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Comparison of covariant and orthogonal Lyapunov vectors

Two sets of vectors, covariant and orthogonal Lyapunov vectors (CLVs/OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover the previous classification of two universality classes works for CLVs as well, i.e. the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs by CLVs. For general dissipative systems with nonhyperbolic dynamics the long wave length structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the micro- reversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs and hyperbolicity under changes of coordinate are discussed in appendices.


I. INTRODUCTION
Chaos means a sensitive dependence on initial conditions. This intrinsic randomness of the fully deterministic systems makes a statistical treatment of them feasible, which is essential for the foundations of statistical mechanics [1]. Besides, chaos plays an important role in a plenty of phenomena which is of relevance to our daily life, for instance the weather forecasting [2].
To characterize the chaoticity of dynamical systems, Lyapunov exponents and vectors are mostly used. One important recent finding of Lyapunov analysis is that for systems with continuous symmetries Lyapunov vectors corresponding to near-zero Lyapunov exponents have long wave-length structures, named hydrodynamic Lyapunov modes (HLMs) [3]. This provides a new possibility to connect the reduced description of a many-body system to the microscopic information of its detailed dynamics. Further investigations showed that HLMs exist in a large number of systems [4][5][6][7] and they have some universal features irrespective of the details of their dynamics [8]. One should mention that localization of Lyapunov vectors corresponding to the largest Lyapunov exponents was also intensively studied [9].
Lyapunov analysis was conventionally undertaken via the so-called Benettin algorithm [11], where Lyapunov vectors are calculated as the set of orthogonal vectors right after reorthogonalization of offset vectors. Recently, the application of another set of vectors called covariant Lyapunov vectors (CLVs) was made feasible via an efficient algorithm proposed by Ginelli et al. [14]. CLVs have been shown suitable for the characterization of hyperbolicity of high dimensional systems since they are expected to span the local stable and unstable subspaces of the investigated systems. In view of the obvious difference, it becomes necessary to study the relation between CLVs and the conventionally used Lyapunov vectors. We denote the latter as orthogonal Lyapunov vectors (OLVs) in order to distinguish them from CLVs.
We first recall in Sec.II the definition and numerical calculation of both sets of vectors.
In practice the Lyapunov exponents and OLVs are calculated via the so-called standard method invented by Benettin and Shimada et al. [11], which was used in most studies of HLMs [3][4][5][6][7][8]. Here the time evolution of a set of offset vectors in tangent space is moni- and the Lyapunov vectors obtained via the standard method is subtle. It is proved that the Lyapunov vectors f (α) obtained via the standard method converge exponentially to the Oseledec eigenvectors for the inverse-time dynamics of the original system [12,13]. In other words, as t → +∞, there is f (α) ∼ g (α) , where g (α) are eigenvectors of the matrix Ref. [12,13] for the details.
Exactly in the same work [10] Oseledec proved also that, for almost all initial conditions x, there is a splitting of the tangent space T M(x) and there exist real numbers λ 1 (x) > λ 2 (x) > · · · > λ s (x) such that where Df is the derivative governing the tangent space dynamics. In contrast to the popularity of OLVs, the use of CLVs was made feasible only recently owing to an efficient algorithm proposed by Ginelli et al. [14]. The new algorithm relies on the information obtained via the standard method of Benettin. One additional integration of the inverse-time dynamics is performed in order to get CLVs and the corresponding fluctuating finite-time Lyapunov exponents. The basic idea is that an arbitrary offset vector will approach asymptotically the most unstable direction corresponding to the largest Lyapunov exponent. It is known that the covariant k-dimensional subspace spanned by the first k CLVs is spanned by the first k OLVs as well. An arbitrary offset vector confined to this subspace will approach asymptotically the k-th CLV if the inverse-time tangent space dynamics is applied. To this aim one needs the effective tangent space dynamics confined in the k-dimensional covariant subspace. A representation of this effective dynamics in the coordinate space of OLVs is given by the R-matrix produced by the reorthonormalization steps of the standard method. Detailed formulas can be found in Ref. [14].
We mention that a different algorithm was used in Ref. [15]. It is demanded to compare the efficiency of the two algorithms.
To characterize Lyapunov vectors of extended systems quantitatively, we introduced in [6] a dynamical variable called LV fluctuation density in the spirit of generalized hydrodynamics, where δ(x) is Dirac's delta function, r l (t) ≡ l · a is the position coordinate of the l-th element taken here as r l (t) ≡ l · a, and {δu (α)l t } is the coordinate or momentum part of the α-th Lyapunov vector at the discrete time t. For simplicity, we set a = 1 in the following discussion. The spatial structure of LVs is characterized by the static LV structure factor defined as which is just the spatial power spectrum of the LV fluctuation density.
As shown in past studies [6][7][8], the quantity k max representing the wave-number of the dominant peak of the structure factor S(k) and S(k max ) can be used to characterize the significance of long wave length structure in Lyapunov vectors.

III. MODELS
CMLs [16] were selected as the main focus of this study because they have, which is essential to HLMs, similar symmetries as many-particle systems but are relatively much simpler.
The two classes of CMLs under investigation have the form and where f (z) is a nonlinear map, t is the discrete time index, l = {1, 2, · · · , N} is the index of the lattice sites and N is the system size. Unless explicitly stated, we use periodic boundary in the numerical simulations below. Two options of the local map are used, the sinusoidal map f C (z) = 1 2π sin(2πz) and the skewed tent map with z ′ = z (mod 1). With the parameter being close to zero Eqs. (5) and (6) with the skewed tent map is highly hyperbolic whereas Eqs. (5) and (6) with the sinusoidal map is nonhyperbolic. Especially the Hamiltonian system Eq. (5) with the two options of the local map are similar to the well-studied cases of hard-core systems and soft-potential systems, respectively. Alternatively, tuning the parameter r of Eq. (7) from 0 to 0.5 leads to a smooth variation of the dynamics of Eq. (5) from hard-core-like to soft-potential like [17].
Obviously, both systems Eq. (5) and (6) are invariant under an arbitrary translation in u-direction. Such a symmetry is known to be responsible for the appearance of HLMs.

IV. EXISTENCE OF HLMS IN CLVS
We show in Fig.1   Hamiltonian systems, respectively. The local map is the skewed tent map with ǫ = 1.3 and r = 0.15. Note the perfect agreement between data from CLVs and OLVs for the highly hyperbolic cases shown here.

V. UNIVERSALITY OF DISPERSION RELATIONS
In Ref. [7,8] we found that the λ-k dispersion relation of HLMs can be classified into two universality classes with respect to the system dynamics. Dissipative systems have a quadratic λ-k dispersion while Hamiltonian systems have a linear one. Now we see whether such a classification is still valid if CLVs are used instead.
The cases with the skewed tent map as local dynamics are shown in Fig.2. As can be seen from the plot, the CLV dispersion relations for dissipative and Hamiltonian systems have the different asymptotic behavior. The former is of the asymptotic form λ ∼ k 2 max while the latter is λ ∼ k max , as reported for OLVs [7]. Moreover, for the used parameter setting the dispersion curves for CLVs agree very well with those of OLVs . Cases with the sinusoidal map as local dynamics are shown in Fig.3. For both dissipative and Hamiltonian systems CLV dispersions are converging to the expected asymptotic forms, even better than OLVs.
Note also that, as shown in Fig  form. Further discussion regarding these differences will be given in the following sections.
Nevertheless, for the two representative cases the investigated CLV dispersions follow well the reported classification of the universality classes of HLMs [7,8].

VI. SIGNIFICANCE OF HLMS
To characterize the significance of long wave length structure in Lyapunov vectors, we use the measure S(k max ), which is the height of the dominant peak in the static LV structure factor S(k) (Eq. (4). OLVs for the highly hyperbolic cases shown.
and Hamiltonian system, clear discrepancies between CLVs and OLVS can be seen in k max and S(k max ). For the dissipative case, the height of the dominant peak S(k max ) for CLVs is much lower than for OLVs (see Fig.5), especially in the regime λ ∼ 0 (α/N ≃ 0.93), which means that for the strongly nonhyperbolic systems as shown here, the use of CLVs reduces the visibility of long wave length structure as compared to OLVs. In contrast, for the Hamiltonian case, the height of the dominant peak S(k max ) is comparable for CLVs and OLVs. Note, however, that the variation of S(k max ) for CLVs is symmetric with respect to the spectral center α/2N = 0.5 while it is asymmetric for OLVs (Fig.5d).
To demonstrate further the influence of hyperbolicity on the significance difference between CLVs and OLVs we tune the parameter r of the skewed tent map Eq. (7). Results for dissipative cases and Hamiltonian cases are shown in Fig. 6 and 7 respectively. In consistence with our previous results in Ref. [17] the weakening of hyperbolicity as increasing r from 0.2 to 0.4 leads to a dramatic reduction of the significance of HLMs, in both CLVs and OLVs, for either dissipative system or Hamiltonian system. In the dissipative system the reduction of CLV significance as increasing r is much faster than the reduction of OLV significance, which leads to an increasing discrepancy between them. In contrast for the Hamiltonian system the significance of CLVs and OLVs is always comparable. Note, however, that the tuning of r has no influence on the symmetry features of Lyapunov vectors.
Owing to the different symmetry properties of CLVs and OLVs of Hamiltonian system, the largest S(k max ) is observed at different α values. This leads to a rather large difference in the static LV structure factors corresponding to the smallest positive Lyapunov exponents.
We show in Fig. 8 two cases with different coupling strength ǫ. As can be seen from the figure the CLV structure factor S(k) diverges quickly as k goes to zero while the OLV structure factor increases relatively slowly and even seems to saturate to a constant. Note that the  same parameter ǫ = 0.6 was used in Ref. [14] (see Fig.3 therein).
Moreover, as increasing the system size N the asymmetrically located peak of S(k max ) for OLVs shifts towards the spectral center point α/2N = 0.5 as shown in Fig. 9, which indicates a gradual reduction of the discrepancy between CLVs and OLVs as approaching the thermodynamic limit. As can be seen from the same plot, CLVs behave differently. The relation δu (α) = ±δu (2L−1−α) and δv (α) = ∓δv (2L−1−α) seems to work well instead. Note also that for CLVs the amplitude of the wave structure in the momentum part is much smaller than in the corresponding coordinate part. Such difference is also reflected in the profiles of S(k max ) in  Fig. 13, were · · · means an average over time.
It shows that for the highly hyperbolic cases, for instance Eq. case with the special skewed tent map is highly hyperbolic while the one with sinusoidal map is nonhyperbolic as most systems. In some sense the Hamiltonian system with the two local maps are corresponding to the often used hard-core system and soft potential system respectively. With the replacement of OLVs by CLVs the formerly detected long wavelength structure in Lyapunov vectors can be seen as well. Moreover the CLV λ-k dispersion relation is linear for Hamiltonian system while quadratic for dissipative system as found for OLVs. The significance of HLMs as measured by the static LV structure factor changes differently for Hamiltonian and dissipative systems with the replacement of OLVs by CLVs.
For Hamiltonian systems the significance of HLMs is always comparable for CLVs and OLVs independent of the variation of hyperbolicity as changing the local maps, besides that the OLVs with the most significant wave structure lie slightly away from the spectral center.
Increasing system size tends to shift them back to the center. For dissipative systems the significance of HLMs is almost the same for CLVs and OLVs if the special skewed tent map is used as local map. Departing from such a highly hyperbolic situation the HLM significance of CLVs reduces much faster than that of OLVs.
In the past there were already discussions regarding the symmetry of the conjugate pair of Lyapunov vectors in Hamiltonian system. It was found that owing to the symplectic feature of Hamiltonian systems the coordinate and momentum parts exchange their position for a conjugate pair of OLVs. A different symmetry is observed, however, for CLVs, namely that two CLVs in one conjugate pair are statistically indistinguishable. As discussed the physical origin of this seemingly unreasonable property is the microscopic reversibility, a general feature of Hamiltonian systems. For the specific issue of HLMs, it implies that the variation of HLM significance for CLVs is symmetric with respect to the spectral center.
Besides that we found for CLVs that the HLM significance is much lower in the momentum part than in the coordinate part.
For the highly hyperbolic cases with the special skewed tent map as the local dynam- between CLVs shows that for those corresponding to near-zero Lyapunov exponents the mutual angles are large and close to π/2. An unexpected observation is that the angle between conjugate pair decreases to zero as approaching the spectral center. Such a feature persists as weakening the hyperbolicity.
It is known that a dynamical system has two sets of OLVs, backward and forward ones.
Only the backward OLVs, which can be calculated numerically via the standard method, are discussed in the main text. Similar results are expected for the forward OLVs except that they bear the similarity to a different part of CLVs compared to the backward OLVs.
A related discussion can be seen in the appendix C.
To mention that a comparison of CLVs and OLVs in systems with hard-core interactions is performed by Posch et al [21], which and the current contribution form a complementary From the calculation algorithm we know that a k-dimensional vector spanned k arbitrary offset vectors will approach asymptotically the most unstable k-dimensional subspace, which can be spanned by k Lyapunov vectors, either CLVs or OLVs, associated with the first k largest Lyapunov exponents. The growth rate Σ k of the volume V k of this k-dimensional subspace can be written as t 2 ) is the growth rate of offset vectors along the i-th CLV, i.e. the i-th FTLE corresponding to this CLV and θ i (t) is the angle between the i-th CLV and the subspace spanned by CLVs with index from 0 to i − 1. Taking into account the mutual orthogonal nature of OLVs, the growth rate Σ k can be expressed as well by using characteristics of OLVs as is the growth rate of offset vectors along the i-th OLV, i.e. the i-th FTLE corresponding to this OLV.

Combining Eq.(A1) and (A2) yields a simple relation between FTLEs
with N the dimension of the considered system. Since CLVs are in general not mutually orthogonal it is obvious from Eq. (A3b) that FTLEs λ i (O) (t 1 , t 2 ) and λ i (C) (t 1 , t 2 ) with i ≥ 1 are normally different.
As approaching the limit t 2 − t 1 = +∞ the contribution of the second term in r.h.s. of Eq. (A3b) becomes negligible since the value of cos θ is bounded. This implies i.e. asymptotic Lyapunov exponents corresponding to CLVs and OLVs are identical. We consider a dynamical system which is written aṡ The time evolution of its trajectory can be expressed as Correspondingly the evolution of an infinitesimal perturbation vector with respect to the reference trajectory x(t) can be written as the governing equation of infinitesimal perturbations becomes δy(t 2 ) = M ′ (t 1 , t 2 )δy(t 1 ).
Here the two variables δy and δx are related via a linear transformation L : δx → δy with It is known that the linear transformation L is determined by the transformation T via where (D x T) ij = ∂T i /∂x j . By using Eq. If e(t 1 ) ≡ e(x(t 1 )) is a CLV in the x-coordinate system, it satisfies the condition | being the Lyapunov exponent corresponding to this CLV. Multiplying the both sides of Eq.(B9) with L and using Eq.(B8) results in Denoting e ′ (t) = Le(t)/ Le(t) one can reformulate Eq.(B10) as Under the condition that one can easily obtain that lim which implies that (i) the unit vector e ′ (t) is a CLV in the y-coordinate system and it is related to e(t) via e ′ (t) = Le(t)/ Le(t) ; (ii) the asymptotic Lyapunov exponent associated with e ′ (t) is identical to the asymptotic Lyapunov exponent corresponding to e(t); (iii) the finite-time Lyapunov exponent λ ′ (t 1 , t 2 ) = 1 t 2 −t 1 (ln |σ(t 1 , t 2 )| + ln Le(t 2 ) Le(t 1 ) ) in the y-coordinate system is different from the one λ(t 1 , t 2 ) = 1 t 2 −t 1 ln |σ(t 1 , t 2 )| in the x-coordinate system. For an invertible transformation T with the assumption that the reference trajectory is bounded in phase space one can easily show the boundedness of L [19], i.e.
for two constant L − ≤ L + < ∞, which implies the validness of the condition stated in Eq.(B12). As discussed in [19] these requirements on T can be weakened such that for noninvertible transformations T one can still get the invariance of Lyapunov exponents and the covariant transformation of CLVs. This is also confirmed by our numerical example below.
As discussed already in Ref. [7], via the transformation Eq. (6) can be mapped to the following diffusively coupled CMLs According to our above arguments CLVs of the two system are related via the transformation where c t is a time-dependent normalization factor. Numerical results shown in Fig. 15 and 16 for a case with the sinusoidal map f (z) confirms our conclusions. Note that the transformation Eq.(B15) is non-invertible.

Invariance of hyperbolicity under diffeomorphisms
In viewing that e ′ (t) = Le(t)/ Le(t) one would expect that the absolute value of angles between CLVs is not invariant under the variable transformation. Whether the angle is zero or not, i.e the feature of hyperbolicity, is expected to be preserved under diffeomorphisms.
This conjecture is supported by the following arguments.
If the variable transformation T given in Eq.(B4) is a diffeomorphism, the corresponding transformation L of the perturbation in Eq.(B4) would be an invertible linear transformation.
Consider now two subspaces S 1 and S 2 spanned by two sets of different CLVs {e i S 1 } and {e i S 2 }. If the angle between the two subspaces is zero in one coordinate system it means that the two sets of CLVs are linearly dependent, i.e. i c i e i S 1 + d i e i S 2 = 0 for certain constants c i and d i . Preservation of collinarity under affine transformations implies that the corresponding transformed CLVs are linearly dependent as well, i.e. the angle between subspaces in the transformed coordinate system is also zero. Similarly one can show that if the angle between two subspaces is nonzero in one coordinate system it would be nonzero in other transformed coordinate systems, too. These arguments show that whether the angle between subspaces is zero or not is invariant under dffeomorphisms, i.e. the property of hyperbolicity is preserved.
Similar to the discussion about the transformation properties of CLVs one can weaken the requirements on the transform T but rather the same conclusion about the hyperbolicity can be reached. A known example is the Kuromato-Sivashinsky equation. It can be written in two different forms as which are related via a non-invertible transformation v = u x . Numerical simulations show that the two forms have the identical hyperbolicity and details will be shown elsewhere [20].
that the fundamental matrix is time independent and thus the eigenvectors of M 2 are CLVs of this system. By using the eigenvectors e (α) of the matrix D L the eigenvectors of the fundamental matrix M 2 can be constructed as { e (α) ; c(k) e (α) }. The associated eigenvalues are where η(k) = −2ǫ(1 − cos k) and the corresponding c(k) can be calculated as The following properties of these eigenvectors/CLVs can be obtained: i) The corresponding eigenvalues satisfy µ + (k)µ − (k) = 1 which indicates the conjugate pair property λ + = −λ − of Lyapunov exponents since λ ≡ ln |µ(k)|.
ii) The group of CLVs { e (α) ; c + (k) e (α) } corresponding to positive Lyapunov exponents are in general not orthogonal to CLVs { e (α) ; c − (k) e (α) } corresponding to the negative branch of the Lyapunov spectrum, although members of either group are mutually orthogonal. This indicates that these eigenvectors/CLVs are not OLVs of that system.
Thus for a conjugate pair of CLVs { e (α) ; √ ǫk e (α) } and { e (α) ; − √ ǫk e (α) } one has In consistence to this, we reported in Sec. VII that for the cases with r close to 0 as shown in Fig. 13 and 14 neighbouring CLVs are nearly orthogonal while those in a conjugate pair tend to be collinear as approaching the spectral center λ = 0.
In consistence to this, as shown in Fig. 4d and 6a, for cases with r close to 0 OLVs and CLVs associated with positive Lyapunov exponents are very similar.
A dynamical system has actually two sets of OLVs, namely backward and forward OLVs, which are eigenvectors of the matrix M n 2 (M T 2 ) n and (M T 2 ) n M n 2 as n goes to infinity, respectively. In above discussions the backward OLVs are used since they are the ones numerically obtained from the standard method [11]. For the forward OLVs one can do the similar calculations and the conclusion is that a half of the forward OLVs are the same as the CLVs associated with negative Lyapunov exponents.