ISSN 1064�5624, Doklady Mathematics, 2011, Vol. 84, No. 1, pp. 551–554. © Pleiades Publishing, Ltd., 2011.Original Russian Text © G.A. Leonov, V. Reitmann, A.S. Slepukhin, 2011, published in Doklady Akademii Nauk, 2011, Vol. 439, No. 6, pp. 736–739.

551

In this paper, Lyapunov�type functions are intro�duced into upper estimates for the Hausdorff dimen�sion of negatively invariant sets of cocycles. For thispurpose, the methods proposed in [1–5] are furtherdeveloped.

1. DIMENSION ESTIMATES FOR COCYCLES

A possible tool for the study of nonautonomous dif�ferential equations is the theory of cocycles and theirattractors [6–9]. Let us describe some developmentsof concepts of this theory.

Let (Θ, ρθ) be a compact complete metric space.The base flow ({σt}t ∈ �, Θ) is a continuous mapping σ:� × Θ → Θ, (t, θ) � σt(θ), satisfying: (i) σ0(·) = idΘ

and (ii) σt + s(·) = σt(·) ° σs(·), ∀t, s ∈ �. The local

cocycle on �+ over the base flow ({σt}t ∈ �, Θ) is a pair({ϕt(θ, ·) , �n), where ϕ(·)(·, ·) is given on the

set D = {(t, θ, u)| (θ, u) ∈ Θ × �n, t ∈ [0, β(θ, u)},where [0, β(θ, u)) is the nonnegative part of the maxi�mal interval on which there exists a mapping ϕt passingthrough the point (θ, u), and where ϕ satisfies the follow�ing conditions: (i) ϕ0(θ, ·) = , ∀θ ∈ Θ and (ii) ϕt + s(θ,

u) = ϕt(σs(θ), ϕs(θ, u)), ∀(θ, u) ∈ Θ × �n, ∀s ∈ [0, β(θ,u)), ∀t ∈ [0, β(σs(θ), ϕs(θ, u))), t + s < β(θ, u). In whatfollows, for brevity, (ϕ, σ) denotes the local cocycle({ϕt(θ, ·) , �n) over the base flow ({σt}t ∈�, Θ).

Given a mapping θ ∈ Θ � Z(θ) ⊂ �n, the set ={Z(θ)}θ ∈ Θ is called nonautonomous. A nonautono�

}θ Θ∈

t 0 β θ ·,( )),[∈

id�

n

}θ Θ∈

t 0 β θ ·,( ) ),[∈

Z

mous set = {Z(θ)}θ ∈ Θ is called compact if, for everyθ ∈ Θ, the set Z(θ) ⊂ �n is compact.

Definition 1. A set is called negatively invariantfor a local cocycle (ϕ, σ) if there exists 0 < τ <

(θ, u) such that

Let (M, ρ) be a metric space and Z ⊂ M be an arbi�trary subset of M. Then dimHZ denotes the Hausdorffdimension of Z.

For any k ∈ {0, 1, …, n}, define

where αi(L) are the singular values of the n × n matrixL arranged in nonincreasing order; i.e., α1(L) ≥α2(L) ≥ … ≥ αn(L).

Let d ∈ [0, n] be an arbitrary number. It can be rep�resented as d = d0 + s, where d0 ∈ {0, 1, …, n – 1} ands ∈ (0, 1]. Then define the function

which is called the singular value function of thematrix L of order d [4].

Let (ϕ, σ) be a local cocycle for which the map�pings ϕt(θ, ·): �n → �n are C1�smooth for all θ ∈ Θ andt ∈ [0, β(θ, ·)).

For the subsequent presentation, we need the fol�lowing assumptions:

(A1) The nonautonomous set = {Z(θ)}θ ∈ Θ iscompact and negatively invariant for the local cocycle(ϕ, σ) with some τ > 0 in the sense of Definition 1.

(A2) For arbitrary (θ, u) ∈ Θ × �n and t ∈ [0, β(θ, ·)),let ∂2ϕt(θ, u): �n → �n denote the differential of ϕt(θ, u)with respect to u that has the following properties:

Z

Z

βθ Θ∈

u Z θ( )∈

min

ϕτ θ Z θ( ),( ) Z στ θ( )( ) for all θ Θ.∈⊃

ωk L( ) := α1 L( )α2 L( )…αk L( ) if k 0>

1 if k 0,=⎩⎨⎧

ωd L( ) := ωd0

L( )1 s– ωd0 1+ L( )sif d 0 n ],(∈

1 if d 0,=⎩⎨⎧

Z

Upper Estimates for the Hausdorff Dimension of Negatively Invariant Sets of Local Cocycles

Corresponding Member of the RAS G. A. Leonov, V. Reitmann, and A. S. Slepukhin

Received March 14, 2011

DOI: 10.1134/S1064562411050103

Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russiae�mail: leonov@math.spbu.ru, vreitmann@aol.com, st_sandro@mail.ru

MATHEMATICS

552

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

LEONOV et al.

(a) For any ε > 0 and 0 < t < (θ, u), the func�

tion

is bounded on Θ and vanishes as ε → 0 for every fixed t.

(b) For any 0 < t < (θ, u),

where ||L||op denotes the operator norm of the n × nmatrix L.

Theorem 1. Let Assumptions (A1) and (A2) hold andthe following conditions be satisfied:

(i) There exists a compact set ⊂ �n such that

(ii) There exist a continuous function �: Θ × �n → �+and a constant d ∈ (0, n] such that

where τ > 0 is the number from Assumption (A1).Then dimHZ(θ) ≤ d for every θ ∈ Θ.

2. COCYCLES GENERATED BY DIFFERENTIAL EQUATIONS

Consider the nonautonomous ordinary differentialequation

(1)

where f: � × �n → �n is a Ck�smooth vector field (k ≥ 2).The hull of f with respect to vector field (1) is defined as

where the closure is taken in the compact�open topol�ogy. It is well known that �( f ) is metrizable with somemetric ρ. As a result, we obtain the complete metricspace (�( f ), ρ), on which the base flow, known as theBebutov flow [10], is given by the shift mapping

for t ∈ � and ∈ �( f ). It is assumed that �( f ) is compact. For this pur�

pose, it is sufficient that, for example, f (t, u) in (1) bea smooth function of u and an almost periodic func�tion of t.

βθ Θ∈

u Z θ( )∈

min

ηε t θ,( )

:= ϕt θ v,( ) ϕt θ u,( )– ∂2ϕt θ u,( ) v u–( )–

v u–���������������������������������������������������������������������������������

v u, Z θ( )∈

0 v u– ε≤<

sup

βθ Θ∈

u Z θ( )∈

min

∂2ϕt θ u,( ) opu Z θ( )∈

supθ Θ∈

sup ∞,<

K

Z θ( )θ Θ∈

∪ K;⊂

� στ θ( ) ϕτ θ u,( ),( )� θ u,( )

�������������������������������������θ u,( ) Θ K×∈

sup ωd ∂2ϕτ θ u,( )( ) 1,<

u· f t u,( ),=

� f( ) f · t+ ·,( ) t, �∈{ },=

t f,( ) � σt f( ) f · t+ ·,( )=

f

The evaluation map : Θ × �n → �n is defined as

Specifically, for θ = f ∈ �( f ) and u ∈ �n,

whence

for all t ∈ � and u ∈ �n.

By using the mapping , system (1) can be associ�ated with the family of vector fields

(2)

where θ ∈ �( f ) is arbitrary. Family (2) includes theoriginal system (1) as a special case.

Let us introduce an additional assumption on (1).(A3) The mapping (t, u) ∈ � × �n � f (t, u) is

almost periodic in t.By applying (A3) and the above assumptions,

we can show for (2) the existence of a local cocycle({ϕt(θ, ·) , �n) over the base flow ({σt}t ∈ �,

�( f )) (see [6]), where ϕt is specified in terms of thesolution operator of system (2) and [0, β(θ, u)) is thenonnegative part of the maximal interval on which thereexists a solution passing through the point (θ, u) ∈ Θ ×�n.

For a point (θ0, u0) ∈ Θ × �n, let w(t, u0) denote thesolution of the variational equation along the cocycletrajectory through the point (θ0, u0), i.e., of the equa�tion

(3)

with the initial condition w(0, w0) = w0 ∈ �n. Then

for 0 ≤ t < β(θ0, u0). Let λ1(θ, u) ≥ λ2(θ, u) ≥ … ≥ λn(θ, u) be the eigen�

values of the matrix [∂2 (θ, u) + ∂2 (θ, u)]T.

Theorem 2. Suppose that the local cocycle (ϕ, σ)generated by differential equation (1) over the Bebutovflow satisfies Assumption (A1) and the following condi�tions:

(i) condition (i) in Theorem 1 with the set ;

(ii) there exists a continuous function V: Θ × �n → �

with derivatives V(σt(θ), ϕt(θ, u0)) along a given tra�

jectory and there exists a number d ∈ (0, n] written as

f

θ u,( ) Θ �n � θ 0 u,( ).×∈

f t u,( ) f 0 u,( ),=

f σt f( ) u,( ) f t u,( )=

f

u· f σt θ( ) u,( ),=

}θ � f( )∈

t 0 β θ ·,( )),[∈

w· ∂2 f σt θ0( ) ϕt θ0 u0,( ),( )w=

∂2ϕt θ0 u0,( )w0 w t w0,( )=

12�� f f

K

ddt����

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

UPPER ESTIMATES FOR THE HAUSDORFF DIMENSION 553

d = d0 + s, where d0 ∈ {0, 1, …, n – 1} and s ∈ (0, 1],such that

for all θ ∈ Θ and u0 ∈ , where τ > 0 is the number fromAssumption (A1).

Then dimHZ(θ) ≤ d for all θ ∈ Θ.Consider the nonautonomous Rössler system [11]

(4)

where a, b: � → �+ are functions given by

Here, a0 and b0 are positive constants, while a1(·) andb1(·) are C1�smooth functions satisfying the inequali�ties

(5)

for all t ∈ �, where ε ∈ (0, 1) is a small parameter.Assume also that there exists � > 0 such that

(6)

for all t ∈ � and that the hull �( f ) with f equal to theright�hand side of (4) is compact. For this purpose, itis sufficient that a and b be almost periodic.

Instead of (4), we consider the family of systems oftype (2):

(7)

where, for brevity, we use the notation

Since system (4) has all the properties of system (1), itgenerates a local cocycle ({ϕt(θ, ·) , �n) over

the base flow ({σt}t ∈ �, �( f )), where t ∈ [0, β(θ, u)) isthe nonnegative part of the maximal interval on whichthere exists solution (7) passing through the point(θ, u) ∈ Θ × �n. Assume that, for this cocycle, there

λ1(σt θ( ) ϕt θ u0,( ), … λd0σt θ( ) ϕt θ u0,( ),( )+ +

0

τ

∫

+ sλd0 1+ σt θ( ) ϕt θ u0,( ),( )

+ ddt����V σt θ( ) ϕt ) θ u0,( ),( ) dt 0<

K

x· y– z,–=

y· x,=

z· b t( )z– a t( ) y y2–( ),+=

a t( ) a0 a1 t( ), b t( )+ b0 b1 t( ).+= =

a1 t( ) εa0, b1 t( ) εb0≤ ≤

b· t( ) ε�≤

x· y– z,–=

y· x,=

z· bθ t( )z– aθ t( ) y y2–( ),+=

aθ t( ) a σt θ( )( ) and bθ t( ) b σt θ( )( ).≡ ≡

}θ � f( )∈

t 0 β θ ·,( )),[∈

exists a compact set = {Z(θ)}θ ∈ �( f ) satisfying condi�

tion (i) in Theorem 1 with a compact and there

exists a time 0 < τ < (θ, u) such that is nega�

tively invariant for the local cocycle in the sense ofDefinition 1.

To estimate the Hausdorff dimension from abovewith the help of Theorem 2, we need to verify the ine�quality

(8)

for all t ∈ [0, τ], (x, y, z) ∈ , and θ ∈ �( f ). Here,

are the eigenvalues of the symmetrized Jacobianmatrix of the right�hand side of (7) arranged in nonin�creasing order (λ1, θ ≥ λ2, θ ≥ λ3, θ) and

is a Lyapunov�type function defined for (x, y, z) ∈ ,θ ∈ �( f ), and t ∈ [0, τ] by the relation

where ξ is a variable parameter.We calculate the eigenvalues λk, θ and the derivative

Vθ and substitute them into (8). Direct computa�

tions with the use of (5), (6), and Theorem 2 finallygive the estimate

for all θ ∈ �( f ), where C is a positive number, whichcan be calculated in terms of the parameters a0, b0, ε,and � of the system and is bounded for all small ε > 0.

REFERENCES

1. G. A. Leonov, Vestn. Leningr. Gos. Univ. Ser. 1 3, 41–44 (1991).

Z

K

βθ � f( )∈u Z θ( )∈

min Z

Z

λ1 θ, t x y z, , ,( ) λ2 θ, t x y z, , ,( ) sλ3 θ, t x y z, , ,( )+ +

+ ddt����Vθ t x y z, , ,( ) 0<

K

λk θ, t x y z, , ,( ) λk σt θ( ) ϕt θ x y z, , ,( ),( ),≡

k 1 2 3, ,=

Vθ t x y z, , ,( ) V σt θ( ) ϕt θ x y z, , ,( ),( )≡

K

V σt θ( ) x z, ,( ) := 12�� 1 s–( )ξ z bθ t( )x–( ),

ddt����

dimHZ θ( )

≤ 32 1 ε–( )b0

1 ε+( )b0 a0 2b0+( )2 b02 1+ + ε C⋅+ +

�������������������������������������������������������������������������������–

554

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

LEONOV et al.

2. G. A. Leonov and V. A. Boichenko, Acta Appl. Math.26, 1–60 (1992).

3. G. A. Leonov, Algebra Anal. 13 (3), 155–170 (2001).

4. V. A. Boichenko, G. A. Leonov, and V. Reitmann,Dimension Theory for Ordinary Differential Equations(Vieweg�Teubner, Wiesbaden, 2005).

5. G. A. Leonov, Strange Attractors and Classical StabilityTheory (St. Petersburg Univ. Press, St. Petersburg,2008).

6. D. R. Wakeman, J. Differ. Equations 17, 259–295(1975).

7. H. Crauel and F. Flandoli, J. Dyn. Differ. Equations 10,449–474 (1998).

8. P. E. Kloeden and D. Stonier, Dyn. Discrete, Continu�ous Impulsive Syst., No. 4, 221–226 (1998).

9. W. A. Veech, J. Modern Dyn. 2, 375–395 (2008).10. M. V. Bebutov, Byull. Mekh.�Mat. Fak. Mosk. Gos.

Univ., No. 5, 1–52 (1941).11. O. E. Rössler, Z. Naturforsch. A 31, 1664–1670 (1976).