Physica D 118 (1998) 167–198

On the concept of stationary Lyapunov basisSergey V. Ershov, Alexei B. Potapov∗

Keldysh Institute for Applied Mathematics, Moscow 125047, Russia

Received 19 March 1997; received in revised form 8 December 1997; accepted 12 January 1998Communicated by F.H. Busse

Abstract

We propose the concept of stationary Lyapunov basis – the basis of tangent vectorse(i)(x) defined at every pointx of theattractor of the dynamical system, and show that one can reformulate some algorithms for calculation of Lyapunov exponentsλi so that eachλi can be treated as the average of a functionSi(x). This enables one to use measure averaging in theoreticalarguments thus proposing the rigorous basis for a number of ideas for calculation of Lyapunov exponents from time series. Wealso study how the Lyapunov vectors in Benettin’s algorithm converge to the stationary basis and show that this convergencerate determines continuity of the field of stationary Lyapunov vectors. © 1998 Elsevier Science B.V.

PACS:05.45Keywords:Dynamical systems; Lyapunov exponents

1. Introduction

Lyapunov exponents are important for quantification of dynamical systems [1–3]. They characterize wholeattractor, not its local properties. To study the latter a number of more detailed characteristics have been proposed,e.g. finite time Lyapunov exponents, local stable and unstable directions, etc.

Calculation of Lyapunov exponents [4] involves the so-called Lyapunov vectors, describing the evolution ofinfinitesimal phase volumes (or, more precisely, their orientation). In the present paper, we show that in a ratherwide class of chaotic systems there exists a field of directions that correspond to distinctaveragestretchings, andthat the evolving Lyapunov vectors exponentially converge to them. We hope, that this field may be helpful inunderstanding the behavior of chaotic systems, as well as in development and substantiation of methods of theirnumerical treatment.

We shall call this local structure for tangent vectors at every point of the attractor, which arise from the algorithm[4], “the stationary Lyapunov basis”. Its vectors at different points of a trajectory are mapped into each other, and theith one is responsible for calculation ofith exponentλi . Within this approach eachλi can be represented as a timeaverage of a “pointwise” functionσi(x) (uniquely determined by the phase space pointx). Usually time average

∗ Corresponding author. E-mail: copal@spp.keldysh.ru.

0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reservedPII S0167-2789(98)00013-X

168 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

coincides with measure (ensemble) average. Whenever this is the case, our ideas help in providing the mathematicalbackground for the numerical algorithms used for calculation of Lyapunov exponents, especially from time series.

We shall focus on iterated maps

xn+1 = f (xn), (1.1)

i.e., systems with discrete time. Indeed, in the opposite case, i.e. for an ordinary differential equationx = F(x),one can easily rewrite the evolution in the form of the “stroboscopic map”

x(t + τ) = Fτ (x(t)),

because for anautonomousdifferential equation the value ofx at time t uniquely determines all the subsequentevolution, includingx(t + τ). Denotingxn ≡ x(nτ) and f (x) ≡ Fτ (x) we obtain nothing but the iteratedmapping (1.1).

The sensitivity to initial conditions, i.e. the evolution ofinfinitesimalperturbationsδx, is governed by the linearizedsystem

δxn+1 = Df (xn)δxn,

whereDf is the Frechet derivative of the nonlinear mappingf :

Df (x) ≡ ∂fi/∂xj .In many systems the growth of the perturbations is exponential:

‖δxn‖ ∼ eλ1n ‖δx0‖ ,and its incrementλ1 is called the first Lyapunov exponent. A far more detailed information about the behavior ofperturbations is given by the complete set of Lyapunov exponentsλi , of which there areN for anN -dimensionaldynamical system. They, in general, characterize divergence/convergence of trajectories along specific directions,which are just the Lyapunov vectors.

The main result of this paper is the proof of the existence of the so-called stationary Lyapunov basis (SLB) – abasis ofN vectorse(1)(x), . . . ,e(N)(x) at almost all pointsx of attractor, such that the Lyapunov exponents can beexpressed as a sum

λi = limK→∞

1

K

K∑n=1

log |(e(i)(f (xn)),Df (xn)e(i)(xn))| ≡ limK→∞

1

K

K∑n=1

σi(xn).

The most important is the fact thatλi in such a representation can be expressed as an ordinary average along thetrajectory. This gives another view on them, enables to apply the usual technique of measure averaging, and gives arigorous basis for some methods ofλ estimating from a time series. Applicability of measure averaging means thatσi(xn) may be calculated in arbitrary order, and the pointsxn even must not belong to the same trajectory. In caseof continuous-time dynamical systemsx = F(x) the corresponding formula is

λi = limT→∞

1

T

T∫0

(e(i)(x(t)),DF(x(t))e(i)(x(t))) dt,

and the use of measure averaging also becomes possible.In Section 2 we shall consider the definition of Lyapunov exponents and Lyapunov vectors along with some

details and notations concerning their calculation. Section 3 contains the proof of stability of the Lyapunov vectors

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 169

(with respect to their perturbation phase space trajectory assumed fixed). In Section 4 we introduce the concept ofthe “stationary Lyapunov basis”, i.e. the orthonormal basis to which the evolving Lyapunov vectors converge. Somepreliminary results concerning its continuity are presented in Section 5. Then we shall discuss some conclusionswhich follow from its existence.

We should also make several remarks.1. The veryexistenceof the stationary Lyapunov basis (SLB) as the limit of the evolving tangent vectors can be

derived from the well-known Oseledec theorem [1] applied to the inverse-time systemx 7→ f−1(x), and theSLB vectors are just the eigenvectors of the corresponding Oseledec matrixΞ for f−1 – see Sections 2.1 and 4.But the approach we present gives more: it enables one to estimate the convergence to this basis along differentdirectionsseparately, which is crucial for studying thecontinuity.

2. Some results on the stability of Lyapunov vectors, including the estimate of the convergence rate, have been alsoobtained by Goldhirsch et. al. [5], but (i) their method can only been applied to ordinary differential equations, notmaps; and (ii) they have considered only infinitesimal perturbations of the Lyapunov vectors, while we presentthe results for finite perturbations as well. Neither the concept of stationary Lyapunov basis nor its continuityproperties, as far as we know, have been considered before.

3. Below we shall assume that the trajectoryxn being analyzed belongs to the system attractor, and therefore maybe infinitely continued ton → ±∞ (backward continuation may not be unique). The same is true for the pointsx, for which we shall state the existence of the stationary Lyapunov basis.

2. Definitions of Lyapunov exponents and notations

2.1. Oseledec approach

Let us consider the matrix

Ξ(n0) ≡ ([Df n0(x)]∗Df n0(x))1/2n0

and its limitΞ ≡ limn0→∞Ξ(n0) [1]. The eigenvaluesΛi of Ξ are called the Lyapunov numbers, and theirlogarithmsλi ≡ logΛi are the Lyapunov exponents. Usually it is assumed thatλ1 ≥ λ2 ≥ · · · ≥ λN . There is nocommon name for the eigenvectorsg(m) of Ξ . Lyapunov exponents are the same for almost all attractor points,though the vectorsg(m) are different. There is no good numerical algorithm forg(m) calculation, and for this reasonthey usually arise only in theoretical arguments.

Using instead of the limit the finite-time matrixΞ(n0), we obtain the so-called “finite-time Lyapunov numbers”and exponents. In contrast to the limiting values, they depend on the pointx andn0.

2.2. The approach of Benettin et al.

The alternative approach is based upon the fact that almost everyk-dimensional infinitesimal volume dΩ(k) growsas

dΩ(k)n ∼ dΩ(k)

0 exp

(k∑i=1

nλi

).

170 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

So, let us consider orthonormal vectorsv(m)n (a basis in tangent space). Their images areDf (xn)v(m)n , which usually

are neither orthogonal nor normalized, thus to obtain thebasisv(m)n+1 at the next time step one must apply theGram–Schmidt orthonormalization:

v(1)n+1 = Df (xn)v(1)n

‖Df (xn)v(1)n ‖,

v(m)n+1 =(1 − Pv(m−1)

n+1) · · · (1 − Pv(1)

n+1)Df (xn)v

(m)n

‖(1 − Pv(m−1)n+1

) · · · (1 − Pv(1)n+1)Df (xn)v

(m)n ‖

, m = 2, . . . , N,

(2.1)

wherePv is the projector onv, i.e.Pvg ≡ v · (v, g). The action of the projectors is to ensure that the resulting vectorsform an orthogonal basis. The Lyapunov exponents are defined as

λm ≡ limn′→∞

1

n′

n′∑n=1

log‖(1 − Pv(m−1)n+1

) · · · (1 − Pv(1)n+1)Df (xn)v(m)n ‖.

One may understand (2.1) like this. First the linearized mapping is applied:v(1)n , . . . , v(N)n 7→ Df (xn)v(1)n , . . . ,Df (xn)v

(N)n and then the resulting basis is reorthonormalized. At the next iteration the procedure is repeated,

and so on; thus reorthonormalizing the vectors atevery time step. It was shown in [8] that the “intermediate”reorthonormalizations can be omitted, so one can obtainv(1)n+n0

, . . . , v(N)n+n0 in justtwosteps: first under the linearized

n0th iterate of the map the basisv(1)n , . . . , v(N)n is transformed intoDf n0(xn)v(1)n , . . . , Df

n0(xn)v(N)n , and then

reorthonormalized – just once:

v(m)n+n0=

(1 − Pv(m−1)n+n0

) · · · (1 − Pv(1)n+n0

)Df n0(xn)v(m)n

‖(1 − Pv(m−1)n+n0

) · · · (1 − Pv(1)n+n0

)Df n0(xn)v(m)n ‖

, (2.2)

and the Lyapunov exponents can be obtained as

λm = limn0→∞

1

n0log‖(1 − Pv(m−1)

n+n0

) · · · (1 − Pv(1)n+n0

)Df n0(xn)v(m)n ‖. (2.3)

Note that (2.1) defines the mapping in the tangent space

v(i)n+n0= Fn0

i (xn, v(1)n , . . . , v

(N)n ). (2.4)

Below we shall show that forn0 large enough it is contracting almost everywhere. Therefore, the initially differentLyapunov vectors converge, which appears to be crucial for the proof of existence of SLB.

If we denotev(m)n ≡ Df n0(xn) v(m)n , the transformation (2.2) can be written in a more explicit form:

v(1)n+n0= v(1)n

‖v(1)n ‖,

v(2)n+n0= v(2)n − ( v(2)n , v(1)n+n0

)v(1)n+n0

‖v(2)n − ( v(2)n , v(1)n+n0)v(1)n+n0

‖,

...

v(i)n+n0= v(i)n −∑i−1

j=1( v(i)n , v(j)n+n0)v(j)n+n0∥∥∥v(i)n −∑i−1

j=1(v(i)n , v(j)n+n0

)v(j)n+n0

∥∥∥ ,(2.5)

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 171

which introducing the upper-triangle matrixR(n0) such that

R(n0)ij ≡

( v(j)n , v(i)n+n0), i = 1, . . . , j − 1,∥∥∥∥∥∥v(i)n −

i−1∑j=1

( v(i)n , v(j)n+n0)v(j)n+n0

∥∥∥∥∥∥ , i = j,

0, i = j, . . . , N,

(2.6)

becomes

v(i)n+n0= 1

R(n0)ii

·v(i)n −

i−1∑j=1

R(n0)j i v(j)n+n0

.So

i∑j=1

R(n0)j i v(j)n+n0

= v(i)n ≡ Df n0(xn)v(i)n . (2.7)

IntroducingN ×N matrixVn whose columns are the vectorsv(m)n , we can rewrite (2.7) in the matrix form:

Vn+n0R(n0) = Df n0(xn)Vn (2.8)

or

Vn+n0 = Df n0(xn)Vn(R(n0))−1. (2.9)

By the definition ofR(n0) and (2.3),

limn0→∞(R

(n0)ii )1/n0 = exp(λi), (2.10)

or n−10 logR(n0)

ii

n0→∞−→ λi . The values

l(n0)i (xn, Vn) ≡ 1

n0lnR(n0)

ii (xn, Vn) (2.11)

can be called the “snapshot Lyapunov exponents”.Though the equivalence of the approaches of both Oseledec and Benettin et al. is physically obvious, its rigorous

proof (which can be found in [7]) is rather tedious.Below we shall also use the fact that the volume of thep-dimensional parallelepiped, formed by the vectors

u(1), . . . ,u(p), can be calculated as a square root of the determinant of thep×pmatrix, whose entries are the innerproducts(u(i), u(j)), i, j = 1, . . . , p (see e.g. [9]). We shall denote

Γp(u(1), . . . ,u(p)) = det

(u(1), u(1)) · · · (u(1), u(p))· · · · · · · · ·

(u(p), u(1)) · · · (u(p), u(p))

, Γ0 = 1. (2.12)

Then, according to the above relations,

R(n0)11 R

(n0)22 · · ·R(n0)

pp =√Γp( v(1)n , . . . , v

(p)n ) ≡ √

Γp

172 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

or

R(n0)pp = √

Γp/Γp−1. (2.13)

We shall use this relation in Section 6 and Appendix A.

3. Decay of perturbations of Lyapunov vectors

Let us return to Eq. (2.9) which governs the evolution of the Lyapunov basis, and assume that at some step theyare perturbed:Vn 7→ Vn+ δVn. In this section we shall consider how this perturbation decays and how it influencesthe values ofl(n0)

i ≡ (1/n0) logR(n0)ii used in numerical calculations as the estimates forλi (3.24).

Note that changing are only the tangent vectors, while the trajectoryxn and therefore, the matricesDf (xn)remain unperturbed. In terms of (2.4), we study the behavior of perturbations of Lyapunov vectors, both infinitesimal

δVn+n0 = DVFn0(xn, Vn)δVn

and finite

1Vn+n0 ≡ Vn+n0 − V ′n+n0

= Fn0(xn, Vn)− Fn0(xn, V′n).

HereDV means differentiation only by the coordinates of the Lyapunov vectors, not by the positionx.

3.1. Estimates forR(n0)ij , finite-time eigenvectorsg(m;n0) ande(m;n0)

Whenever one uses matrix elements, a coordinate system is necessary. It is convenient to use the system ofeigenvectorsg(m;n0) of the matrix [Df n0(x)]∗Df n0(x):

[Df n0(x)]∗Df n0(x)g(m;n0) = γm;n0g(m;n0). (3.1)

They certainly depend onx; later we shall write

g(m;n0)n ≡ g(m;n0)(xn), γm;n0 ≡ γm;n0(xn)

in short if it does not cause a confusion. According to the Oseledec theorem, for almost all starting pointsx

(γm;n0)1/2n0

n0→∞−→ exp(λm). (3.2)

Since the matrix is symmetric, its eigenvectors are orthogonal; besides, it is natural to assume that they are normal-ized:‖g(m;n0)(x)‖ = 1. We also assume that they are numbered so thatγ1;n0 ≥ γ2;n0 ≥ · · · ≥ γN;n0.

Besidesg(m;n0) we shall use another set of vectors

e(m;n0)n ≡ 1√

γm;n0

Df n0(xn)g(m;n0)n , (3.3)

which form orthonormal basis in tangent space at the pointf n0(xn) = xn+n0. Later, in Section 4 we shall show thatthey are the eigenvectors of the matrix

Df n0(xn)[Dfn0(xn)]

∗,

as well as its inverse

[Df n0(xn)]−∗[Df n0(xn)]

−1 = [Df−n0(xn+n0)]∗Df−n0(xn+n0).

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 173

These eigenvectors are also assumed to be normalized:

‖e(m;n0)n ‖ = 1. (3.4)

In most cases it is assumed that all Lyapunov exponents are distinct, and therefore one can make the ratiosγj ;n0/γi;n0 ≈ en0(λj−λi) for j > i as small as necessary by choosingn0 large enough (see (3.2)).

All relations including the tangent vectors do not change if one inverts the sign of (some) of them. For this reasonwe shall regard two bases in tangent space, related to the samexn, as the same if they can be transformed one intoanother by reflectionsv(m)n 7→ −v(m)n . Therefore, it is always possible to ensure(v(m)n , g(m;n0)

n ) ≥ 0.There are two relations, which are important for the arguments presented below, their proof can be found in

Appendix A:1. For a “typical” basisVn the matrixR(n0) satisfies

R(n0)ij = O(1)

√γi;n0, i = 1, . . . , j,

and so can be written as

R(n0) = DR′, (3.5)

whereD = diag√γi;n0 is diagonal, andR′ = D−1R(n0) is upper triangle matrix, such that

Dii = √γi;n0, R′

ij = O(1), i ≤ j, R′ij = 0, i > j. (3.6)

The word “typical” means that some generic conditions must be satisfied; namely the vectorsv(i)n , i = 1, . . . , k,should not be orthogonal to the span ofg(1;n0)

n , . . . ,g(k;n0)n for anyk ≤ N . This can be expressed as

det

∣∣∣∣∣∣(v(1)n , g

(1;n0)n ) · · · (v(1)n , g

(k;n0)n )

· · · · · · · · ·(v(k)n , g(1;n0)

n ) · · · (v(k)n , g(k;n0)n )

∣∣∣∣∣∣ 6= 0, k = 1, . . . , N. (3.7)

2. There is the relation between the vectorsv(i)n+n0ande(j ;n0)

n :

(v(i)n+n0, e(j ;n0)n ) =

O(1)

√γj ;n0/γi;n0 for j > i,

1 − O(1)

(∑k<i

γi;n0

γk;n0

+∑k>i

γk;n0

γi;n0

)for j = i,

O(1)√γi;n0/γj ;n0 for j < i.

(3.8)

This relation plays the key role in the proof of existence of SLB.

3.2. Infinitesimal perturbationsδV

Perturbing the Lyapunov vectors (and therefore, the matrixR(n0)) in (2.8), one obtains

δVn+n0R(n0) + Vn+n0δR

(n0) = Df n0(xn)δVn

or

δVn+n0 = Df n0(xn)δVn(R(n0))−1 − Vn+n0δR

(n0)(R(n0))−1. (3.9)

174 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

To calculate the variationδR we can use the relation

(R(n0))∗R(n0) = V ∗n ([Df

n0(xn)]∗Df n0(xn))Vn, (3.10)

which can be derived if one multiplies the relationVn+n0R(n0) = Df n0(xn)Vn by its conjugate and takes into

account that the matricesVn representorthonormalbases and soV ∗n Vn = 1. Taking the variation of (3.10) one

obtains

δ(R(n0))∗R(n0) + (R(n0))∗δR(n0)

= δV ∗n ([Df

n0(xn)]∗Df n0(xn))Vn + V ∗

n ([Dfn0(xn)]

∗Df n0(xn))δVn. (3.11)

Below it is convenient to introduce two infinitesimal matricesρ andε:

ρ ≡ δR(n0)(R(n0))−1, ε ≡ V ∗n δVn. (3.12)

in which notations (recall thatV ∗n Vn = 1)

δR(n0) = ρR(n0), δVn = Vnε. (3.13)

Since the perturbed matrixVn + δVn is also orthogonal, [Vn + δVn]∗[Vn + δVn] = 1. This immediately impliesthat for infinitesimal perturbations

V ∗n δVn + VnδV

∗n = 0,

i.e.

ε∗ + ε = 0,

which means thatε is an antisymmetric matrix and soεii = 0.Substituting (3.13) into (3.11) we get

(R(n0))∗(ρ + ρ∗)R(n0) = ε∗V ∗n ([Df

n0(xn)]∗Df n0(xn))Vn + V ∗

n ([Dfn0(xn)]

∗Df n0(xn))Vnε,

which in view of (3.10) becomes

(R(n0))∗(ρ + ρ∗)R(n0) = ε∗(R(n0))∗R(n0) + (R(n0))∗R(n0)ε,

from what it follows that

ρ + ρ∗ = [(R(n0))∗]−1ε∗(R(n0))∗ + R(n0)ε(R(n0))−1

= (R(n0)ε(R(n0))−1)∗ + R(n0)ε(R(n0))−1.

DenotingZ ≡ R(n0)ε(R(n0))−1 we can write this equation for matrix elements:

ρij + ρji = Zij + Zji. (3.14)

By definition,R is upper triangle, and so are(R(n0))−1, δR(n0) andρ. Therefore, fori < j ρji vanishes and (3.14)becomesρij = Zij + Zji , while for i = j it givesρii = Zii . Combining these cases, one has

ρij =

0, i > j,

Zii, i = j,

Zij + Zji, i < j.

(3.15)

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 175

Now let us rewrite (3.9) in terms ofε andρ. Substituting (3.13) in (3.9), one obtains (it is easy to check the resultby direct calculation) that

δVn+n0 = Vn+n0(Z − ρ).

Introducing

ε = V ∗n+n0

δVn+n0, (3.16)

which is the analog of the matrixε, corresponding to timen+ n0, we rewrite this equation in the form

ε = Z − ρ,

and applying (3.15) one arrives at

εij =Zij , i > j,

0, i = j,

−Zji, i < j.

(3.17)

In view of (3.5) we can writeZ ≡ R(n0)ε(R(n0))−1 ≡ DR′ε(R′)−1D−1, or

Z = DZ′D−1, whereZ′ ≡ R′ε(R′)−1.

An upper triangle matrix with bounded elements is invertible, and the estimate holds

‖(R′)−1‖ ≤ N ·(

maxi

maxj |R′ij |

|R′ii |

)N,

which can be easily derived from the usual Gauss successive elimination procedure when applied to an upper trianglematrix. In view of (A.16) this gives

‖(R′)−1‖ ≤ O(1)

and therefore

‖Z′‖ ≤ O(1) · ‖ε‖ .Then, by definition

Zij ≡ (Z′)ijDii

Djj= (Z′)ij

√γi;n0

γj ;n0

,

so ∣∣Zij ∣∣ ≤ O(1) · ‖ε‖ ·√γi;n0

γj ;n0

. (3.18)

Substituting this in (3.17) we have

∣∣εij ∣∣ ≤ ‖ε‖ ·

O(1)√γj ;n0/γi;n0 for j > i,

0 for j = i,

O(1)√γi;n0/γj ;n0 for j < i.

According to the definition ofε andε Eqs. (3.12) and (3.16),

εij = (v(j)n , δv(i)n ), εij = (v(j)n+n0, δv(i)n+n0

),

176 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

so

|(v(j)n+n0, δv(i)n+n0

)| ≤ maxm

‖δv(m)n ‖ ·

O(1)√γj ;n0/γi;n0 for j > i,

0 for j = i,

O(1)√γi;n0/γj ;n0 for j < i.

(3.19)

For largen0, using the asymptotics (3.2), it becomes

|(v(j)n+n0, δv(i)n+n0

)| ≤ O(1)maxm

‖δv(m)n ‖ ·

exp(−n0|λi − λj |) for j 6= i,

0 for j = i.(3.20)

3.3. Finite perturbations

Now let us derive the estimate forfinite perturbations. Unfortunately, we cannot base upon that forinfinitesimalperturbations – mainly because (3.19) and (3.20) expand them in thevaryingbasis, and one should also take intoaccount its “rotation” if one wants to treat finite deviations. Fortunately, the necessary estimate can be derived, andrather simply, from our earlier auxiliary relation (3.8).

Indeed, let us take some different initial basesv(i)n andv(i)n . Quite obviously, for their images aftern0 iterationswe can write

(v(i)n+n0, v(j)n+n0

) =N∑k=1

(v(i)n+n0, e(k;n0)

n )(v(j)n+n0, e(k;n0)

n ).

Now let j < i; in this case, splitting the summation domain into five parts:

(v(i)n+n0, v(j)n+n0

)=∑k<j

(v(i)n+n0, e(k;n0)

n )(v(j)n+n0, e(k;n0)

n )+ (v(i)n+n0, e(j ;n0)

n )(v(j)n+n0, e(j ;n0)

n )

+∑j<k<i

(v(i)n+n0, e(k;n0)

n )(v(j)n+n0, e(k;n0)

n )+ (v(i)n+n0, e(i;n0)

n )(v(j)n+n0, e(i;n0)

n )

+∑k>i

(v(i)n+n0, e(k;n0)

n )(v(j)n+n0, e(k;n0)

n )

and estimating the inner products by means of (3.8) which holds for both bases, we have

|(v(i)n+n0, v(j)n+n0

)| ≤ O(1)√γi;n0

γj ;n0

+∑k<j

O(1)√γi;n0

γk;n0

√γj ;n0

γk;n0

+∑k>i

O(1)√γk;n0

γi;n0

√γk;n0

γj ;n0

= O(1)√γi;n0

γj ;n0

·1 +

∑k<j

γj ;n0

γk;n0

+∑k>i

γk;n0

γi;n0

= O(1)

√γi;n0/γj ;n0.

In quite a similar way (suffice it to swapi andj ) one finds that forj > i

(v(i)n+n0, v(j)n+n0

) ≤ O(1)√γj ;n0/γi;n0,

thus combining these two cases we arrive at

(v(i)n+n0, v(j)n+n0

) ≤

O(1)√γj ;n0/γi;n0 for j > i,

O(1)√γi;n0/γj ;n0 for j < i.

(3.21)

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 177

The casej = i is now obvious: orthonormality of the basesv(i)n+n0 andv(i)n+n0

implies that

N∑j=1

(v(i)n+n0, v(j)n+n0

)2 = 1,

so

|(v(i)n+n0, v(i)n+n0

)| =√

1 −∑j 6=i(v(i)n+n0

, v(j)n+n0)2.

Estimating(v(i)n+n0, v(j)n+n0

) by means of (3.21), one obtains that for largen0, when the fractions in (3.21) are verysmall,

1 ≥ |(v(i)n+n0, v(i)n+n0

)| ≥ 1 − O(1)

(∑k<i

γi;n0

γk;n0

+∑k>i

γk;n0

γi;n0

).

Combining all these three possible cases, and choosing the bases so that(v(i)n , v(i)n ) > 0 for anyn, we finallyarrive at

|(v(i)n+n0, v(j)n+n0

)| ≤

O(1)√γj ;n0/γi;n0 for j > i,

1 − O(1)

(∑k<i

γi;n0

γk;n0

+∑k>i

γk;n0

γi;n0

)for j = i,

O(1)√γi;n0/γj ;n0 for j < i.

Obviously, the deviation1v(i)n+n0≡ v(i)n+n0

− v(i)n+n0satisfies

(1v(i)n+n0, v(j)n+n0

) ≡ (v(i)n+n0− v(i)n+n0

, v(j)n+n0) = (v(i)n+n0

, v(j)n+n0)−

1 if j = i,

0 if j 6= i,

and thus

|(1v(i)n+n0, v(j)n+n0

)| ≤

O(1)

√γj ;n0/γi;n0 for j > i,

O(1)

(∑k<i

γi;n0

γk;n0

+∑k>i

γk;n0

γi;n0

)for j = i,

O(1)√γi;n0/γj ;n0 for j < i.

(3.22)

Using the asymptotics (3.2)γi,n0

n0→∞−→ exp(n0λi), one can write this as

|(1v(i)n+n0, v(j)n+n0

)| ≤

O(1) exp(−n0|λi − λj |) for j 6= i,

O(1)

∑k 6=i

exp(−n0|λi − λk|) for j = i.

(3.23)

Therefore, if all Lyapunov exponents are different, both infinitesimal and finite perturbations of Lyapunov basisdecay exponentially.

178 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

3.4. The estimate for variations of “snapshot” Lyapunov exponents

The “snapshot” Lyapunov exponents were defined as (see (2.11))

l(n0)i (xn, Vn) ≡ 1

n0logR(n0)

ii (xn, Vn).

So, their deviations due to the deviation of the Lyapunov vectorsv(m)n are

δl(n0)i (xn, Vn) = 1

n0δ(logR(n0)

ii (xn, Vn) = 1

n0δR

(n0)ii /R

(n0)ii .

Recalling thatρ ≡ δR(n0) · (R(n0))−1 while δR(n0) and(R(n0))−1 are upper triangle matrices, we see thatρii =δR

(n0)ii /R

(n0)ii , so

δl(n0)i (xn) = 1

n0ρii .

Meanwhile, due to (3.15)ρii = Zii , and thusδl(n0)i = Zii which using (3.18) becomes

|δl(n0)i | ≤ 1

n0O(‖ε‖), (3.24)

whereεij ≡ (δv(i)n , v(j)n ) and so‖ε‖ is a measure of the deviation of the Lyapunov vectorsv(m)n which form thebasisVn.

4. Stationary Lyapunov basis (SLB)

According to (3.8), if startedmorethann0 iterations ago, the Lyapunov vectors at the given pointxn are veryclose toe(m;n0)

n−n0:

‖v(m)n − e(m;n0)n−n0

‖ n0→∞−→ 0,

where

e(m;n0)n−n0

≡ 1√γm;n0

Df n0(xn−n0)g(m;n0)n−n0

,

see (3.3). Therefore, if the latterconvergeasn0 → ∞, the evolving Lyapunov vectors (if started long before) willonly depend on the pointx where they are observed. In other words, in such a case the “extended” phase space,introduced in [8], reduces to the original phase space. The limiting vectors are naturally called “stationary Lyapunovvectors”.

As well asg(m;n0)n andγm;n0, thesee(m;n0)

n depend onn only implicitly, via xn:

e(m;n0)n−n0

≡ e(m;n0)(xn).

To study this dependence we shall base upon the fact thate(m;n0)(x) are the eigenvectors of the matrix

An0(x)≡ ([Df n0(f−n0(x))]−1)∗[Df n0(f−n0(x))]−1

= [Df−n0(xn)]∗Df−n0(xn).

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 179

Indeed, denotingx−n0 ≡ f−n0(x) and using the definition ofe(m;n0)n in the form

e(m;n0)(x) ≡ 1√γm;n0(x−n0)

Df n0(x−n0)g(m;n0)(x−n0),

one easily obtains

An0(x)e(m;n0)(x)= 1√

γm;n0(x−n0)([Df n0(x−n0)]

−1)∗g(m;n0)(x−n0)

= 1√γm;n0(x−n0)

Df n0(x−n0)([Dfn0(x−n0)]

∗Df n0(x−n0))−1g(m;n0)(x−n0),

and since by definition

[Df n0(x)]∗Df n0(x)g(m;n0)(x) = γm;n0(x) · g(m;n0)(x),

we have

An0(x)e(m;n0)(x)= 1

γm;n0(x−n0)√γm;n0(x−n0)

·Df n0(x−n0)g(m;n0)(x−n0)

= 1

γm;n0(x−n0)· e(m;n0)(x). (4.1)

Also, we shall use the following obvious factorization of the matrixAn(x):

An(x) ≡ ([Df n(x−n)]−1)∗[Df n(x−n)]−1 = (J ∗0 J

∗1 · · · J ∗

n )(JnJn−1 · · · J0), (4.2)

wherex−n ≡ f−n(x) andJn ≡ Df−1(x−n) is the Jacobian of the inverse transformation at the pointx−n. Herethe absence of a subscript means the moment when the trajectory hits the pointx, and the subscript “−n” means “niterations before it had hit that point”.

So we conclude that if the evolution starts from the pointx−n, and the set of Lyapunov vectorsv(m;n)−n =

g(m;n0)(x−n), then aftern iterations they will be mapped intox = f n(x−n) andv(m;n) = e(m;n)(x), respectively,which using notation (2.4) can be written as

e(m;n)(x) = Fnm(x−n, g(1;n0)(x−n), . . . ,g(N;n0)(x−n)). (4.3)

So the basise(1;n)(x), . . . ,e(N;n)(x) is the set of Lyapunov vectors (at this point) corresponding to some particularchoice of their “initial” orientationn iterations before.

Now let us take some greatern > n. Again we have

e(m;n)(x) = F nm(x−n, g(1;n)(x−n), . . . ,g(N;n)(x−n)),

which, according to [8] (see also the beginning of Section 3), can be as well achieved in two steps: first we passfrom x−n to f n−n(x−n) = x−n while the Lyapunov vectors are transformed into

E (m) = F n−nm (x−n, g(1;n)(x−n), . . . ,g(N;n)(x−n)),

and then fromx−n to f n(x−n) = x, this time the Lyapunov vectors being transformed into the “final”v(m;n):e(m;n)(x) = Fn

m(x−n, E (1), . . . , E (N)). (4.4)

In other words, inn iterations the basisE (m) is transformed intoe(m;n)(x).

180 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

Comparing this with (4.3) we see that the difference betweene(m;n) ande(m;n) can be thought of as being causedby the difference between their predecessors (g(m;n)(x−n) andE (m), respectively)n iterations before. Therefore,it can be estimated using (3.22), only the number of iterations done is now denotedn instead ofn0:

|(e(i;n) − e(i;n), e(j ;n))| ≤

O(1)

√γj ;n/γi;n for j > i,

O(1)

(∑k<i

γi;nγk;n

+∑k>i

γk;nγi;n

)for j = i,

O(1)√γi;n/γj ;n for j < i.

(4.5)

Applying the asymptotics (3.2)γi,n0

n0→∞−→ exp(n0λi), one can rewrite this a bit more graphically:

|(e(i;n) − e(i;n), e(j ;n))| ≤O(1)

∑k 6=i

exp(−n |λk − λi |) if j = i,

O(1) exp(−n ∣∣λj − λi∣∣) if j 6= i.

(4.6)

Remark 1.Note that as well as (3.22), these estimates hold not for all orientations of Lyapunov vectors. Namely,they fail if (g(m;n)(x−n), E (m)) = 0 for somem. Since bothg(m;n)(x−n) andE (m) are uniquely determined byx, we conclude that, though (4.5) and (4.6) are valid for most pointsx, there can exist some exceptional points atwhich they fail.

Therefore, there is a convergence asn → ∞, and one may define thelimiting vectors which are just the stationaryLyapunov basis at the pointx as

e(m)(x) = limn→∞ e(m;n)(x),

wheree(m;n)(x) are the eigenvalues of the matrixAn(x) (4.2). According to (4.1), its eigenvalues areαm;n(x) =1/γm;n, and thusα1;n(x) ≤ α2;n(x) ≤ · · · ≤ αN;n(x). They obviously have the following asymptotics asn → ∞:

α1/nk;n (x)

n→∞−→ exp(−2λk),

whereλ1 ≥ λ2 ≥ · · · ≥ λN are the Lyapunov exponents of the mappingf .

Remark 2.Other stationary bases may exist, but converging to them are only very specific initial vectors (thosewhichdo notsatisfy (3.7)). In other words, thesestationarybases areunstable, and tiny deviation of the initial basisresults in its convergence to the stablee(m). Further, the latter isuniquein view of the results of Section 3 (up tochanging the directions of the vectors or reflection transformations of the basis).

Remark 3.The very existence of the limiting basise(m)(x) = limn0→∞ e(m;n0)(x) can be derived from the Oseledectheorem applied to the inverse dynamical systemf−1(x), provided the latter exists, but our approach givesseparateestimates for the convergence along different directions. That proves to be crucial when studying continuity of thestationary Lyapunov vectors.

Remark 4.For non-invertible systems there are several trajectories through the same point (merging at it), and eachof them can have its own stationary Lyapunov basis. But if we consider only one of the trajectories, then againthe stationary Lyapunov basis becomes unique. In particular, in time series analysis we usually deal with a singletrajectory, and therefore it is possible to apply the concept of SLB for both invertible and non-invertible dynamicalsystems.

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 181

5. Continuity of the field of stationary Lyapunov vectors

5.1. Why we need equicontinuity. Infinitesimal perturbationsδx

Since the matrixAn(x) depends on the pointx, so do its eigenvectors, and therefore the stationary Lyapunovbasis, their limit, also changes from point to point. It is interesting to find out, whether the vectors of SLB changecontinuously withx or not, i.e. to study the continuity with respect to the perturbations of basic trajectory. Let usconsider two infinitely close pointsx andx + δx, such that (1) both of these points belong to the same attractor, (2)there exists the SLB in both of them, (3) the trajectories through these points are “typical”, i.e. the sets ofλi forthem coincide and (4) all exponentsλi are different.

According to the theory of perturbations of symmetric operators, forany finiten the eigenvectors ofAn(x) changecontinuously withx. This does not guarantee thattheir limit is continuous, though. Indeed,if e(m;n), as a functionof x, becomes more and more ragged, though remaining continuous, its limit may be discontinuous. To ensure the

smoothness of the limiting vectors, they should beequicontinuous, i.e. there must exist some functionq(r)r→∞−→ 0

such that

|e(m;n)(x + δx)− e(m;n)(x)| ≤ q(‖δx‖)for anyn large enough. To find if this inequality holds, and under which conditions, we must have an estimate fordeviation of the eigenvectors for an arbitraryn. Note that the matrixAn(x) itself hasno limit asn → ∞.

In the remainder of this section we will writee(m;n) andαm;n instead ofe(m;n)(x) andαm;n(x)wherever this doesnot cause a confusion.

Denoting the perturbed pointx′ ≡ x + δx, we can write the perturbationδAn ≡ An(x′)− An(x) of the matrix

An = (J ∗0 J

∗1 · · · J ∗

n )(JnJn−1 · · · J0)

as

δAn =n∑k=0

J ∗0 · · · J ∗

k−1 · δJ ∗k · J ∗

k+1 · · · J ∗n · Jn · · · J0

+n∑k=0

J ∗0 · · · J ∗

n · Jn · · · Jk+1 · δJk · Jk−1 · · · J0, (5.1)

where

δJk ≡ Df−1(x−k)−Df−1(x′−k) ≡ Df−1(f−k(x))−Df−1(f−k(x′)).

The corresponding perturbations of the eigenvectors are (see, e.g. Landau and Lifshitz, “Quantum Mechanics”or application of this technique in the context of Lyapunov exponents in [5]):

δe(i;n) =∑j 6=i

(e(j ;n), δAne(i;n))αi;n − αj ;n

e(j ;n). (5.2)

The main question is: when the deviation‖δe(i;n)‖ remains bounded forn → ∞?We must note that applicability of this relation requires more slightly stronger condition than the degeneracy

of of Lyapunov specturm. We shall assume that no quasidegeneracy occurs, i.e. the shift in the specturm ofαi;nis less than the spectral difference. In Appendix B it is shown that with the help of the above formulas and the

182 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

estimate exp[n · (λi + ϕ1n)] ≤ αi;n ≤ exp[n · (λi + ϕ2n)], whereϕjnn→∞−→ +0, it is possible to obtain the

relation

‖δe(i;n)‖ ≤ O(1) · ‖δx‖∑j<i

n∑k=0

exp[k(λi − λj − λN + ψk)] +∑j>i

n∑k=0

exp[k(λj − λi − λN + ψk)]

.(5.3)

Now let us assume that there exists suchq > 0 thatλi − λj − λN ≤ −q for j < i, andλj − λi − λN ≤ −q forj > i. In this case, it follows from (5.3) that

‖δe(i;n)‖ ≤ O(1) · ‖δx‖n∑k=0

exp[−k(q − ψk)].

Beginning with suchk∗ thatψk∗ < q, its terms decay exponentially, so the series converges, andδe(i;n) is boundedfor n → ∞.

Due to the arbitrariness of theq, we can say that the deviations‖δe(i;n)‖ are bounded forn → ∞ if

λj > λi − λN for j < i, λj < λi + λN for j > i. (5.4)

In other words, in this case the eigenvectore(i;n)(x) as a function ofx is equicontinuous, and so its limit, the staticLyapunov vectore(i)(x) = limn→∞ e(i;n)(x), continuously depends onx.

Remark 5.It should be emphasized that the above consideration, based on the estimates from above, may onlyprovidesufficient, butnot necessaryconditions for the continuity of the stationary Lyapunov vector field. In otherwords, there may be continuity even when inequalities (5.4) cease to hold.

In the framework of our approach,all of the stationary Lyapunov vectors are continuous if (5.4) holds for everyi. In this case it implies

λi−1 > λi − λN, λi+1 < λi + λN,

which means that

λN−1 > 0, λN−2 > −λN, . . . , λN−k > (−k + 1)λN .

and therefore,unlessN = 2,

N∑m=1

λm > λN + λN−1 + λN−2 > 0,

which is impossibleif the dynamics isinvertible. So, our approachneverstates continuity ofall vectors if thedimension is greater than 2.

Now let us discuss two simple examples and consider the conditions, under which some of vectors of SLB ofdissipative systems for which

∑i λi < 0 are continuous.

Example 1.N = 2. Directly from (5.4) we see that for both vectors to be continuous it suffices thatλ1 be positive.That is,any invertible 2D chaotic map (e.g. the Henon map) possesses a continuous Lyapunov basis.

Example 2.N = 3. Again directly from (5.4) we see that:– e(1) is continuous ifλ1 > 0, 2λ2 < λ1 + λ2 + λ3 < 0, λ3 < 0;

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 183

– continuity ofe(2) can neither be proved nor rejected using the approach presented;– e(3) is continuous ifλ1, λ2 > 0, λ3 < 0.

That is, there are three cases: continuity can be proved for (a)e(1); (b) e(3); (c) none.For the Lorenz system (σ = 10, r = 28,b = 8/3, λ1 = 0.9, λ2 = 0, λ3 = −14.8) our experiments show that

e(3) is continuous even despiteλ2 = 0.We should emphasize again that all these aresufficient, notnecessaryconditions, and our approachneverenables

one to provediscontinuityof the stationary Lyapunov vectors.

5.2. Continuity and finite perturbations

Unfortunately, it is impossible to extend our approach directly tofiniteperturbations.Indeed, at first glance we could have proceeded as follows. Take two pointsx andx′ and connect them by a

smooth curveξ(t), 0 ≤ t ≤ 1 of the lengthl, ξ(0) = x, ξ(1) = x′. Then, assuming thate(i;n)(x) is differentiable,we can write

e(i;n)(x′)− e(i;n)(x) =1∫

0

dξ∂e(i;n)(ξ)∂ξ

and therefore

‖e(i;n)(x′)− e(i;n)(x)‖≤1∫

0

dξ

∥∥∥∥∥∂e(i;n)(ξ)∂ξ

∥∥∥∥∥ ≤ l supx∈ξ(t)

‖∇e(i;n)(x)‖.

However, this obviously requires that (1)ξ(t) also belong to attractor and (2) in all its points there exists thestationary Lyapunov basis. Moreover since chaotic attractors usually have very complex geometry, this curve mayprove very long even for very close pointsx andx′. In other words, some points may be close, even arbitrary close, ineuclidean space, i.e.‖x−x′‖ small, but remote in the “attractor’s metric”, i.e.l may be large, e.g. because of fractalstructure of the attractor. In such situations our approach cannot guarantee the closeness ofe(i)(x) ande(i)(x′).

Indeed, our calculations for the Henon mapping confirmed that at most pointsx ‖e(i)(x′) − e(i)(x)‖ → 0 as‖x−x′‖ → 0. But at some points the deviations ofe(i) are O(1) for ‖1x‖ as small as 10−5. The origin of this effectis not completely clear, since the estimates of‖∇e(i)(x)‖ show that usually it does not exceed 10, but sometimes,at rare points, may reach the values of 1010, and even greater.

On the contrary, for the Lorenz system the continuity ofe(3)(x) can be easily detected: maxx′ ‖e(3)(x′)−e(3)(x)‖goes to 0 approximately as‖x − x′‖.

Therefore, the numerical verification of the continuity of SLB can be a very difficult problem.

6. Stationary Lyapunov basis and calculation of Lyapunov exponents

6.1. Lyapunov exponents as measure averages

Now let us consider the consequences which follow from the existence of stationary Lyapunov basis and theestimates of the rate of convergence towards it. But first let us return to the algorithm of Benettin et al. of calculationof Lyapunov exponents and introduce the “one-step contribution” terms for them.

184 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

Let us denote

S(n0)i (xn, Vn) = logR(n0)

ii (xn, Vn)

(see (2.6)). As mentioned in Section 2.2, orthonormalization of the evolving vectors can be performed after everyiteration, which in matrix notation means that the matrixR(n0)(xn, Vn) is the product of single-step matricesR(1)(xk, Vk):

R(n0)(xn, Vn) = R(1)(xn, Vn)R(1)(xn+1, Vn+1) · · ·R(1)(xn+n0−1, Vn+n0−1),

which, in its turn, means thatS(n0)i is the sum of stretchings per one iterationS(1)i (xk, Vk) which later we shall

denote simply asSi(xk, Vk). By definition,

Si ≡ logR(1)ii = log |(v(i)n+1,Df (xn)v(i)n )|.

With these notations, the Lyapunov exponents are

λi = limn0→∞

1

n0

n0∑k=0

Si(xk, Vk) ≡ limn0→∞

1

n0

n0∑k=0

log |(v(i)k+1,Df (xk)v(i)k )|. (6.1)

This means thatλi is the time-average of a function which depends both on the phase space pointx and the orientationof the tangent vectorsv(i)k . It is the latter dependence that makes it impossible to replace the time averaging withthe measure averaging without passing to the “extended phase space” which incorporates both usual and tangentcoordinatesx andV [8].

But if the SLB exists, the situation is quite different. Indeed, now the evolving Lyapunov vectors converge totheir stationary directionse(m)(x), thus after some transient periodVk = E(xk), i.e. their orientation is uniquelydetermined by the phase space point. Therefore, (6.1) becomes

λi = limn0→∞

1

n0

n0∑k=0

σi(xk), (6.2)

where (xk+1 is replaced byf (xk))

σi(xk) = Si(xk, E(xk)) = log |(e(i)(f (xk)),Df (xk)e(i)(xk))|.Now the averaged function depends only on the phase space pointx, and one can replace the time averaging withthe measure averaging in the original phase space:

λi =∫σi(x)µ(dx). (6.3)

Note that in the sum (6.2), which can be considered as a numerical approximation to this integral, the terms canbe calculatedindependentlyfrom each other and in anarbitrary order. Sometimes this may be very important fornumerical implementation.

For continuous time dynamical systemsx = F(x), when the time stepτ , corresponding to one iteration, maybe arbitrarily small, we can obtain the relation for dS/dt instead ofS. In continuous time systems the Lyapunovexponents are related totime, not to number of iterations – thus (6.1) becomes

λi = limn0→∞

1

τn0

n0∑k=0

Si(xk, Vk),

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 185

which can be written as

λi = limT→∞

1

T

T∫0

Si(x(t), V (t))

τdt.

For τ 1 we can use approximationsf (x) ≈ 1 + τF (x), v(i)n+1 ≈ v(i)n which leads to

Si(x(t), V (t))

τ≈ (v(i), DF(x)v(i)),

thus

λi = limT→∞

1

T

T∫0

(v(i)(t),DF(x(t))v(i)(t)) dt.

This relation, but without stationary basis context, has been obtained in [5,6]. Again the expression for Lyapunovexponents can be rewritten as a measure average

λi =∫(e(i)(x),DF(x)e(i)(x))µ(dx).

6.2. Some consequences of SLB existence

Relations (6.1)–(6.3) are convenient for showing the role of SLBE(x).(1) Since during the evolution the basisVk converges toE(xk), the first iterations, whenVk has not yet converged,

produce an error in “snapshot” Lyapunov exponents. Approximately, it can be estimated using (3.24). This errordecays very slowly as

1λi ∼ 1

n‖V0 − E‖ ,

where‖V0 − E‖ denotes the deviation of the starting Lyapunov vectors from their stationary directions. Sincethis deviation decays rapidly, the accuracy will be substantially improved if one starts averaging a bit later, thusdiscarding the very first iterations. This means that several first terms must be excluded from the sum (6.1);their optimal number depends on the convergence rate. Everyone who has calculated Lyapunov exponentsnumerically usually knows this fact from one’s own experience, though.

(2) The relation (6.3) enables one to propose another approach to the calculation of Lyapunov exponents. It isknown that usually most initial probability distributions converge to the invariant measure. Let us takeM pointsx0j , j = 1, . . . ,M, on the attractor of the dynamical system, and let them evolve during some timeT . The lattershould be large enough for convergence of distribution to invariant and for convergence of Lyapunov vectorsto the stationary Lyapunov basis. After that has occurred, at the end of each trajectory we calculate the singlevalueσi(f T (x0j )). Then, according to (6.3)

1

M

M∑j=1

σi(fT (x0j )) → λi asM → ∞, T → ∞. (6.4)

This method is of little practical importance for calculation of Lyapunov exponents from equations of motion,while it has been successfully applied for time-series analysis [11,12], when trajectories (which are actuallyjust different parts of the same long trajectory) remain close only for some limited time.

186 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

(3) Now let us briefly show, how this idea can be used in time-series analysis. Let there be a vector time seriesx1, . . . , xM , it may be either points in original phase space or delay reconstructions of a scalar time series. Letus consider a pointxi and its neighborhood, the pointsxjk , ‖xjk − xi‖ < ε, k = 1, . . . , K. If ε is small, thevectorsv0,k(xi) = xjk − xi can be considered as approximately tangent vectors. The same usually will be truefor nf forward andnb backward iterates:vn,k(xi) = xjk+n − xi+n, −nb ≤ n ≤ nf . The old idea is to use thevectorsvn,k to obtain the estimates of Lyapunov exponents.

Suppose thatnb iterates are enough for a tangent vector to converge with good accuracy to the direction ofe(1)(xi). Then the ratio‖vn,k‖/‖v0,k‖ can be considered as an approximation forR

(n)11 (see (2.6)). For better

performance of the method it is possible to average this ratio over allK neighbors, and therefore,

σ(n)1 (xi) ' log

(1

K

K∑k=1

‖vn,k(xi)‖‖v0,k(xi)‖

)(the upper index(n) means that we estimate theσ1 for thenth iterate of our map, i.e.f n(x)). Then, accordingto (6.2),

nλ1 ' 1

M

M∑i=1

σ(n)1 (xi) = 1

M

M∑i=1

log

(1

K

K∑k=1

‖vn,k(xi)‖‖v0,k(xi)‖

). (6.5)

Analogs of this formula with some differences, which are inessential for us now, has been proposed in [11,12] forestimatingλ1 from a time series. Our results form the rigorous mathematical background for this approach. Theimportant feature of this approach is that it does not require approximation of the matrixDf and it is insensitiveto the dimension of the phase space, which is essential for attractor reconstruction from a scalar time series.

Note that without SLB one has to follow the usual definitions of Lyapunov exponents, and to ensure consis-tency of the “tangent” vectorsv at successive points. This creates additional problems with vectors selection,accuracy etc. – see [10] for the details.

In a similar way it is possible to propose methods for estimatingλ2. We assume that duringnb iterates thepairsvn,k1(xi), vn,k2(xi) converge with good accuracy to spane(1)(xi), e(2)(xi). Then one should calculate2D volumes formed by these pairs. This can be done for example by using the Gram determinant – see (2.12):

Γ(n)2,k1,k2

(xi) = det

((vn,k1(xi), vn,k1(xi)) (vn,k1(xi), vn,k2(xi))

(vn,k2(xi), vn,k1(xi)) (vn,k2(xi), vn,k2(xi))

).

According to (2.13),

R(n)11 R

(n)22 '

√Γ(n)2 /Γ

(0)2 ,

and, as in the case of (6.5), we obtain

n(λ1 + λ2) ' 1

M

M∑i=1

log

1

N(pairs)

∑k1,k2

√√√√Γ(n)2,k1,k2

(xi)

Γ(0)2,k1,k2

(xi)

. (6.6)

In [10] the attempts to estimateλ2 have been done by other methods also using 2D phase volumes, and it waspointed out that estimates for it are worse than forλ1. It is natural to expect similar problems for (6.6) as well,though averaging over many pairs may improve situation.

In principle, it is possible to propose generalization of this approach for otherλi .Therefore, the SLB concept gives new and interesting viewpoint on the Lyapunov exponents and the problem

of their computation. Though it does not give practical outcome at once, it may serve as a basis for a number ofalgorithms.

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 187

7. Conclusion

In this paper we have studied the mathematical background of some algorithms for calculation of Lyapunovexponents.

(i) We have shown that the evolving basis of Lyapunov vectors, used in the algorithm of Benettin et al. [4],converges to the stationary Lyapunov basis, which is unique at every pointx of the attractor.

(ii) We estimated the rate of convergence and stability of this basis against trajectory perturbation(iii) In contrast with the paper [5], our results on convergence rate are applicable for dynamical systems with both

continuous and discrete time. This is also important for applications in time-series analysis, since usually thereconstructed system is a mapping.

Acknowledgements

This work has been partially supported by grants 96-01-01161 and 96-02-18689 from Russian Foundation forBasic Research.

Appendix A. Proof of (3.5) and (3.8)

A.1. Estimation of the diagonal elementsR(n0)ii

Since the eigenvectorsg(m;n0)n form an orthonormal basis in the tangent space at the pointxn, we can expand

another element of this space, the Lyapunov vectorv(i)n as

v(i)n =N∑j=1

cijg(j ;n0)n . (A.1)

Obviously,

v(i)n = Df n0(xn)v(i)n =N∑k=1

cikDfn0(xn)g(k;n0)

n =N∑k=1

cik√γk;n0e(k;n0)

n

and

(v(i)n , v(j)n ) =

N∑k=1

cikcjkγk;n0.

The coefficientscik form an orthogonal matrix – the transition matrix from one orthogonal basis to another, i.e.∑Nk=1 cikcjk = δij .

Using the Gram matrix forv(i)n and the definition of det(·), we can write

Γp = det‖(v(i)n , v(j)n )‖

= det

∣∣∣∣∣∣ c11 · · · c1N

· · · · · · · · ·cp1 · · · cpN

γ1;n0

· · ·γN;n0

c11 · · · cp1

· · · · · · · · ·c1N · · · cpN

∣∣∣∣∣∣=

∑1≤i1<i2···<ip≤N

(γi1;n0γi2;n0 . . . γip;n0) · (Ji1i2···ip )2,

188 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

where

Ji1i2···ip = det

c1i1 · · · c1ip· · · · · · · · ·cpi1 · · · cpip

.Therefore,

Γp

γ1;n0γ2;n0 . . . γp;n0

=∑

1≤i1<i2···<ip≤N

(γi1;n0γi2;n0 . . . γip;n0

γ1;n0γ2;n0 · · · γp;n0

)· (Ji1i2···ip )2 ≥ (J12···p)2.

By choosingn0 large enough it is possible to make all ratios in the first brackets≤ 1. Then, for thisn0

Γp

γ1;n0γ2;n0 . . . γp;n0

≤∑

1≤i1<i2···<ip≤N(Ji1i2···ip )

2

= det

∥∥∥∥∥∥ c11 · · · c1N

· · · · · · · · ·cp1 · · · cpN

1· · ·

1

c11 · · · cp1

· · · · · · · · ·c1N · · · cpN

∥∥∥∥∥∥ = 1.

This means that forn0 large enough and allp = 1, . . . , N ,

(J12...p)2γ1;n0γ2;n0 . . . γp;n0 ≤ Γp ≤ γ1;n0γ2;n0 . . . γp;n0,

and, using the relationRpp = √Γp/Γp−1,

|J12···p|√γp;n0 ≤ Rpp ≤ 1

|J12···p−1|√γp;n0.

Therefore, one obtains that

O(1)√γp;n0 ≤ R(n0)

pp ≤ O(1)√γp;n0 (A.2)

as long as

J12...k = det

∣∣∣∣∣∣∣∣c11 c12 · · · c1k

c21 c22 · · · c2k

· · · · · · · · · · · ·ck1 ck2 · · · ckk

∣∣∣∣∣∣∣∣ 6= 0, k = 1, . . . , p, cij = (v(i)n , g(j ;n0)n ),

i.e. as long as the projections of the vectorsv(1)n , . . . , v(p)n onto the span ofg(1;n0)

n , . . . ,g(p;n0)n are linearly

independent.

A.2. Estimation of the off-diagonal elementsR(n0)ij

Now assume that conditions (3.7) hold. Using expansion (A.1) one obtains

v(i)n ≡ Df n0(xn)v(i)n =N∑j=1

cijDfn0(xn)g

(j ;n0)n ≡

N∑j=1

√γj ;n0cije

(j ;n0)n , (A.3)

where the vectorse(m;n0)n have been defined in (3.3), and likeg(m;n0)

n , they form orthonormal basis, but in tangentspace atxn+n0.

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 189

Let us also expand ine(m;n0)n the vectorv(i)n+n0

:

v(i)n+n0=

N∑j=1

Cij e(j ;n0)n (A.4)

and calculate the coefficientsCij . Substituting (A.3) in (2.7) and comparing with (A.4), we have

Cij =√γj cij −∑i−1

m=1Cmj∑Nk=1

√γk;n0cikCmk

R(n0)ii

. (A.5)

Now taken0 1. In this case, according to (3.2)√γm;n0 ≈ en0λm , so

√γk;n0 √

γk′;n0 for k′ > k. For i = 1,

using the estimate forR(n0)11 (A.2) one obtains

C1j ≤ O(1) ·√γj ;n0/γ1;n0.

Let us show, by induction, that it implies

Cij ≤ O(1) ·√

γj ;n0/γi;n0 for j ≥ i,√γi;n0/γj ;n0 for j ≤ i.

(A.6)

Assume that it holds fori = 1 , . . . , I − 1. Will it then be true fori = I?Denoting

C′ij ≡ Cij ·

√γi;n0/γj ;n0 for j ≥ i,√γj ;n0/γi;n0 for j ≤ i,

we can write our supposition as∣∣C′ii′∣∣ ≤ O(1) for i < I and anyi′. (A.7)

In terms ofC′ij , (A.5) becomes forj ≥ i

C′ij =

√γi;n0

R(n0)ii

cij −i−1∑m=1

C′mj

m∑k=1

cikC′mk +

N∑k=m+1

γk;n0

γm;n0

cikC′mk

,and substituting in it the estimate forR(n0)

ii (A.2), one obtains

C′ij = O(1)− O(1)

i−1∑m=1

C′mj

m∑k=1

cikC′mk − O(1)

i−1∑m=1

C′mj

N∑k=m+1

γk;n0

γm;n0

cikC′mk for j ≥ i.

Now let j ≥ I andi = I . Sincem < i = I , (A.7) implies thatC′mj ≤ O(1) andC′

mk ≤ O(1), so

|C′Ij | ≤ O(1) for j ≥ I. (A.8)

Now let us estimateC′Ij for j < I .

The basisv(m)n+n0 is orthogonal:(v(i)n+n0

, v(j)n+n0) = 0, which using expansion (A.4) and the orthogonality of the

eigenvectorse(m;n0)n can be written as

N∑k=1

Ci,kCj,k = 0 for j 6= i.

190 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

Splitting the summation domain 1≤ k ≤ N into three parts and using our notation ofC′ij , one rewrites the above

equality as

j−1∑k=1

C′i,kC

′j,k

√γi;n0

γk;n0

γj ;n0

γk;n0

+√γi;n0

γj ;n0

i−1∑k=j

C′i,kC

′j,k +

N∑k=i

C′i,kC

′j,k

√γk;n0

γi;n0

γk;n0

γj ;n0

= 0 for j < i,

which after multiplying by√γj ;n0/γ i becomes

j−1∑k=1

C′i,kC

′j,k

γj ;n0

γk;n0

+i−1∑

k=j−1

C′i,kC

′j,k + C′

i,iC′j,i +

N∑k=i+1

C′i,kC

′j,k

γk;n0

γi;n0

= 0 for j < i. (A.9)

Now let us takei = I . Denoting

ξj ≡ C′ij /C

′ii

we can rewrite (A.9) in the form

j−1∑k=1

γj ;n0

γk;n0

C′j,kξk +

I−1∑k=j

C′j,kξk = −C′

j,I −∆j , j = 1, . . . , I − 1, (A.10)

where

∆j ≡ 1

C′II

N∑k=I+1

γk;n0

γI ;n0

C′j,kC

′I,k.

In our case, i.e. forj < I and largen0, it is negligible. Indeed, now according to (A.8)C′I,k ≤ O(1); and also

C′j,k ≤ O(1) because of (A.7). Therefore,

|∆j | =∣∣∣∣∣∣ 1

C′II

N∑k=i+1

C′i,kC

′j,k

γk;n0

γi;n0

∣∣∣∣∣∣ ≤ O(1)N∑

k=i+1

γk;n0

γi;n0

≤ O(1)γi+1;n0

γi;n0

1 for j < I.

If not for this tiny term∆j in the right-hand side, (A.10) would be an(I −1)× (I −1) linear system with respectto I − 1 variablesξ1, . . ., ξI−1. Moreover, it would be nearly an upper-triangle system, because the lower-trianglematrix elements,(γj ;n0/γk;n0)C

′j,k, are very small: hereC′

j,k = O(1) according to (A.7), while forn0 1

γj ;n0

γk;n0

≤ γj ;n0

γj−1;n0

1.

Therefore, the first approximation to the solution,ξ (1), can be found from the upper triangle system

i−1∑k=j−1

C′j,kξ

(1)k = −C′

j,I , j = 1, . . . , I − 1, (A.11)

which matrix elements and right-hand side are all≤ O(1). Moreover ifn0 1, for its diagonal elements we havethe estimateC′

jj ≈ 1. Indeed, the condition

‖v(j)n+n0‖ = 1

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 191

implies∑Nk=1C

2jk = 1 or

|Cjj | =

√√√√√1 −j−1∑k=1

C2jk −

N∑k=j+1

C2jk. (A.12)

Recalling the definition ofC′ij , we derive from this

|C′jj | =

√√√√√1 −j−1∑k=1

(C′jk)

2γj ;n0

γk;n0

−N∑

k=j+1

(C′jk)

2γk;n0

γj ;n0

≥√√√√1 −

(γj ;n0

γj−1;n0

+ γj+1;n0

γj ;n0

) N∑k 6=j(C′jk)

2. (A.13)

For j < I we haveC′jk ≤ O(1) according to (A.7). Therefore,

|C′jj |

n0→∞−→ 1. (A.14)

Thus we conclude that the matrix of the linear system (A.11) is well-defined and invertible; and so is itssmallperturbation, the matrix of (A.10). Therefore, its solution is of the same order of magnitude as the right-hand side:

|ξk| ≤ O(1).

By our definition,ξj ≡ C′Ij /C

′II whileC′

II is not estimated as yet. Though, similarly to (A.13) we can write

(C′II )

2 ≥ 1 −(γI ;n0

γI−1;n0

+ γI+1;n0

γI ;n0

)I−1∑k=1

(C′Ik)

2 +N∑

k=I+1

(C′Ik)

2

= 1 −(γI ;n0

γI−1;n0

+ γI+1;n0

γI ;n0

)(C′II )

2I−1∑k=1

(ξk)2 +

N∑k=I+1

(C′Ik)

2

.The values ofξk for k < I were estimated just above; andC′

Ik for k > I satisfy (A.8). Thus,

(C′II )

2 ≥ 1 − O(1) ·(γI ;n0

γI−1;n0

+ γI+1;n0

γI ;n0

)− O(1) ·

(γI ;n0

γI−1;n0

+ γI+1;n0

γI ;n0

)(C′II )

2,

so

|C ′II | ≥

√1 − O(1) · (γI ;n0/γI−1;n0 + γI+1;n0/γI ;n0)

1 + O(1) · (γI ;n0/γI−1;n0 + γI+1;n0/γI ;n0).

On the other hand, by definitionC′ii = Cii while, according to (A.12),|Cii | ≤ 1. Therefore,|C′

II | ≤ 1 and so√1 − O(1) · (γI ;n0/γI−1;n0 + γI+1;n0/γI ;n0)

1 + O(1) · (γI ;n0/γI−1;n0 + γI+1;n0/γI ;n0)≤ ∣∣C′

II

∣∣ ≤ 1

from what it follows that

C′IIn0→∞−→ 1,

and thus

C′Ij ≡ ξjC

′II = O(1) for j < I.

192 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

Combining this with the estimate (A.8), we conclude that for largen0

|C′Ij | ≤ O(1) for anyj,

which means that

CIj ≤ O(1) ·√

γj ;n0/γI ;n0 for j ≥ I,√γi;n0/γI ;n0 for j ≤ I.

Therefore, we have shown that if the estimate (A.6) holds fori = I − 1, then it is also true fori = I . Repeatingthe process, we conclude that it is valid for allanyi.

The diagonal elements can now be estimated substituting (A.6) in (A.12) which gives

|Cjj | ≥

√√√√√1 − O(1)

∑k<j

γj ;n0

γk;n0

+∑k>j

γk;n0

γj ;n0

.For largen0 when the fractions are very small, it reduces to

∣∣Cjj ∣∣ ≥ 1 − O(1)

∑k<j

γj ;n0

γk;n0

+∑k>j

γk;n0

γj ;n0

,and recalling that(v(i)n+n0

, e(j ;n0)n ) ≡ Cij we finally come to (3.8):

(v(i)n+n0, e(j ;n0)n ) =

O(1)

√γj ;n0/γi;n0 for j > i,

1 − O(1)

(∑k<i

γi;n0

γk;n0

+∑k>i

γk;n0

γi;n0

)for j = i,

O(1)√γi;n0/γj ;n0 for j < i.

Now writingR(n0)ij as

R(n0)ij ≡ ( v(j)n , v(i)n+n0

)

=N∑k=1

( v(j)n , e(k;n0)n )(v(i)n+n0

, e(k;n0)n )

=N∑k=1

cjk√γk;n0(v

(i)n+n0

, e(k;n0)n )

and using the estimate (3.8), one obtains fori = 1, . . . , j − 1 and largen0:

|R(n0)ij | ≤ √

γi;n0 · i∑k=1

O(1) · cjk +N∑

k=i+1

O(1) · cjk γk;n0

γi;n0

= O(1) · √γi;n0. (A.15)

By definition,R′ij ≡ R

(n0)ij /

√γi;n0. So combining (A.15) with (A.2), one obtains that

R′ij =

O(1) for j ≥ i,

0 for j < i.(A.16)

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 193

so its upper triangle (which only is distinct from zero) contains neither too large nortoo smallelements. The latterfact, which follows from (A.2), is crucial for obtaining the estimate of the inverse matrix,(R′)−1.

Appendix B. Estimates for perturbations of stationary Lyapunov vectors

Using our expansion ofδAn (5.1) we see that

(e(j ;n), δAne(i;n)) = Vij + Vji,

where

Vij =n∑k=0

(δJkJk−1 · · · J0e(j ;n), J ∗k+1 · · · J ∗

n Jn · · · J0e(i;n))

=n∑k=0

(δJkJk−1 · · · J0e(j ;n), [J ∗0 · · · J ∗

k ]−1Ane(i;n))

= αi;nn∑k=0

([Jk · · · J0]−1δJkJk−1 · · · J0e(j ;n), e(i;n))

= αi;nn∑k=0

(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n)),

and so

(e(j ;n), δAne(i;n))= αi;nn∑k=0

(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))

+αj ;nn∑k=0

(Df k(x−k)δJkJk−1 · · · J0e(i;n), e(j ;n)). (B.1)

B.1. Estimation of the inner product(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n)) for largek

To estimate the magnitude ofDf k(x−k)δJkJk−1 · · · J0e(j ;n) we shall use the eigenvectors ofAk, i.e.e(m;k), andthe vectors

e(m;k)−k ≡ JkJk−1 · · · J0e(m;k)

‖JkJk−1 · · · J0e(m;k)‖ ≡ Df−k(x)e(m;k)

‖Df−k(x)e(m;k)‖ ,

which are defined at the pointx−k. They are the eigenvectors of the matrixAk ≡ [Df k(x−k)]∗Df k(x−k) (seeSection 4) and are therefore orthogonal.

Let us expandδJkJk−1 · · · J0e(j ;n) in the basise(m;k)−k :

δJkJk−1 · · · J0e(j ;n) =N∑m=1

bjme(m;k)−k ,

194 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

which gives

(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))

=(Df k(x−k)

N∑m=1

bjme(m;k)−k , e(i;n)

)=

N∑m=1

bjm(Dfk(x−k)e(m;k)

−k , e(i;n))

=N∑m=1

bjm

(e(m;n), Ake(m;n))1/2· (e(m;k), e(i;n)).

At the same time,bjm = (e(m;k)−k , δJkJk−1 · · · J0e(j ;n)), so

|bjm| ≤ ‖δJkJk−1 · · · J0e(j ;n)‖which writing δJkJk−1 · · · J0e(j ;n) = (δJkJ

−1k )Jk · · · J0e(j ;n) can be estimated as

|bjm| ≤ ‖δJk‖ · ‖J−1k ‖ · ‖JkJk−1 · · · J0e(j ;n)‖

= ‖δJk‖ · ‖J−1k ‖ · (e(j ;n), Ake(j ;n))1/2

and thus, assuming that the map has finite derivative so thatJ−1k ≡ Df (x−k) is bounded, one arrives at

|(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))|

≤ O(1) · ‖δJk‖N∑m=1

(e(j ;n), Ake(j ;n))1/2

(e(m;n), Ake(m;n))1/2· (e(m;k), e(i;n))

≡ O(1) · ‖δJk‖N∑m=1

(e(j ;n), Ake(j ;n))1/2

(e(m;n), Ake(m;n))1/2· βim, (B.2)

where

βim ≡ (e(i;n), e(m;k)). (B.3)

Expandinge(j ;n) in the series of the eigenvectors ofAk:

e(j ;n) =N∑m=1

(e(j ;n), e(m;k))e(m;k) ≡N∑m=1

βjme(m;k)

and using orthogonality of the latter, we obtain

(e(j ;n), Ake(j ;n)) =N∑m=1

β2jmαm;k. (B.4)

These values ofβjm can be estimated by means of (4.5) (note that in this expression we should use the eigenvalues,corresponding to theleastbetweenn andk, i.e.γj ;min(n,k) = γj ;k):

βjm =

O(1)√γj ;k/γm;k for j > m,

O(1)

(∑`<i

γm;kγ`;k

+∑`>m

γ`;kγm;k

)for j = m,

O(1)√γm;k/γj ;k for j < m,

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 195

or, recalling thatαm;k = 1/γm;k,

βjm =

O(1)

√αm;k/αj ;k for j > m,

O(1)

(∑`<m

α`;kαm;k

+∑`>m

αm;kα`;k

)for j = m,

O(1)√αj ;k/αm;k for j < m.

(B.5)

Therefore,

(e(j ;n), Ake(j ;n)) =N∑m=1

β2jmαm;k

=j−1∑m=1

β2jmαm;k + β2

jjαj ;k +N∑

m=j+1

β2jmαm;k

= αj ;k ·

j−1∑m=1

O(1)

(αm;kαj ;k

)2

+ O(1)

∑`<j

α`;kαj ;k

+∑`>j

αj ;kα`;k

2

+N∑

m=j+1

O(1)

.Sinceαi;k ≤ αi′;k for i ≤ i′, all the fractions here cannot exceed 1, and so

(e(j ;n), Ake(j ;n)) ≤ O(1) · αj ;k.Similarly we can derive the estimate from below:

(e(j ;n), Ake(j ;n)) =N∑m=1

β2jmαm;k ≥

N∑m=j+1

β2jmαm;k = αj ;k

N∑m=j+1

O(1),

so finally

O(1) · αj ;k ≤ (e(j ;n), Ake(j ;n)) ≤ O(1) · αj ;k.Substituting this inequality together with (B.5) in (B.2), one obtains

|(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))|

≤ O(1) · ‖δJk‖N∑m=1

βim

√αj ;kαm;k

= O(1) · ‖δJk‖ ·√αj ;kαi;k

i−1∑m=1

O(1)+ O(1)

(∑`<i

α`;kαi;k

+∑`>i

αi;kα`;k

)+

N∑m=i+1

O(1)αi;kαm;k

.Again we note that all the fractions here cannot exceed 1, so

|(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))| ≤ O(1) · ‖δJk‖ ·√αj ;kαi;k

.

The deviation of the Jacobian satisfies

δJk ≡ ∂2f−1(f−k(x))Df−k(x)δx, (B.6)

where∂2f−1 is the second derivative, and if it is bounded,

‖δJk‖ ∼ ‖Df−k(x)‖ · ‖δx‖ ≤ O(1) · ‖δx‖ · √αN;k,

196 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

because the growth of the norm of the matrix is determined by its largest eigenvalueαN;k (associated with thesmallestLyapunov exponentλN ). So finally

|(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))| ≤ O(1) · ‖δx‖√αN;kαj ;kαi;k

(B.7)

Remark B.1.In case when the system inN1-dimensional phase space possesses anN2-dimensional stable invariantmanifold containing its attractor, in the above estimate we should takeN = N2, because the dynamics on theattractor is completely determined byN2 first Lyapunov exponents; the others only control convergence towardsthe attractor and so are unessential in the stationary regime.

That can be easily understood from the following example. Consider the three-dimensional dynamical system(i.e.,N1 = 3):

xn+1 = 1 − a |xn| + yn, yn+1 = bxn, zn+1 = κzn.

The first two equations form the Lozy mapΦ; and the third one is “self-consistent”. The matrixDf is

Df (x, y, z) =(DΦ(x, y) 0

0 κ

),

so

[Df −n]∗Df −n =(

[DΦ−n]∗DΦ−n 00 κ−2n

).

Obviously,oneof its eigenvalues isκ−2n; for κ 1 it is thegreatestone and soα3;n = κ−2n. The attractor is theinvariant manifoldz = 0, so the displacementδx ≡ δx, δy, δz = δx, δy,0 andDf −k(x)δx in (B.6) is

Df −k(x)δx =DΦk

(δx

δy

), 0

.

Evidently, it grows not like√α3;k, but like

√α2;k, whereα2;k corresponds to the smallest Lyapunov exponent of

the Lozy mapΦ; i.e. the smallest Lyapunov exponent among those describing the growth of displacementsalongthe attractor. So in this example we should takeN = 2, notN = 3.

Quite similarly (suffice it to exchangei with j ),

|(Df k(x−k)δJkJk−1 · · · J0e(i;n), e(j ;n))| ≤ O(1) · ‖δx‖√αN;kαi;kαj ;k

. (B.8)

B.2. Estimation of the inner product(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n)) for smallk

For smallk the asymptoticsαm;k ≈ exp(−2kλm) that Section B.1 was based upon is no longer valid. But thenatural assumption that bothDf andDf−1 are bounded immediately gives that fork = O(1)

|(Df k(x−k)δJkJk−1 · · · J0e(j ;n), e(i;n))| ≤ O(1) · ‖δx‖ ,and similarly

|(Df k(x−k)δJkJk−1 · · · J0e(i;n), e(j ;n))| ≤ O(1) · ‖δx‖ .This coincides with what the estimates (B.7) and (B.8) would have becomeif they had been true for such smallk.Therefore, we can use themfor anyvalue ofk.

S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198 197

B.3. Deviations of the stationary Lyapunov vectorsδe(i;n)

Substituting (B.7) and (B.8) in (B.1) one obtains

|(e(j ;n), δAne(i;n))| ≤ O(1) · ‖δx‖ ·n∑k=0

[αi;n

√αN;kαj ;kαi;k

+ αj ;n√αN;kαi;kαj ;k

],

which together with expansion (5.2) gives

‖δe(i;n)‖ ≤ O(1) · ‖δx‖ ·∑j 6=i

αi;n|αi;n − αj ;n|

n∑k=0

√αN;kαj ;kαi;k

+∑j 6=i

αj ;n|αi;n − αj ;n|

n∑k=0

√αN;kαi;kαj ;k

= O(1) · ‖δx‖ ·

∑j<i

1

|1 − αj ;n/αi;n|n∑k=0

√αN;kαj ;kαi;k

(1 + αj ;n

αi;nαi;kαj ;k

)

+∑j>i

1

|1 − αi;n/αj ;n|n∑k=0

√αN;kαi;kαj ;k

(1 + αi;n

αj ;nαj ;kαi;k

) . (B.9)

For anyfiniten the deviation of the stationary vectors is obviously finite; so actually we are only interested in itsgrowth whenn → ∞. In this case the eigenvalues differ drastically, and so forn large enough

1

|1 − αj ;n/αi;n| ≤ 2, j < i,1

|1 − αi;n/αj ;n| ≤ 2, j > i.

Thus

‖δe(i;n)‖ ≤ O(1) · ‖δx‖ ·∑j<i

n∑k=0

√αN;kαj ;kαi;k

(1 + αj ;n

αi;nαi;kαj ;k

)+∑j>i

n∑k=0

√αN;kαi;kαj ;k

(1 + αi;n

αj ;nαj ;kαi;k

) .Moreover, for largen

1 + αj ;nαi;n

αi;kαj ;k

≈ 1 + exp(2(n− k)(λi − λj )), 1 + αi;nαj ;n

αj ;kαi;k

≈ 1 + exp(2(n− k)(λj − λi)),

and sinceλ1 ≥ λ2 ≥ · · · ≥ λN , for n large enough, one has

1 + αj ;nαi;n

αi;kαj ;k

≤ 2, j < i, 1 + αi;nαj ;n

αj ;kαi;k

≤ 2, j > i.

Substituting these estimates in (B.9), we arrive at

‖δe(i;n)‖ ≤ O(1) · ‖δx‖ ·∑j<i

n∑k=0

√αN;kαj ;kαi;k

+∑j>i

n∑k=0

√αN;kαi;kαj ;k

. (B.10)

The eigenvaluesα have the exponential asymptoticsα−1/ki;k

k→∞−→ exp(2λi), which readily implies that√αN;kαj ;kαi;k

≤ exp[k · (λi − λj − λN + ψk)],√αN;kαi;kαj ;k

≤ exp[k · (λj − λi − λN + ψk)]

198 S.V. Ershov, A.B. Potapov / Physica D 118 (1998) 167–198

for some functionψkk→∞−→ +0. Applying these estimates to (B.10), one obtains (5.3)

‖δe(i;n)‖ ≤ O(1) · ‖δx‖∑j<i

n∑k=0

exp[k(λi − λj − λN + ψk)] +∑j>i

n∑k=0

exp[k(λj − λi − λN + ψk)]

.This formula is only valid for “typical” pointsx. It may cease to hold at exceptional points, see Remark 1.

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