturbation theories of dense gases and liquids [4, 5]. Finally, the combination of a Lyapunovanalysis with novel statistical methods for rare events [6] seems particularly promising for the study of such raretransformations in systems, for which hard core interactions are at the root.The paper is organized as follows. After an introduction of the basic concepts for the dynamics of phase spaceperturbations in Section II, we summarize in Section III the features and our numerical implementation of thealgorithm of Ginelli et al. [1] for the computation of covariant vectors and covariant subspaces. In Section IV, theH´enon map serves as a simple two-dimensional illustration. The hard-disk model is introduced in Section V. In thiswork we restrict ourselves to 198 disks, a number which is dictated by computational economy, but still large enough toallow the study of Lyapunov modes. In Section V A we study the relative orientations of Gram-Schmidt and covariantvectors, which give rise to the same Lyapunov exponents. Next, in Section V B, we compare the localization propertiesin physical space for these two sets of perturbation vectors. The configuration and momentum space projections ofthe perturbation vectors – Gram-Schmidt or covariant – are the topic of Section V C. The central manifold (or nullsubspace) and its dependence on the intrinsic continuous symmetries – translation invariance with respect to time ∗ Electronic address: [email protected] † Electronic address: [email protected] and space – is discussed in Section V D. Although the null subspace is completely orthogonal to the unstable andstable subspaces, it is essential for a proper understanding of the Lyapunov modes [7, 8]. Section V E is devoted to adiscussion of these modes and how they are represented by the covariant vectors. In Subsection V F we compute theangles between the covariant modes and test for tangency between covariant Oseledec subspaces. In Section VI weconclude with a summary.
II. PHASE SPACE AND TANGENT SPACE DYNAMICS
The dynamics of a system of hard disks is that of free flight, interrupted by elastic binary collisions. If Γ denotesthe state of the system at time 0, the state at time t is given by Γ t = φ t ( Γ ), where φ t : X → X defines the flow inthe phase space X . Similarly, if δ Γ is a vector in tangent space TX at Γ , at time t it becomes δ Γ t = Dφ t | Γ · δ Γ ,where Dφ t defines the tangent flow. It is represented by a D × D matrix, where D is the dimension of phase space.A subspace E ( i ) of the phase space is said to be covariant if Dφ t | Γ E ( i ) ( Γ ) = E ( i ) ( φ t ( Γ )) . (1)This definition also applies to covariant vectors, if E ( i ) is one-dimensional. Loosely speaking, covariant subspaces(vectors) are co-moving (co-rotating in particular) with the tangent flow. An analogous relation holds for the time-reversed flow.Next we consider the decomposition of the tangent space into subspaces according to the multiplicative ergodictheorem of Oseledec [9–11]. Here, we closely follow Ref. [7].The first part of the multiplicative ergodic theorem asserts that the real and symmetric matricesΛ ± = lim t →±∞ (cid:16)(cid:2) Dφ t | Γ (cid:3) T Dφ t | Γ (cid:17) / | t | (2)exist for (almost all) phase points Γ . Here, T denotes transposition. The eigenvalues of Λ + are ordered accordingto exp( λ (1) ) > · · · > exp( λ ( ℓ ) ), where the λ ( j ) are the Lyapunov exponents, which appear with multiplicity m ( j ) .For symplectic systems as in our case, λ ( j ) = − λ ( ℓ +1 − j ) , which is referred to as conjugate pairing. Similarly, theeigenvalues of Λ − are exp( − λ ( ℓ ) ) > · · · > exp( − λ (1) ). The eigenspaces of Λ ± associated with exp( ± λ ( j ) ) are denotedby U ( j ) ± . They are pairwise orthogonal but not covariant. If the λ ( j ) are degenerate with multiplicity m ( j ) = dim U ( j ) ± ,all multiplicities sum to D , the dimension of the phase space. Since the matrices Λ ± are symmetrical, each of the twosets of eigenspaces, { U ( j ) ± } , completely span the tangent space, TX ( Γ ) = U (1) ± ( Γ ) ⊕ · · · ⊕ U ( ℓ ) ± ( Γ ) . (3)The eigenspaces U ( j ) ± are not covariant, but the subspaces U ( j )+ ⊕ · · · ⊕ U ( ℓ )+ and U (1) − ⊕ · · · ⊕ U ( j ) − , j ∈ { , . . . , ℓ } , (4)are. They are, respectively, the most stable subspace of dimension P ℓi = j m ( i ) of Λ + , and the most unstable subspaceof dimension P ji =1 m ( i ) of Λ − (corresponding to the most stable subspace of that dimension in the past).The second part of Oseledec’ theorem asserts that for (almost) every phase-space point Γ there exists anotherdecomposition of the tangent space into covariant subspaces E ( j ) ( Γ ) referred to as Oseledec splitting, TX ( Γ ) = E (1) ( Γ ) ⊕ · · · ⊕ E ( ℓ ) ( Γ ) . (5)For δ Γ ∈ E ( j ) ( Γ ) the respective Lyapunov exponent follows fromlim t →±∞ | t | log k Dφ t | Γ · δ Γ k = ± λ ( j ) ∀ j ∈ { , . . . , ℓ } . (6)The subspaces E ( j ) are covariant (see Eq. (1)) but, in general, not orthogonal. According to Ruelle [10], they arerelated to the eigenspaces U ( j ) ± of Λ ± : E ( j ) = (cid:16) U (1) − ⊕ · · · ⊕ U ( j ) − (cid:17) ∩ (cid:16) U ( j )+ ⊕ · · · ⊕ U ( ℓ )+ (cid:17) . (7)This equation is at the heart of the construction of covariant vectors according to Ginelli et al. as described in thenext section. Furthermore, one can show that F ( j ) ≡ E (1) ⊕ · · · ⊕ E ( j ) = U (1) − ⊕ · · · ⊕ U ( j ) − (8)are covariant subspaces. III. NUMERICAL CONSIDERATIONS
Numerical methods probe the tangent space by a set of D tangent vectors, such that the Lyapunov exponents arerepeated with multiplicities, λ ≥ · · · ≥ λ D . Here, the lower index is referred to as the Lyapunov index. The relationbetween the λ ( j ) and λ i is given by λ ( j ) = λ f ( j − +1 = · · · = λ f ( j ) , where f ( j ) = m (1) + · · · + m ( j ) is the sum of all subspace dimensions up to j .For notational convenience in the following, the vectors g jn , j = 1 , . . . , D spanning the tangent space at time t n ,are arranged as column vectors of a D × D matrix G n ≡ ( g n | . . . | g Dn ). The same convention is used below for otherspanning vector sets such as G n ≡ ( g n | . . . | g Dn ) and V n ≡ ( v n | . . . | v Dn ).In the classical algorithm of Benettin et al. [12] and Shimada et al. [13] for the computation of Lyapunov exponents,an orthonormal set of tangent vectors G n − at time t n − is evolved to a time t n ≡ t n − + τ , ( τ > G n = J τn − G n − , where J τn − is the Jacobian of the evolution map taking the phase space point Γ n − at time t n − to Γ n at time t n .The column vectors of G n at time t n generally are not orthonormal any more and need to be re-orthonormalized witha Gram-Schmidt procedure. This gives the matrix G n with column vectors { g j } n , which form the next orthonormalGram-Schmidt (GS) basis at time t n . These vectors are pairwise orthogonal but not covariant. Each GS renormal-ization step is equivalent to a so-called QR decomposition of the matrix G n , G n = G n R n , where the matrix R n isupper triangular [14]. The diagonal elements of R n are required for the accumulative computation of the Lyapunovexponents. This procedure is iterated until convergence for the Lyapunov exponents is obtained.For the computation of a covariant set of vectors { v j } spanning the tangent space for the phase point Γ ≡ Γ (0)at, say, time t , Ginelli et al. [1] start with a well-relaxed set of GS vectors at t and follow the dynamics forward fora sufficiently long time up to t ω = t + ωτ , storing G n and G n (or, equivalently, R n ) for t n = t + nτ, n = 0 , · · · , ω along the way. At t ω a set of unit tangent vectors (cid:8) v j (cid:9) ω is constructed according to v jω ∈ S jω ≡ span (cid:8) g ω , . . . , g jω (cid:9) ∀ j ∈ { , . . . , D } , (9)which serve as starting vectors for a backward iteration from t ω to time t . The vector v jn will stay in S jn atany intermediate time t n , because S jn is the most stable subspace of dimension j for the time-reversed iteration.Arranging these vectors again as column vectors of a matrix V n and expressing them in the GS basis at time t n , onehas V n = G n C n , where the matrix C n is again upper triangular with elements [ C n ] i,j = g in · v jn . If, at any step n , C n − is constructed from C n according to C n − = [ R n ] − C n , Ginelli et al. have shown that V n = J n − V n − and, hence, the respective column vectors of this matrix follow the natural tangent space dynamics without re-orthogonalization. They are covariant but not orthogonal in general. At this stage of the algorithm, renormalizationof v jn − is still required to escape the exponential divergence of the vector norms without affecting their orientation.After reaching t at the end of the iteration, the vectors v j point into their proper orientations in tangent spacesuch that, according to Eq. (7), span (cid:16) v , · · · , v f ( j ) (cid:17) = E (1) ( Γ ) ⊕ · · · ⊕ E ( j ) ( Γ ) is the most-unstable subspace ofdimension f ( j ) ≡ m (1) + · · · + m ( j ) of the tangent space at the space point Γ , going forward in time. If there aredegeneracies (as in the presence of Lyapunov modes to be discussed below), the Oseledec subspace E ( j ) is spannedaccording to E ( j ) = v f ( j − +1 ⊕ · · · ⊕ v f ( j ) , (10)where, as in the following, we omit the arguments for the phase-space point. If there are no degeneracies, v f ( j ) = E ( j ) .Similarly, the Gram-Schmidt vectors may be expressed in terms of the eigenspaces of Λ − , U ( j ) − = g f ( j − +1 ⊕ · · · ⊕ g f ( j ) . For nondegenerate subspaces one finds U ( j ) − = g f ( j ) [7, 15, 16].The drawback of this algorithm for many-particle systems is the large storage requirement for the matrices G n and G n (or, equivalently, R n ) for the intermediate times t n = t + nτ, n = 0 , · · · , ω , because τ must not be chosen toolarge (containing not more than, say, 20 particle collisions). At the expense of computer time, this can be bypassedby storing the matrices only for times separated by, say, 100 τ intervals and recomputing the forward dynamics inbetween when required during the time-reversed iteration. In this case, also the phase-space trajectory needs to bestored. IV. A SIMPLE EXAMPLE: THE H´ENON MAP -2-1 0 1 2 -2 -1 0 1 2 y x ω FIG. 1: The H´enon attractor (black line) and a finite-length approximation of its stable manifold (dotted line) are shown. Thered vectors are the covariant vectors at the phase point 0 as explained in the main text. The blue vectors are Gram-Schmidtvectors.
To illustrate the foregoing algorithm, we apply it to a simple two-dimensional example, the H´enon map [17], x n +1 = a − x n + b y n ,y n +1 = x n , with a = 1 . b = 0 .
3. In Fig. 1 the H´enon attractor is shown (black line), which is known to coincide with itsunstable manifold. An approximation of the stable manifold is shown by the dotted lines. At the point 0 the initialGS basis is indicated by the two orthogonal vectors in blue, where one, as required, points into the direction of theunstable manifold. If these vectors are evolved forward in time with the GS method for a few hundred steps, thetwo orthogonal GS vectors at the point ω are obtained. Taking these vectors as the initial vectors v ω and v ω , theconsecutive backward iteration yields the covariant vectors at point 0 indicated in red. As expected, one is parallelto the unstable manifold, the other parallel to the stable manifold at that point. V. SYSTEMS OF HARD DISKS
Now we turn to the study of a two-dimensional system of hard disks in a box with periodic boundaries, where theparticles suffer elastic hard collisions (without roughness), and move along straight lines in between collisions. Thecase of rough hard disks is the topic of a forthcoming publication [18].The Lyapunov instability of hard disk systems has been studied in detail in the past [19–23]. Here we are mainlyconcerned with the differences encountered with the GS and covariant vectors, which, as we have seen, give rise toidentical Lyapunov spectra. To facilitate comparison with our previous work, we consider reduced units for whichthe particle diameter σ , the particle mass m and the kinetic energy per particle, K/N , are unity. Here, K isthe total energy, which is purely kinetic, and N denotes the number of particles. Lyapunov exponents are givenin units of p K/N mσ . If not otherwise stated, our standard system consists of N = 198 particles at a density ρ ≡ N/ ( L x L y ) = 0 . L y /L x = 2 /
11, which is periodic in x and y . Thechoice of such a small aspect ratio facilitates the observation of the Lyapunov modes to be discussed later. As usual,the total momentum is set to zero.The state of the system is given by the coordinates and momenta of all the particles, Γ = { q n , p n ; n = 1 , · · · , N } . Similarly, an arbitrary tangent vector δ Γ - either a Gram-Schmidt vector g or a covariant vector v - consists of therespective coordinate and momentum perturbations, δ Γ = { δ q n , δ p n ; n = 1 , · · · , N } . (11)The time evolution of these vectors and the construction of the map from one Gram-Schmidt step to the next hasbeen discussed before [19, 24]. -6-4-2 0 2 4 6 0 0.2 0.4 0.6 0.8 1 λ i / 4N GS cov FIG. 2: Lyapunov spectrum for the 198 disk system described in the main text. The spectrum calculated in the forwarddirection with the GS method is shown by the blue line, the one calculated in the backward direction with the covariant vectorsby the red line. Reduced indices i/ N are used on the abscissa. Although the spectrum is defined only for integer i , solid linesare drawn for clarity. Fig. 2 shows the Lyapunov spectrum for this system computed both in forward direction with the GS vectors(blue line) and in backward direction with the covariant vectors (red line). The time of the simulation in the forwarddirection is for t + t ω = 2 . × τ , where τ = 0 . t ω − t = 2 . × τ .The time t (usually of the order of 1 × τ ) is required for the preparation of the relaxed initial state at t . It canbe observed in the figure that the unstable directions in the future correspond well to the stable directions in thepast and vice versa. Of course, if the sequence of covariant vectors is followed in the forward direction of time, thespectrum is identical to the classical GS results (blue line in Fig. 2). A. Covariant versus Gram-Schmidt vectors
Whereas the time evolution of the GS vectors is determined by the exponential growth of infinitesimal volumeelements belonging to subspaces g ⊕ · · · ⊕ g i for i ∈ { , . . . , D } according to exp (cid:16) t P ij =1 λ j (cid:17) , the growth of aninfinitesimal perturbation representing a covariant vector v i is directly proportional to exp ( t λ i ), for all i . Thus, it is D | g i . v i | E i D | g i . v i | E i FIG. 3: Plot of the scalar product norm, h| g i · v i |i , for GS and covariant vectors giving rise to the same Lyapunov exponent λ i , as a function of i . The line is a time average as discribed in the main text. Left panel: full range of Lyapunov exponents;Right panel: enlargement of the central part. interesting to compare the relative orientations of respective vectors giving rise to the same exponent. In the left panelof Fig. 3 the difference in orientation of the two types of vectors is demonstrated by a plot of | g i · v i | as a functionof i . The black line is an average over 100 frames separated by time intervals of 250 τ . Since for tangent vectors onlytheir direction and not the sense of direction is important, an absolute value is taken (here and for analogous casesbelow), otherwise the scalar product might average to zero over long times, with equal numbers of vectors pointinginto opposite directions. For the unstable directions in the left half of the left panel, one observes a rapid decreaseof the scalar product with i and, hence a rapid increase of the angle between respective covariant and GS vectors.This decrease is repeated for the stable directions in the right half of the figure. These two parts are separated bythe mode region, an enlargement of which is shown in the right panel of Fig. 3 and which will be dealt with in moredetail below.In Fig. 4 we show similar projections (time averages of absolute values of scalar products as before) for selectedcovariant vectors with the whole Gram-Schmidt vector set. One observes that the covariant vectors v j belong to theGS subspace g ⊕ · · · ⊕ g j , for all j ∈ { , . . . , N } and, thus, give rise to the upper-triangular property of the matrix R in the QR-decomposition mentioned above. The curves in the figure strongly depend on the choice of j : • If it belongs to the unstable subspace E u but does not represent a Lyapunov mode (top-left panel for j = 370),there is no obvious orientational correlation with any of the GS vectors with index i < j . For i = j = 1 correspondingto the maximum exponent, the covariant and GS vectors are identical. If, however, the covariant vector representsa Lyapunov mode as in the bottom-left panel for j = 393, then its angle with the respective GS vector may becomesmaller, giving rise to a scalar product closer to unity. • If the covariant vector belongs to the stable subspace E s but does not represent a mode as for j = 700 in thetop-right panel of Fig. 4, it has non-vanishing components in the GS basis for all i ≤ j with the exception of the zerosubspace 2 N − ≤ i ≤ N + 3, which is strictly orthogonal. With the exception of the step at the conjugate index i = 4 N + 1 − j = 93, the origin of which is not fully understood, there is no indication of orientational correlationsbetween the covariant vector with any of the GS vectors for i ≤ j . If, however, the covariant vector represents amode as for j = 400 in the lower-right panel of the figure, there is strong orientational correlation not only with therespective GS vector with i = 400, but also with its conjugate pair at 4 N + 1 − i (= 393 in our example).It is interesting to note that the leading GS and covariant vectors in the null subspace are always identical (up toan irrelevant sign): v N − = g N − . B. Localization
The maximum (minimum) Lyapunov exponent is the rate constant for the fastest growth (decay) of a phase-spaceperturbation and is dominated by the fastest dynamical events, a locally-enhanced collision frequency. It is not toosurprising that the associated tangent vector components are significantly different from zero for only a few strongly-interacting particles at any instant of time. Thus, the respective perturbations are strongly localized in physical space.This property persists in the thermodynamic limit such that the fraction of tangent-vector components contributing tothe generation of λ follows a power law ∝ N − η , η >
0, and converges to zero for N → ∞ [21, 25–27]. The localizationbecomes gradually worse for larger indices i >
1, until it ceases to exist and (almost) all particles collectively contribute D | g i . v j | E i j = 370 D | g i . v j | E i j = 700 D | g i . v j | E i j = 393 D | g i . v j | E ij = 400 FIG. 4: Time averaged absolute value of the scalar product of selected covariant vectors v j (as indicated by the labels) withthe whole set of Gram-Schmidt vectors g i as a function of i . to the coherent Lyapunov modes to be discussed below. Similar observations for spatially extended systems have beenmade by various authors [22, 23, 28–30], which were consequently explained in terms of simple models [31, 32]. Wealso mention Ref. [33], where the tangent-space dynamics of the first Lyapunov vector g for various one-dimensionalHamiltonian lattices is compared to that for the Kardar-Parisi-Zhang model of spatio-temporal chaos. The unexpecteddifferences found for the scaling properties are traced back to the existence of long-range correlations, both in spaceand time, in the Hamiltonian chains, the origin of which, however, could not be fully disclosed. The same correlationsare conjectured to be responsible for a slow 1 / √ N convergence of λ towards its thermodynamic limit [33], which isalso observed for hard-disk systems [19].Up to now, all considerations concerning localization were based on the Gram-Schmidt vectors. Here, we demon-strate the same property for the covariant vectors. According to Eq. (11) we define the contribution of an individualdisk n to a particular perturbation vector as the square of the projection of δ Γ onto the subspace pertaining to thisdisk, µ n = ( δ q n ) + ( δ p n ) . Since δ Γ is either a GS vector or a covariant vector both of which are normalized, one has P Nn =1 µ n = 1, and µ n maybe interpreted as a kind of action probability of particle n contributing to the perturbation in question. It shouldbe noted that for the definition of µ n the Euclidean norm is used and that all localization measures depend on this W i / 4N
373 396 4200.51.0 W i FIG. 5: Localization spectra W for the complete set of Gram-Schmidt vectors (blue) and covariant vectors (red). The detailsof the hard-disk system are given in Section V. Reduced indices i/ N are used on the abscissa. In the inset a magnification ofthe central mode region is shown. choice. Qualitatively, this is still sufficient to demonstrate localization. From all the localization measures introduced[25, 30], the most common is due to Taniguchi and Morriss [22, 23], W = 1 N exp[ S ] , S = * − N X n =1 µ n ln µ n + . Here, S is the Shannon entropy for the ”probability” distribution µ n , and h· · · i denotes a time average. W is boundedaccording to 1 /N ≤ W ≤
1, where the lower and upper bounds apply to complete localization and delocalization,respectively. In Fig. 5, we compare W obtained for the full set of Gram-Schmidt vectors (blue curve) to that of allthe covariant vectors (red curve). The spectra are obtained by identifying δ Γ with all vectors of the respective sets, i = 1 , · · · , N . Not too surprisingly, the localization is stronger for the covariant vectors, whose direction in tangentspace is solely determined by the tangent flow and is not affected by renormalization constraints. Another interestingfeature is the symmetry W i = W N +1 − i , which is a direct consequence of the symplectic nature of the flow [34]. C. Tangent space projections
It is interesting to see how much the coordinate and momentum subspaces contribute to a particular tangent vector δ Γ (see Eq. 11), which may be a Gram-Schmidt vector g i or a covariant vector v i , both associated with the sameLyapunov exponent λ i . The time-averaged squared projections of δ Γ onto the coordinate and momentum subspaces Q and P , respectively, are given by η q = * N X n =1 δ q n + , η p = * N X n =1 δ p n + (12)and are plotted in Fig. 6 for the whole set of Gram-Schmidt vectors, and in Fig. 7 for the whole set of covariantvectors, i = 1 , · · · , N . One notes that for the Gram-Schmidt case the contributions of η q and η p to a vector g i andits conjugate g N +1 − i are interchanged, whereas for the covariant vectors v i and v N +1 − i they are the same. This isparticularly noticeable for the expanded central regions in the respective right panels of Figs. 6 and 7. η q G S , η p G S i / 4N qp
373 385 396 408 4200.00.20.40.60.81.0 i pq FIG. 6: Mean squared projections for the full Gram-Schmidt vector set, i = 1 , · · · , N , onto the coordinate subspace Q , η GSq (green line), and the momentum subspace P , η GSp (black line), for the 198-particle system defined above. Left panel: fullspectrum; Right panel: magnification of the central mode-carrying region. η q c ov , η p c ov i / 4N qp
373 385 396 408 4200.00.20.40.60.81.0 i pq FIG. 7: Mean squared projections for the full covariant vector set, i = 1 , · · · , N , onto the coordinate subspace Q , η covq (greenline), and the momentum subspace P , η covp (black line), for the 198-disk system defined above. Left panel: full spectrum; Rightpanel: magnification of the central mode-carrying region. D. Central manifold and vanishing exponents
The dynamics of a closed particle system such as ours is strongly affected by the inherent continuous symme-tries, which leave the Lagrangian and, hence, the equations of motion invariant. The symmetries relevant for ourtwo-dimensional system with periodic boundaries are the homogeneity of time (or invariance with respect to timetranslation), and the homogeneity of space (or invariance with respect to space translations in two independent di-rections). Each of these symmetries is associated with two vector fields with sub-exponential growth (or decay) and,therefore, gives rise to two vanishing Lyapunov exponents [35]. At any phase-space point Γ , the six vectors span asix-dimensional subspace N ( Γ ) of the tangent space TX ( Γ ), which is referred to as null space or central manifold.This subspace is covariant. If the 4 N components of the state vector are arranged as Γ = (cid:0) q x , q y , . . . , q Nx , q Ny ; p x , p y , . . . , p Nx , p Ny (cid:1) , (13)0the six orthogonal spanning vectors, which are the generators of the elementary symmetry transformations, are givenby [7, 21] e = 1 √ K ( p x , p y , . . . , p Nx , p Ny ; 0 , , . . . , , , (14) e = 1 √ N (1 , , . . . , , , , . . . , , , (15) e = 1 √ N (0 , , . . . , , , , . . . , , , (16) e = 1 √ K (0 , , . . . , , p x , p y , . . . , p Nx , p Ny ) , (17) e = 1 √ N (0 , , . . . , , , , . . . , , , (18) e = 1 √ N (0 , , . . . , , , , . . . , , . (19) e corresponds to a change of the time origin, e to a change of energy, e and e to an (infinitesimal) uniformtranslation of the origin in the x and y directions, respectively, and e and e to a perturbation of the total momentumin the x and y directions, respectively. The six vanishing Lyapunov exponents are located in the center of the Lyapunovspectrum with indices 2 N − ≤ i ≤ N + 3. The first three of these vectors have non-vanishing components only forthe position perturbations in the 2 N -dimensional configuration subspace Q , the remaining only for the momentumperturbations in the 2 N -dimensional momentum subspace P . They are related by e k = J e k +3 for k ∈ { , , } , where J is the symplectic (skew-symmetric) matrix.Let us consider the projection matrices α and β of the GS and covariant vectors, respectively, onto the naturalbasis, α i,k = g i · e k ; β i,k = v i · e k , k ∈ { , . . . , } i ∈ { N − , · · · , N + 3 } . For i / ∈ { N − , . . . , N + 3 } these components vanish. Without loss of generality, we consider in the followingexample a system with only N = 4 particles in a periodic box, which is relaxed for t r = 10 time units, followed by aforward and backward iteration lasting for t ω − t = 10 time units. Very special initial conditions for the backwarditeration v iω = g iω for i = 1 , · · · , N (= 16) are used. The projections at the time t are given in Table I for the GSvectors, in Table II for the covariant vectors. TABLE I: Instantaneous projection matrix α of Gram-Schmidt vectors (for i ∈ { N − , · · · , N + 3 } ) onto the natural basis { e k , ≤ k ≤ } of the central manifold. The system contains N = 4 disks. The powers of 10 are given in square brackets. i α i, α i, α i, α i, α i, α i, N − − . −
6] 0 . −
6] 0 . − N − . − − . −
6] 0 . − N . −
6] 0 . − − . − N + 1 − . −
6] 0 . − − . −
6] 0.611 -0.782 0.1212 N + 2 0 . −
6] 0 . −
6] 0 . −
6] -0.575 -0.544 -0.6112 N + 3 − . − − . −
6] 0 . −
6] 0.543 0.304 -0.783
A comparison of the two tables reveals the following: • The six orthogonal GS vectors g i ; i = 2 N − , · · · , N + 3 completely span the null subspace (the squared elementsfor each rows add up to unity in Table I). The same is true for the six non-orthogonal covariant vectors v i ; i =2 N − , · · · , N + 3 in Table II. • The first three covariant and Gram-Schmidt vectors completely agree. This is a consequence of the special initialconditions for the former at the time t ω as mentioned above. During the backward iteration the three covariant vectorsstay in their respective subspaces and remain parallel to the GS vectors (which were stored during the forward phase ofthe algorithm). At t they are still identical to their GS counterparts. The first vectors always agree, v N − = g N − ,if less special initial conditions conforming to Eq. 9 are used.1 TABLE II: Instantaneous projection matrix matrix β for the the six central covariant vectors onto the natural basis { e k , ≤ k ≤ } of the central manifold. The system contains of N = 4 particles. The powers of 10 are given in squarebrackets. i β i, β i, β i, β i, β i, β i, N − − . −
6] 0 . −
6] 0 . − N − . − − . −
6] 0 . − N . −
6] 0 . − − . − N + 1 -0.611 0.782 -0.121 0.611 [-5] -0.782 [-5] 0.121 [-5]2 N + 2 0.575 0.544 0.611 -0.575 [-5] -0.544[-5] -0.611[-5]2 N + 3 -0.543 -0.304 0.783 0.543 [-5] 0.304 [-5] -0.783[-5] • Equivalent components have the same mantissa but may differ by a factors of 10 or 10 , which are related to theduration of the relaxation phase t r and of the forward-backward iteration time t ω − t .The explanation for this behavior [34] is obtained by a repeated explicit application of the linearized maps for thefree streaming and consecutive collision of particles [19, 24] to the six basis vectors e i . One finds that Dφ t Γ · e j ( Γ ) = e j ( Γ t ) , (20) Dφ t Γ · e j +3 ( Γ ) = t e j ( Γ t ) + e j +3 ( Γ t ) , (21)for j ∈ { , , } . Eq. (21) implies that any perturbation vector with non-vanishing components parallel to e , e ,or e will rotate towards e , e , and e , respectively. It follows i) that the null subspace N ( Γ ) is covariant; ii) thatthe subspaces N = span { e } , N = span { e } and N = span { e } are separately covariant (from Eq. (20)); that,as was already noted in Ref. [7], N can be further decomposed into the three two-dimensional covariant subspaces N p = span { e , e } , N x = span { e , e } , and N y = span { e , e } . E. Lyapunov modes
We have seen in Section V B that the perturbation vectors are less and less localized, the smaller the Lyapunovexponents become, until they are coherently spread out over the physical space and form periodic spatial patterns witha well-defined wave vector k . This collective patterns are referred to as Lyapunov modes. The modes were observedfor hard particle systems in one, two and three dimensions [7, 20, 22, 23, 36], for hard planar dumbbells [25, 37, 38]and for one and two-dimensional soft particles [27, 39, 40]. A formal classification of the modes is given in Ref. [7].Physically, they are interpreted as periodic modulations with wave number ( k = 0) of the null modes associated withthe elementary continuous symmetries and conservation laws. Since this modulation involves the breaking of suchsymmetries, the modes have been interpreteted as Goldstone modes [8]. Theoretical approaches are based on randommatrix theory [41, 42], periodic orbit expansion [43], and kinetic theory [8, 44, 45].So far the numerical work on Lyapunov modes has been exclusively concerned with the orthonormal Gram-Schmidtvectors { g i } , i = 1 , · · · , N . The purpose of this section is to point out some differences one encounters, if the modesfor the Gram-Schmidt and covariant vectors are compared.Fig. 8 shows an enlargement of the mode-carrying region for the Lyapunov spectrum of Fig. 2. In order toemphasize the conjugate pairing symmetry λ i = − λ N +1 − i for symplectic systems, conjugate exponent pairs areplotted with the same index i on the abscissa, where now i ∈ { , · · · , N } . The open circles are computed from theGram-Schmidt vectors in the forward direction of time, the dots from the covariant vectors during the time-reversediteration. Considering the size of the system ( N = 198), the agreement is excellent.The steps in the spectrum due to degenerate exponents is a clear indication for the presence of Lyapunov modes.According to the classification in our previous work [7], the steps with a two-fold degeneracy are transverse (T)modes – T(1,0), T(2,0) and T(3,0) from right to left in Fig. 2. Similarly, the steps with a four-fold degeneracy ofthe exponents are longitudinal-momentum (LP) modes – LP(1,0), LP(2,0) and LP(3,0) again from the right. Thearguments ( n x , n y ) account for the number of periods of the sinusoidal perturbations in the x and y directions. Sinceour simulation cell is rather narrow, only wave vectors k parallel to the x axis of the (periodic) cell appear, leaving 0for the second argument [7]. As usual, “transverse” and “longitudinal” refer to the spatial polarization with respectto k of the wave-like pattern.One of our early observations, which greatly facilitates the classification of the modes for the Gram-Schmid vectors[7], is that in the limit N → ∞ the cosine of the angle Θ between the 2 N -dimensional vectors of the position2 -2.0-1.5-1.0-0.50.0 0.51.0 1.52.0 370 375 380 385 390 396 λ i FIG. 8: Enlargement of the mode regime for the Lyapunov spectrum depicted in Fig. 2. The open symbols indicate exponentscomputed from the Gram-Schmidt vectors, the full dots are for exponents obtained from the covariant vectors.TABLE III: Basis vectors of ( n x ,
0) modes for a hard disk system in a rectangular box with periodic boundaries. We use thenotation c x = cos( k x x ), and s x = sin( k x x ), where the wave vector is given by k = ( k x , k y ) = (2 πn x /L x , n x ∈ { , , } . n Basis of T ( n ) Basis of L ( n ) Basis of P ( n ) n x ! c x ! , s x ! c x ! , s x ! p x p y ! s x , p x p y ! c x perturbations and momentum perturbations converges to +1 for the smallest positive, and to -1 for the smallestnegative exponents. See the blue line in Fig. 9. Furthermore, the relation δ p = C ± δ q (22)holds with known constants C ± . This means that these vectors are nearly parallel or anti-parallel for large N andthat the mode classification may be based solely on δ q . Somewhat surprisingly, this property does not strictly holdanymore for the covariant vectors. This is shown by the red line in Fig. 9, where cos(Θ) is seen to differ significantlyfrom ± i outside of the null subspace (for which 394 ≤ i ≤ i = 393, namely plots of δq y as a function of q x (top left), andof δp y as a function of q x (bottom left). The respective plots for the x components fluctuate around zero, as expected,and are not shown. Analogous plots for the mode T(2,0) with i = 387 are shown in the panels on the right-hand side.The blue points are for GS vectors, the red squares for the respective covariant vectors. It is interesting to note thatthe scatter of the points for the position perturbations is smaller for the covariant modes (red squares) than for theGS modes (blue dots). A fit shows that the residuals for the covariant modes are smaller by about a factor of twoin comparison to Gram-Schmidt. Quite the opposite is true for the momentum perturbations in the bottom row ofpanels. Although the proportionality of Eq. (22) still holds, the scatter of the red squares for the covariant vectors islarger than that of the blue dots for the GS vectors. Such a behavior is always observed and is not simple numerical3 -1.0-0.50.00.51.0 0 0.25 0.5 0.75 1 D c o s ( θ ) E i / 4N i (a)
397 417 -1.0-0.9-0.8-0.7 i (b) FIG. 9: Time averaged value of cos(Θ) = ( δ q · δ p ) / ( | δ q || δ p | ) as a function of the Lyapunov index i for a system with N = 198hard disks. Here, δ q ∈ Q and δ p ∈ P are the 2 N -dimensional vectors of all position perturbations respective all momentumperturbation for the Gram-Schmidt vectors g i (blue line) and the covariant vectors v i (red line). The insets are magnificationsof the mode-carrying region. noise. The reason for this behavior is related to the previous discussion in connection with Fig. 9 and needs furtherclarification.Longitudinal (L) and associated momentum (P) modes share the same degenerate Lyapunov exponent λ ( i ) , andgenerally appear superimposed in experimental vectors. With a rectangular box and periodic boundaries, they formfour-dimensional LP perturbations. The superposition varies periodically with time. This “dynamics” has beenidentified as a rotation of the pure L and P vectors in the standard frame. For details we refer to previous work inRef. [7]. The patterns for the pure L mode are easily recognizable as sine and cosine functions, but those for the Pmodes are not. As is evident from the spanning vectors for L(1,0) and P(1,0) also listed in Table III, the P modesare proportional to the instantaneous velocities of all particles, which does not at all constitute a smooth vector field.For a pattern to be recognizable, these velocities need to be “divided out”. A full mode reconstruction is requiredas is described for the case of Gram-Schmidt vectors in Ref [7]. Here we carry out an analogous reconstruction interms of the covariant vectors and compare them to the GS modes. In Fig. 11 two of the reconstructed patterns forL and P modes belonging to the four-dimensional LP(1,0) subspace with indices i ∈ { , , , } are shown.The blue dots are for GS modes, the red squares for covariant modes. To judge the quality of the reconstruction, wehave included in the top-right panel also the δq y -versus- q x curve, which vanishes nicely as required.For comparison, Fig. 12 gives results for a completely analogous reconstruction, where instead of the positionperturbations as in Fig. 11, the corresponding momentum perturbations are used. For this example cosine patternswere selected, whereas in Fig. 11 sine patterns were used. As before, blue dots refer to GS vectors, red squares tocovariant vectors. F. Transversality
From the Lyapunov spectrum of Fig. 2 and the magnification of its central part in Fig. 8, the following inequalitiesare read off, λ > · · · ≥ λ N − > h λ (0) i > λ N +4 ≥ · · · > λ N , (23)where the equal sign applies for the degenerate exponents belonging to modes. (cid:2) λ (cid:3) = 0 is sixfold degenerate in ourcase. Conjugate pairing assures that λ i = − λ N +1 − i . The Oseledec splitting provides us with the following structure4 -0.10.00.1-20 -10 0 10 20 δ q y q x -0.10.00.1-20 -10 0 10 20 δ p y q x -0.10.00.1-20 -10 0 10 20 δ q y q x -0.10.00.1-20 -10 0 10 20 δ p y q x FIG. 10: Instantaneous transverse Lyapunov modes T(1,0) for index i = 393 (left panels), and T(2,0) for index i = 387(right panels) for the 198-disk system. In the panels at the top (bottom) the y -coordinate perturbations δq y ( y -momentumperturbations δp y ) of all particles are plotted as a function of their x coordinate, q x , in the simulation cell. The wave vector isparallel to the x axis. The blue dots are for Gram-Schmidt vectors, the red squares for covariant vectors. of the tangent space, TX = E u ⊕ N ⊕ E s , (24)where E u = v ⊕· · ·⊕ v N − and E s = v N +4 ⊕· · ·⊕ v N are the covariant stable and unstable subspaces, respectively,and N is the null subspace or central manifold. The question arises whether the system is hyperbolic, which impliesthat the angles between the stable manifold E s and the unstable manifold E u are bounded away from zero for allphase points (Due to the existence of a central manifold this is referred to as partial hyperbolicity in the mathematicalliterature [46]). Even more, we may ask whether the angles between all Oseledec subspaces and, hence between allcovariant vectors, are bounded away from zero for all phase space points. To find an answer to that question, wecompute in the following the scalar products for all covariant vector pairs and present representative results. (Thisprocedure reminds us of the so-called coherence angles introduced by d ’Alessandro and Tenenbaum [47, 48], measuringthe angular distance between a physically interesting direction and the direction of maximum perturbation expansion).The lines in Figure 13 depict the product norms h| v j · v i |i for selected covariant vectors v j with all other covariantvectors v i , i = j . As before, a time average is performed. The panels on the left-hand side provide three examples for v j from the unstable manifold outside of the mode regime ( j = 1 , j = 420 , N is orthogonal to both E u and E s . For two covariant vectors from the same subspace, E u or E s , however, the scalar product does not vanish indicating considerable nonorthogonality. But at the same time it isalso well bounded away from unity, which means that the two vectors do not become parallel either. However, onepossible exception may be the covariant vector pairs ( v j , v j +1 ) for adjacent Lyapunov exponents in the spectrum. Inthese cases, the scalar product reaches a pronounced maximum in all of the six panels of Fig. 13 which may still allowthese vectors to become parallel occasionally. This will be discussed further below.5 -0.10.00.1-20 -10 0 10 20 δ q x q x -0.10.00.1-20 -10 0 10 20 δ q y q x L(1,0) -0.10.00.1-20 -10 0 10 20 δ q x / p x q x -0.10.00.1-20 -10 0 10 20 δ q y / p y q x P(1,0)FIG. 11: Reconstructed position perturbations of the pure L(1,0) mode (top panels) and P(1,0) mode (bottom panels). Onlythe patterns proportional to sin(2 πq x /L x ) are shown. Blue dots: Gram-Schmidt vectors; Red squares: covariant vectors. Fordetails we refer to the main text. So far we have considered only vectors v j outside of the mode regime. The case of covariant vectors representingmodes is treated separately in Fig. 14, where, as before, time-averaged scalar product norms h| v j · v i |i for i = j are plotted as a function of i . The standard deviation is too small to be included in the plots. The panels on theleft-hand side are for j belonging to unstable transversal modes, the panels on the right-hand side for j belongingto unstable LP pairs. The curves for the conjugate stable modes just look like the mirror images around the centralindex. Each vector representing a T or LP-mode has significant contributions to the scalar product only for covariantvectors belonging to the same degenerate exponent and – to a lesser extent – the corresponding conjugate (negative)exponents (where the latter is not true anymore for the LP(3,0) modes in the bottom-right panel of Fig. 14, whereno peak around i = 414 is discernible).The covariant vectors belonging to transverse (or to LP) modes span covariant Oseledec subspaces E ( i ) with adimension m ( i ) equal to 2 (respective 4). To ease the notation, we refer to them as E ( X ) in the following, where X iseither T( n x ,
0) or LP( n x ,
0) with n x ∈ { , , } . The conjugate Oseledec subspaces, E ( X ) ∗ , have the same dimensionand are spanned by the respective conjugate covariant vectors. Fig. 14 shows that the covariant vectors spanningany of the subspaces E ( X ) or E ( X ) ∗ have a rather small but finite angular distance and, thus, are transversal. TheOseledec subspaces representing modes are themselves transversal to all other subspaces of the Oseledec splitting, butto a varying degree. The angular distances in tangent space are generally large except between conjugate subspaces E ( X ) and E ( X ) ∗ , for which the scalar products of their spanning vectors may become surprisingly large.To check more carefully for transversality even in this case, we show in Fig. 15 the probability distribution forthe minimum angle between the conjugate subspaces E ( X ) and E ( X ) ∗ , where X stands for the T and LP modesas indicated by the labels. This angle Φ is computed from the smallest principal angle between the two subspaces[49, 50]. If the covariant vectors belonging to E ( X ) and E ( X ) ∗ are arranged as the column vectors of matrices V and V ∗ , respectively, the QR decompositions V = QR and V ∗ = Q ∗ R ∗ of the latter provide matrices Q and Q ∗ ,with which the matrix M = Q T Q ∗ is constructed. The singular values of M are equal to the cosines of the principalangles, of which Φ is the minimum angle. Since Φ is never very small, this method works well and does not need6 -0.10.00.1-20 -10 0 10 20 δ p x q x -0.10.00.1-20 -10 0 10 20 δ p y q x L(1,0) -0.10.00.1-20 -10 0 10 20 δ p x / p x q x -0.10.00.1-20 -10 0 10 20 δ p y / p y q x P(1,0)FIG. 12: Reconstructed momentum perturbations for the pure L(1,0) mode (top panels) and P(1,0) mode (bottom panels)depicted already in Fig. 11. Only the patterns proportional to cos(2 πq x /L x ) are shown. Blue dots: Gram-Schmidt vectors;Red squares: covariant vectors. more complicated refinements [49–51]. It is seen that all distributions are well bounded away from zero indicatingtransversality for the respective subspaces.Finally, we concentrate on the minimum angle between the full unstable subspace E u = v ⊕ · · · ⊕ v and itsconjugate stable counterpart E s = v ⊕ · · · ⊕ v , using the same method as before. These subspaces include themode-carrying vectors studied before. The probability distribution for the minimum angle is denoted by Ψ and isalso shown in Fig. 15 (red line). Also this distribution is well bounded away from zero and indicates transversalitybetween E s and E u . We conclude that for finite N the hard-disk systems are (partially) hyperbolic in phase space.In Fig. 13 it was observed that the scalar products between covariant vectors with adjacent indices are rather largeand possibly may allow tangencies. To study this point more carefully, we follow a suggestion of G. Morriss andconsider the angle Θ = cos − | v i · v j | between the vectors v i ∈ F ( J ) and v j ∈ F ( J ) , for which i − j is a specifiedpositive integer. The probability distributions for angles with i − j = 1 , , , ,
50 are shown in Fig. 16, and for i − j = 100 , ,
300 in the inset of the same figure. Whereas the probabilities for i − j > i − j = 1 seems to converge to zero for Θ → i − j > i − j . The inset provides a magnification of the most interesting region. One observes that the minimum of theangle Θ between covariant vectors specifying Oseledec subspaces with i − j = 1 may indeed become very small, butthis happens with extremely small probability. Our numerical evidence is consistent with the assumption that theangle becomes zero with vanishing probability.7 D | v i . v j | E i j = 1 D | v i . v j | E ij = 792 D | v i . v j | E i j = 200 D | v i . v j | E ij = 600 D | v i . v j | E i j = 370 D | v i . v j | E ij = 420 FIG. 13: The lines are time averaged (100 frames separated by 150 time units) absolute values of the scalar product of aspecified covariant vector v j with all the remaining covariant vectors v i = j as a function of i . Left panels from top to bottom: j = 1 , j = 420 ,
370 380 390 400 410 420 D | v i . v j | E i T(1,0) j = 393 j = 392
370 380 390 400 410 420 D | v i . v j | E i LP(1,0) j = 391 j = 390 j = 389 j = 388
370 380 390 400 410 420 D | v i . v j | E i T(2,0) j = 387 j = 386
370 380 390 400 410 420 D | v i . v j | E i LP(2,0) j = 385 j = 384 j = 383 j = 382
370 380 390 400 410 420 D | v i . v j | E i T(3,0) j = 381 j = 380
350 370 390 410 430 D | v i . v j | E i LP(3,0) j = 379 j = 378 j = 377 j = 376 FIG. 14: The lines are time averaged (100 events separated by 150 time units) absolute values for the scalar product of covariantvectors v j (as specified by the label j ) with all the remaining covariant vectors v i = j as a function of i . The abscissa is restrictedto the mode regime. j is for modes from the unstable subspace with positive exponents only. For the respective conjugatemodes from the stable subspace, the curves in all panels are just the mirror images around the center as in Fig. 13. π /8 π /4 3 π /8 π /2 P ( Φ ) Φ Ψ T(1,0) LP(1,0) T(2,0) LP(2,0)T(3,0)LP(3,0)
FIG. 15: Probability distributions for the minimum angle between the Oseledec subspace E ( X ) and its conjugate subspace E ( X ) ∗ . Here, X ∈ { T( n x , , LP( n x , } with n x = 1 , , E s and E u is shown by the red line. π /8 π /4 3 π /8 π /2 P ( Θ ) Θ π /8 π /2 | i - j | =1 2 4 20 50 π /8 π /2 | i - j | =1 2 4 20 50 | i - j | =100 200300 FIG. 16: Probability distributions for the angles Θ between all covariant vectors v i and v j from F (370) = v ⊕ · · · ⊕ v withprescribed separation i − j of their indices as indicated by the labels. π /8 π /43 π /8 π /2
1 50 100 150 200 250 300 350 m i n ( Θ ( t ) ) i - j π /16 π /8
1 3 5 7 9
FIG. 17: Plot of the minimum angle between different covariant vectors v i and v j from the unstable subspace without themode-affected directions, F (370) = v ⊕ · · · ⊕ v , as a function of their index difference i − j . VI. CONCLUSION
A comparison of the covariant vectors with corresponding orthonormal Gram-Schmidt vectors reveal similarities, butalso significant differences. The vectors associated with the maximum Lyapunov exponent are identical, v = g , andalso the leading vectors in the central manifold agree, v N − = g N − . All the other corresponding vectors generallypoint into different tangent space directions. Whereas the GS vectors are pairwise orthogonal by construction, thecovariant vectors are not. Most notably, the perturbation contributions from the particles’ positions and momenta aresignificantly different and even exhibit a different symmetry between vectors from the stable and unstable manifold asin Fig. 6. For the covariant vectors these contributions agree in accordance with the time-reversal symmetry requiredfor them, whereas for the Gram-Schmidt vectors these contributions are interchanged. Another significant differenceis the degree of localization in physical space for the non-degenerate perturbations. As Fig. 5 shows, the covariantvectors are much more localized than the GS vectors in accordance with the fact that they are not dynamicallyconstrained by re-orthogonalization.From a theoretical point of view, an interesting result is that no tangencies occur between the respective unstableand stable manifolds E u and E s . In Fig. 15 the probability distribution Ψ of the minimum angle between stable andunstable subspaces (including the Lyapunov modes) is well bounded away from zero, and even more so for the vectorsbelonging to unstable respective stable modes. Thus, a hard disk system with N = 198 particles as in our case is(partial) hyperbolic for all points in phase space. We even find that all Oseledec subspaces are pairwise transversalwith non-vanishing angles between them.We speculate that for N → ∞ the distribution for the minimum angle between E u and E s may possibly reach theorigin in Fig. 15. To clarify this point further studies are required [34].The concept of hyperbolicity is closely linked with the notion of dominated Oseledec splitting for all phase spacepoints [46]. We may rewrite Eq. (6) for the Lyapunov exponents, expressed in terms of the covariant vectors, accordingto λ ℓ = lim N →∞ N N − X n =0 τ ln (cid:13)(cid:13) Dφ τ | Γ ( t n ) v ℓ ( Γ ( t n )) (cid:13)(cid:13) , (25)where t n ≡ nτ , and τ is the short time interval between consecutive re-normalizations of the covariant vectors. Here, λ ℓ is expressed as a time average of a quantityΛcov ℓ ( Γ ( t n )) = 1 τ ln (cid:13)(cid:13) Dφ τ | Γ ( t n − ) v ℓ ( Γ ( t n − )) (cid:13)(cid:13) , (26)which is referred to as local or (time-dependent) Lyapunov exponent, and is a function of the instantaneous phasepoint Γ ( t ). The Oseledec splitting is said to be dominated, if the local Lyapunov exponents, when averaged over afinite time ∆, do not change their order in the spectrum for any ∆ larger than some finite ∆ > VII. ACKNOWLEDGEMENTS
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