Fractal dimension estimates for flow-invariant sets on manifolds

Wolfram Heineken

Wolfram.Heineken@Student.Uni-Magdeburg.de

Volker Reitmann^{∗}

reitmann@rcs.urz.tu-dresden.de

Abstract. This paper is concerned with upper bounds for the fractal or upper capacitive dimension of invariant sets of flows on Riemannian manifolds. The main theorems generalize several results from [10, 11, 15]. To obtain upper bounds for the fractal dimension of flow invariant sets conditions for the eigenvalues of the symmetric part of the covariant derivative of the associated vector field are formulated. Furthermore it is shown that the results can be applied to the investigation of the Lorenz attractor.

Keywords: Riemannian manifolds, fractal dimension estimates, flow invariant sets, Lyapunov exponents, Lorenz equation AMS subject classification: Primary 58F12, secondary 58F08

1 Introduction

The dimension of the attractor of a dynamical system is some quantity which is closely related to systems properties like stability or chaotic behaviour. Various definitions of dimension have been established which may also take fractal values. The most important ones are Hausdorff and fractal dimensions. Considerable work has been done to estimate upper bounds for the Hausdorff dimension [6, 10, 11, 13, 14, 15].

In the present paper we focus our attention on the fractal dimension also known as upper capacity or box-counting dimension. To estimate the fractal dimension several techniques have been developed ([1, 7, 15]). Two of them yield the basic statements we are dealing with.

The first technique follows an approach in [15] which is achieved to estimate the fractal dimension of invariant sets of a dynamical system in a linear space. In Theorem 3.1 this result will be extended to invariant sets on a Riemannian manifold. This theorem turns out to be more suitable for higher dimensional problems as appearing in the investigation of Navier-Stokes equations.

The second basic statement is Theorem 3.2 which was developed in [1]. In Section 4 we will adapt these techniques to vector fields on Riemannian manifolds. Introducing various types of Lyapunov exponents we will derive a fractal dimension estimate which can be considered as the fractal dimension version of the well-known Kaplan-Yorke formula [9].

∗Supported by the DFG-Schwerpunktprogramm ”Dynamik: Analysis, effiziente Simulation und Ergodentheorie”

Later, by changing the Riemannian metric tensor, we will develop an approach which allows to improve the estimates for concrete systems. This approach uses frequency domain conditions forming an interesting link to stability theory methods for feedback systems ([2, 11, 12, 16]). An application of this approach to the Lorenz attractor leads to the remarkable result that our upper bound of its fractal dimension is smaller than other estimates known so far and is equal to the upper Hausdorff dimension estimate in [11].

2 Some basic notations

Suppose that (X, ρ) is a metric space and K ⊂ X a compact set. For any given ε > 0 denote by n_{K}(ε) the smallest number of balls of radius ε needed to cover K. The upper limit

dim_{F} K = lim sup_{ε→0+0} \frac{ln n_{K}(ε)}{ln \frac{1}{ε}}

is the fractal dimension of K. (dim_{F} K is also called upper box or upper capacity dimension of K.)

For arbitrary ε > 0 and d ≥ 0 the Hausdorff d-measure at level ε of K is given by

μ_{H}(K, d, ε) = inf ∑_{i} r_{i}^{d},

where the infimum is taken over all countable covers of K by open balls of radii r_{i} ≤ ε. For fixed d and K, the limit (which always exists)

μ_{H}(K, d) = lim_{ε→0+0} μ_{H}(K, d, ε)

defines the Hausdorff d-measure of K. For any compact set K there exists an unique critical number d^{∗} ≥ 0 such that

μ_{H}(K, d) = \{ \begin{matrix}+∞ & for & 0 ≤ d < d^{∗}, \\ 0 & for & d > d^{∗}. \end{matrix}

This critical number dim_{H} K = d^{∗} is called the Hausdorff dimension of K. It can be shown that for any compact set K the inequality dim_{H} K ≤ dim_{F} K holds.

If L : E → F is a linear operator between m-dimensional Euclidean spaces E and F with scalar products ⟨·, ·⟩_{E} and ⟨·, ·⟩_{F} , respectively, the adjoint operator is the linear operator L^{∗} : F → E, uniquely determined by the relation ⟨Lξ, η⟩_{F} = ⟨ξ, L^{∗}η⟩_{E} for all ξ ∈ E, η ∈ F. The m eigenvalues σ_{1}(L) ≥ … ≥ σ_{m}(L) ≥ 0 of the operator \sqrt{L^{∗}L} are the singular values of L ordered will respect to their size and multiplicity.

For an arbitrary number d ∈ [0, m], for d > 0 written in the form d = d_{0} + d_{1} with d_{0} ∈ {0, 1, …, m − 1} and d_{1} ∈ (0, 1] we define by

ω_{d}(L) = \{ \begin{matrix}1 & for & d = 0, \\ σ_{1}(L) · . . . · σ_{d_{0}}(L)σ_{d_{0}+1}^{d_{1}} & for & d ∈ (0, m] \end{matrix}

the singular value function of order d. Let E be an ellipsoid in the m-dimensional Euclidean space E and let α_{1}(E) ≥ … ≥ α_{m}(E) ≥ 0 denote the length of its semi-axes. For d ∈ [0, m] written as above we introduce the d-dimensional ellipsoid volume by

ω_{d}(E) = \{ \begin{matrix}1 & for & d = 0, \\ α_{1}(E) · . . . · α_{d_{0}}(E)α_{d_{0}+1}^{d_{1}}(E) & for & d ∈ (0, m]. \end{matrix}

Let be (M, g) a smooth m-dimensional Riemannian manifold of class C^{3}, U ⊂ M an open set, φ : U → M a C^{1}-map and let d_{u}φ denote the tangent map of φ at the point u ∈ U. Then for an arbitrary set K ⊂ U and any d ∈ [0, m] we define

ω_{d,K}(φ) = sup_{u∈K} ω_{d}(d_{u}φ).

We briefly recall some notations from Riemannian geometry. By d(u, v) we denote the geodesic distance between two points u and v on M, by B_{M}(u, ε) the open and by \overline{B}_{M}(u, ε) the closed ball on M with centre at u and radius ε. Let us fix some point u ∈ M and define a chart x : D(x) → R(x) in a neighbourhood of u. We set g_{ij,k}(u) = \frac{∂(g_{ij}∘x^{−1})}{∂x^{k}}(x(u)) and introduce by Γ_{ij}^{k}(u) = \frac{1}{2}g^{ks}(u)(−g_{ij,s}(u)+g_{js,i}(u)+g_{si,j}(u)) the Christoffel symbols of second kind on (M, g) with respect to the chart x. We can now define the covariant derivative of a C^{1}-vector field f : M → T M in u as the linear map ∇f(u) : T_{u}M → T_{u}M given in the chart x by

∇_{v}f(u) = (\frac{∂f^{k}(u)}{∂x^{i}}v^{i} + Γ_{ij}^{k}(u)f^{j}(u)v^{i}) ∂_{k}(u)

for all v = v^{i}∂_{i}(u) ∈ T_{u}M. Here {∂_{s}(u)} is the canonical basis of T_{u}M in the chart x and f(u) = f^{i}∂_{i}(u) is the representation of the vector field in this chart. For a C^{1}-curve α : [a, b] → M let x^{i}(t) denote the local coordinates of α(t) in the chart x. With respect to a vector field X(t) along α, i.e., for a vector field with X(t) ∈ T_{α(t)}M for t ∈ [a, b], the absolute derivative of X along α is defined by

\frac{DX(t)}{dt} ≡ ∇α̇X(t) = (\frac{dX^{k}}{dt} + Γ_{ij}^{k}X^{j}α̇^{j}) ∂_{k}(α(t)).

For a given C^{1}-curve α : [a, b] → M, any v ∈ T_{α(a)}M and s ∈ [a, b] there exists an unique vector field X_{v} along α such that X_{v} is parallel along α (i.e., ∇α̇X_{v} ≡ 0) and X_{v}(a) = v. This defines the parallel transport τ_{α(s)}^{α(t)} : T_{α(s)}M → T_{α(t)}M along α from α(s) to α(t) for any s, t ∈ [a, b] with s ≤ t which relates to any v ∈ T_{α(s)}M the vector X_{v}(t) ∈ T_{α(t)}M. Recall that a geodesic on (M, g) is a C^{2}-curve α : [a, b] → M satisfying \frac{Dα̇}{dt} ≡ 0. For any p ∈ M and v ∈ T_{p}M we denote by α_{p,v} the maximal geodesic and by Ω ⊂ T M the set of all points (p, v) such that α_{p,v}(1) exists. The exponential map is the map exp : Ω → M given by exp((p, v)) = α_{p,v}(1). The restriction exp |_{T_{p}M∩Ω} we denote by exp_{p} . It is well-known, that Ω is open in T M, that exp : Ω → M is C^{1}-smooth and for any p ∈ M there exists an open set Ω_{p} ⊂ T_{p}M such that exp_{p} is a diffeomorphism on Ω_{p} and isometric, i.e., d(p, exp_{p} v) = ∥v∥_{T_{p}M} for any v ∈ T_{p}M.

The two following properties of the exponential map will be essential for our purposes.

Lemma 2.1 Suppose K ⊂ M is a compact set and φ : M → M is a C^{1}-map. Then there exists an ε_{0} > 0, such that for arbitrary u ∈ K and v ∈ B_{M}(u, ε) with 0 < ε ≤ ε_{0} the relation

exp_{φ(u)}^{−1}(φ(v)) = ∫_{0}^{1}(τ_{φ(u_{s})}^{φ(u)}d_{u_{s}}φτ_{u}^{u_{s}}) exp_{u}^{−1}(v)ds

holds. ( Here u_{s} = α(s), s ∈ [0, 1], denotes the geodesic with α(0) = u and α(1) = v.)

Proof. Let V be an Euclidean space. For an arbitrary differentiable map F : U ⊂ M → V and a smooth curve α with α(0) = u and α(1) = v the formula

F (v) − F (u) = ∫_{0}^{1} Ḟ (α(t))dt = ∫_{0}^{1} F′(α(t))α̇(t)dt

holds. If we set V = T_{φ(u)}M and F = exp_{φ(u)}^{−1} ∘ φ : M → T_{φ(u)}M we obtain

exp_{φ(u)}^{−1}(φ(v)) − exp_{φ(u)}^{−1}(φ(u)) = ∫_{0}^{1}(τ_{φ(u_{s})}^{φ(u)}d_{u_{s}}φ τ_{u}^{u_{s}})(exp_{u}^{−1}(v))ds.

With exp_{φ(u)}^{−1}(φ(u)) = 0 the assertion follows. ■

Lemma 2.2 Let the exponential map exp_{u} on (M, g) be defined in a neighbourhood U of u ∈ M. Let K ⊂ M be a compact set, u ∈ K be an arbitrary point and ε > 0 be such that \overline{B}(u, ε) ⊂ U holds. For any v, w ∈ \overline{B}(u, ε) there exists a constant C(ε) such that the inequality

d(v, w) ≤ C(ε) ∥exp_{u}^{−1}(v) − exp_{u}^{−1}(w)∥_{T_{u}M}

is fulfilled.

Proof. Since the map exp_{v}^{−1} ∘ exp_{u} : \overline{B}_{T_{u}M}(0, ε) → T_{v}M is of class C^{1}, it satisfies a Lipschitz condition on the compact set \overline{B}_{T_{u}M}(0, ε). Thus, there exists a constant C(u, v, ε) such that the inequality

∥(exp_{v}^{−1} ∘ exp_{u})(x) − (exp_{v}^{−1} ∘ exp_{u})(y)∥_{T_{v}M} ≤ C(u, v, ε)∥x − y∥_{T_{u}M}

for all x, y ∈ \overline{B}_{T_{u}M}(0, ε) holds. Setting x = exp_{u}^{−1}(v), y = exp_{u}^{−1}(w), C(ε) = sup_{u∈K} sup_{v∈B_{M} (0,ε)} C(u, v, ε) and taking the suprema over v ∈ \overline{B}_{M}(0, ε) and u ∈ K we obtain

d(v, w) ≤ C(ε)∥ exp_{u}^{−1}(v) − exp_{u}^{−1}(w)∥_{T_{u}M}. ■

The essence of Lemma 2.2 is that the constant C(ε) does not depend on the choice of u ∈ K.

3 Fractal dimension estimates for map invariant sets

Let (M, g) be, as above, an m-dimensional Riemannian C^{3}-manifold, u ∈ M an arbitrary point, x : D(x) → R(x) a chart around u and G_{x}(u) the symmetric and positive definite matrix representation of the Riemannian fundamental tensor in the chart x. Suppose that on M there is given some differentiable map φ : M → M which has a compact invariant set K, i.e., φ(K) = K.

3.1 Lemmas on covering

Now we consider several general covering problems arising in a linear space when we investigate maps with respect to the cover of a set K with metric balls. The properties, we first derive, form an essential part of the proof of Theorem 3.1 given later.

Lemma 3.1 Let E be an ellipsoid in the m-dimensional Euclidean space (E, ⟨·, ·⟩_{E}) with lengths α_{1}(E) ≥ … ≥ α_{m}(E) > 0 of semi-axes. Then for arbitrary k = 1, …, m and η > 0 the set E + B(0, η), where B(0, η) denotes the ball with radius η and centre in the origin, is contained in an ellipsoid E′ with the lengths of semi-axes

α_{1}(E′) = a α_{1}(E), …, α_{k}(E′) = a α_{k}(E) and

α_{k+1}(E′) = … = α_{m}(E′) = a α_{k}(E), where a = 1 + \frac{η}{α_{k}(E)}.

Proof. The proof is similar to that of a lemma given in [13]. ■

Corollary 3.1 Under the assumptions of Lemma 3.1 the ellipsoid E′ can be chosen such that ω_{d}(E′) = a^{d}ω_{d}(E) holds for all d ∈ (0, k].

We now recall the two approaches we will apply to cover an ellipsoid in an Euclidean space with balls. In general it depends on the shape of the given ellipsoid which of these two techniques is the more appropriate. The first approach was proposed in [15], the second one was introduced in [1].

Lemma 3.2 Let E be an ellipsoid in the m-dimensional Euclidean space E with lengths of semi-axes α_{1}(E) ≥ … ≥ α_{m}(E). For an arbitrary number k ∈ {0, 1, …, m − 1} let r be a number satisfying α_{k}(E) ≤ r. Then the number of balls of radius r\sqrt{k + 1} needed to

cover E is not greater than N = ∏_{i=1}^{k} ([\frac{α_{i}(ε)}{r}] + 1).

([.] denotes the integer part of a real number.)

Remark 3.1 If for the number r introduced in Lemma 3.2 the inequality α_{l+1}(E) < r ≤ α_{l}(E) holds then number N of balls needed to cover the ellipsoid E can be estimated by

N ≤ 2^{k} \frac{ω_{l}(E)}{r^{l}}.

Remark 3.2 The conclusion of Lemma 3.2 remains true if E is a degenerated ellipsoid of dimension n < m. Setting α_{n+1}(E) = … = α_{m}(E) = 0 the ellipsoid E can be covered by N balls, where N is given as in Lemma 3.2.

The following Lemma was proved in [1].

Lemma 3.3 Any ellipsoid E in the m-dimensional Euclidean space E with lengths of semi-axes α_{1}(E) ≥ … ≥ α_{m}(E) > 0 can be covered by N balls of radius \sqrt{mr}, where

0 < r < α_{m}(E) and N ≤ \frac{2^{m}ω_{m}(E)}{r^{m}} hold.

3.2 A first fractal dimension estimate

On the basis of the covering lemmas given above we are now able to present a first estimate for the fractal dimension of map invariant sets. The idea follows closely the approach of [15] for invariant sets of maps in a linear space. Using Lemma 2.2 we extend the result of [15] to invariant sets of maps on a Riemannian manifold.

Theorem 3.1 Let (M, g) be an m-dimensional Riemannian C^{3}-manifold and K ⊂ M be a compact set invariant under the C^{1}-map φ : M → M. Let d ∈ (1, m] be a number written as d = d_{0} +d_{1} with d_{0} ∈ {1, …, m−1} and d_{1} ∈ (0, 1] and suppose σ_{d_{0}+1}(d_{u}φ) > 0 on K. Suppose further

ω_{j,K}(φ)[ω_{d_{0}+1,K}(φ)]^{\frac{d−j}{d_{0}+1}} < 1 for j = 1, …, d_{0}.  (3.1)

Then dim_{F} K ≤ d.

Proof. For brevity we introduce for u ∈ K, p ∈ ℕ and i = 1, …, m the no-

tations σ_{i}(p, u) = σ_{i}(d_{u}φ^{p}), \overline{σ}_{i}(p) = sup_{u∈K} σ_{i}(d_{u}φ^{p}) and ω_{i}(p) = ω_{i}(d_{u}φ^{p}) and ω̄_{i}(p) =

sup_{u∈K} ω_{i}(d_{u}φ^{p}). For an arbitrary p ∈ ℕ and certain positive numbers η and δ let ε > 0 be

small enough such that the following three conditions are satisfied:

1. For arbitrary u ∈ K and v ∈ B_{M}(u, ε) the inequality

∥τ_{φ(v)}^{φ(u)}d_{v}φ^{p}τ_{u}^{v} − d_{u}φ^{p}∥ ≤ η  (3.2)

holds. ( ∥.∥ is the operator norm in T_{u}M.)

2. The inequality

2 ε\overline{σ}_{1}(p) ≤ δ  (3.3)

is satisfied.

3. For arbitrary u ∈ K the exponential map exp_{u} is defined on B_{M}(u, ε) and Lemma 2.2 is applicable for all v ∈ B_{M}(u, ε).

Let us consider an ε-cover K ⊂ ⋃_{i=1}^{n_{K}(ε)} B_{M}(u_{i}, ε) of K consisting of the minimal num-

ber n_{K}(ε) of balls B_{M}(u_{i}, ε). The image of an arbitrarily chosen ball B_{M}(u_{i}, ε) under the inverse exponential map exp_{u_{i}}^{−1} is the ball B_{T_{u_{i}}M}(0, ε). Lemma 2.1 provides the representation

exp_{φ^{p}(u_{i})}^{−1}(φ^{p}(v)) = d_{u_{i}}φ^{p}(exp_{u_{i}}^{−1}(v)) + ∫_{0}^{1}(τ_{φ^{p}(u_{s})}^{φ^{p}(u_{i})}d_{u_{s}}φ^{p}τ_{u_{i}}^{u_{s}} − d_{u_{i}}φ^{p})(exp_{u_{i}}^{−1}(v))ds

for v ∈ B_{M}(u_{i}, ε), where u_{s} is defined as in Lemma 2.1. Thanks to condition (3.2) the norm of the latter term can be estimated by

∥∫_{0}^{1}(τ_{φ^{p}(u_{s})}^{φ^{p}(u_{i})}d_{u_{s}}φ^{p}τ_{u_{i}}^{u_{s}} − d_{u_{i}}φ^{p})(exp_{u_{i}}^{−1}(v))ds∥_{T_{φ^{p}(u_{i})}M}

≤ sup_{{u,v∈K \atop d(u,v)≤ε}} ∥τ_{φ^{p}(v)}^{φ^{p}(u)}d_{v}φ^{p}τ_{u}^{v} − d_{u}φ^{p}∥ ∥ exp_{u_{i}}^{−1}(v)∥_{T_{u_{i}}M} ≤ ηε,

resulting in the inclusion

φ^{p}(B_{M}(u_{i}, ε)) ⊂ exp_{φ^{p}(u_{i})}(d_{u_{i}}φ^{p}(B_{T_{u_{i}}M}(0, ε)) + B_{T_{φ^{p}(u_{i})}M}(0, ηε)).

The set d_{u_{i}}φ^{p}(B_{T_{u_{i}}M}(0, ε)) represents an ellipsoid with lengths of semi-axes εσ_{j}(p, u_{i}) in the tangent space T_{φ^{p}(u_{i})}M. Applying Lemma 3.1 to the case k = d_{0} + 1, we obtain that the set d_{u_{i}}φ^{p}(B_{T_{u_{i}}M}(0, ε)) + B_{T_{iu}M}(0, ηε) is contained in an ellipsoid E_{i} with lengths of semi-axes

α_{j}(E_{i}) = (1 + \frac{η}{σ_{d_{0}+1}(p, u_{i})}) εσ_{j}(p, u_{i}) for j = 1, …, d_{0} + 1

and, if d_{0} ≤ m − 2,

α_{j}(E_{i}) = (1 + \frac{η}{σ_{d_{0}+1}(p, u_{i})}) εσ_{d_{0}+1}(p, u_{i}) for j = d_{0} + 2, …, m.

(3.4)

Now we choose η > 0 such that every u ∈ K satisfies the inequality η ≤ σ_{d_{0}+1}(p, u). With (3.4) this leads to ω_{d}(E_{i}) ≤ (2 ε)^{d}ω_{d}(p, u_{i}) and α_{d_{0+1}}(E_{i}) ≤ 2 εσ_{d_{0+1}}(p, u_{i}). Applying Lemma 3.2 with r = 2 ε\overline{σ}_{d_{0}+1}(p) we have the existence of a covering of E_{i}, with N_{i} =

2^{d_{0}} max_{1≤l≤d_{0}} \frac{ω_{l}(E_{i})}{r^{l}} balls B_{T_{φ^{p}(u_{i})}M}(v_{j}, r\sqrt{d_{0} + 1}), which we briefly denote by B_{ij}. Taking the

supremum we obtain the estimate

N_{i} ≤ 2^{d_{0}} max_{1≤l≤d_{0}} \frac{\overline{ω}_{l}(p)}{\overline{σ}_{d_{0}+1}(p)^{l}} = N.  (3.5)

Thanks to condition (3.3) the inequality α_{1}(E_{i}) = 2 εσ_{1}(p, u) ≤ δ holds. Without loss of generality, we may assume that the radius of all balls B_{ij} does not exceed the value δ. Thus, for arbitrary x, y ∈ B_{ij} Lemma 2.2 gives

d (exp_{φ^{p}(u_{i})}(x), exp_{φ^{p}(u_{i})}(y)) ≤ C∥x − y∥_{T_{φ^{p}(u_{i})M}} ,

where C is a certain constant depending only on δ. Obviously there exists a ball B_{M}(w_{ij}, Cr\sqrt{d_{0} + 1}) ⊂ M satisfying the inclusion exp_{φ^{p}(u_{i})}(B_{ij}) ⊂ B_{M}(w_{ij}, Cr\sqrt{d_{0} + 1}).

We conclude, that φ^{p}(B_{i}) ⊂ ⋃_{j=1}^{N_{i}} B_{M}(w_{ij}, Cr\sqrt{d_{0} + 1}) and therefore a new covering of

K is given by

K ⊂ ⋃_{i=1}^{n_{K^{(ε)}}} ⋃_{j=1}^{N_{i}} B_{M}(w_{ij}, Cr\sqrt{d_{0} + 1}).

Since (3.5) holds for all i this cover consists of no more than Nn_{K}(ε) balls of radius Cr\sqrt{d_{0} + 1}. Hence,

n_{K}(Cr\sqrt{d_{0} + 1}) ≤ Nn_{K}(ε).

A short calculation shows that

n_{K}(2Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d + 1}) ≤ n_{K}(2Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d_{0} + 1}) = n_{K}(Cr\sqrt{d_{0} + 1}) ≤ Nn_{K}(ε)

= n_{K}(ε)2^{d_{0}} max_{1≤l≤d_{0}} \frac{\overline{ω}_{l}(p)}{\overline{σ}_{d_{0}+1}(p)^{l}} ≤ n_{K}(ε)2^{d} max_{1≤l≤d_{0}} \frac{\overline{ω}_{l}(p)}{\overline{σ}_{d_{0}+1}(p)^{l}} .

Multiplying the last inequality by (2Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d + 1})^{d} we get

(2Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d + 1})^{d}n_{K}(2Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d + 1})

≤ n_{K}(ε)2^{2d}C^{d}ε^{d}\overline{σ}_{d_{0}+1}(p)^{d}(d + 1)^{d/2} max_{1≤l≤d_{0}} \frac{\overline{ω}_{l}(p)}{\overline{σ}_{d_{0}+1}(p)^{l}}

= n_{K}(ε)2^{2d}C^{d}ε^{d}(d + 1)^{d/2} max_{1≤l≤d_{0}} \overline{ω}_{l}(p)\overline{σ}_{d_{0}+1}(p)^{d−l}

≤ n_{K}(ε)2^{2d}C^{d}ε^{d}(d + 1)^{d/2} max_{1≤l≤d_{0}} \overline{ω}_{l}(p)\overline{ω}_{d_{0}+1}(p)^{\frac{d−ℓ}{d_{0}+1}} . (3.6)

Here, we have used the estimate \overline{σ}_{d_{0}+1}(p) ≤ \overline{ω}_{d_{0}+1}(p)^{\frac{1}{d_{0}+1}} . If we set now

α = 2C\sqrt{d + 1}σ_{d_{0}+1}(p) and ϑ = 2^{2d}C^{d}(d + 1)^{d/2} max_{1≤l≤d_{0}} \overline{ω}_{l}(p)\overline{ω}_{d_{0}+1}(p)^{\frac{d−ℓ}{d_{0}+1}}

inequality (3.6) can be written as (αε)^{d}n_{K}(αε) ≤ ϑε^{d}n_{K}(ε). From assumption (3.1) it follows that

\overline{σ}_{d_{0}+1}(p) → 0 for p → ∞ and \overline{ω}_{j}(p)\overline{ω}_{d_{0}+1}(p)^{\frac{d−j}{d_{0}+1}} → 0 for p → ∞

and therefore α, ϑ → 0 for p → ∞ holds. Now let p be large enough to ensure that α, ϑ ≤ 1/2. If we introduce the function g(ε) = ε^{d}n_{K}(ε), we see that g(αε) ≤ \frac{1}{2}g(ε) and g(ε) → 0 for p → ∞. If ε is small enough we have ε^{d}n_{K}(ε) ≤ 1. This results immediately

in d ≥ lim sup_{ε→0+0} \frac{ln n_{K}(ε)}{ln 1/ε} = dim_{F} K. ■

3.3 A second fractal dimension estimate

As stated above the essential idea of Theorem 3.1 in the case of a linear space refers to an estimate in [15] which was developed for rather high dimensional objects, e.g. for sets in Hilbert space. Dealing with dynamical systems in low dimensional spaces the following estimate from [1], which we need in the following, is often more appropriate.

Theorem 3.2 Let (M, g) be an m-dimensional C^{3}-manifold, U ⊂ M be an open set, φ : U → M a C^{1}-map and K ⊂ U a compact set which is invariant under φ, i.e. φ(K) = K. Define ρ_{K}(φ) = min_{u∈K} σ_{m}(d_{u}φ) and assume ρ_{K}(φ) > 0. Suppose also that

there exists a number d ∈ [0, m] with

ω_{m,K}(φ)ρ_{K}(φ)^{d−m} < 1.  (3.7)

Then dim_{F} K ≤ d.

3.4 Fractal dimension estimates using Lyapunov exponents

Fractal dimension estimates are often based on various types of Lyapunov exponents of a map instead of its singular values. Lyapunov exponents, not assuming the existence of an invariant measure, were introduced for dynamical systems in linear spaces in [3, 4]. These quantities give some information about the distortion of infinitesimal small volume elements under the linearized flow for large times.

The uniform Lyapunov exponents of φ on the invariant set K are defined as

μ_{1} = lim_{p→∞} \frac{1}{p} ln ω_{1,K}(φ^{p}) and

μ_{j} = lim_{p→∞} \frac{1}{p} ln \frac{ω_{j,K}(φ^{p})}{ω_{j−1,K}(φ^{p})} for j = 2, …, m.

Furthermore we introduce the upper Lyapunov exponents of φ on K by

\overline{μ}_{j} = lim_{p→∞} \frac{1}{p} ln [sup_{u∈K} σ_{j}(d_{u}φ^{p})] for j = 1, 2, …, m and

the lower Lyapunov exponent of φ on K by

\underline{μ}_{m} = lim_{p→∞} \frac{1}{p} ln [ inf_{u∈K} σ_{m}(d_{u}φ^{p})] .

Although the Lyapunov exponents are defined as functions of singular values in a number of cases the singular values are not available but the Lyapunov exponents can be estimated directly. On the other hand the assumptions of Theorem 3.1 and 3.2 can sometimes be weakened by using Lyapunov exponents instead of singular values. Therefore it is worth deriving fractal dimension estimates on Riemannian manifolds based on Lyapunov exponents as done in the following two theorems. Similar results for dynamical systems in linear spaces can be found in [5, 15].

Theorem 3.3 Let (M,g) be an m-dimensional Riemannian C^{3}-manifold, φ : M → M a C^{1}-map and K ⊂ M a compact φ-invariant set. Assume that the upper Lyapunov exponent \overline{μ}_{n+1} of φ on K with n ∈ \{1, 2, …, m − 1\} satisfies \overline{μ}_{n+1} < 0. Then

dim_{F} K ≤ n + \frac{max_{j=1,... ,n}(μ_{1} + … + μ_{j})_{+}}{|\overline{μ}_{n+1}|} .

((·)_{+} denotes here the positive part of a real number.)

Proof. Without loss of generality we assume that

n + \frac{max_{j=1,... ,n}(μ_{1} + … + μ_{j})_{+}}{|\overline{μ}_{n+1}|} < m.

With the proof of Theorem 3.1 and estimate (3.6) we have

(2 Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d + 1})^{d}n_{K}(2 Cε\overline{σ}_{d_{0}+1}(p)\sqrt{d + 1})

≤ n_{K}(ε)2^{2d}C^{d}ε^{d}(d + 1)^{d/2} max_{1≤l≤d_{0}} \overline{ω}_{l}(p)\overline{ω}_{d_{0}+1}(p)^{\frac{d−l}{d_{0}+1}} . (3.8)

Note that this inequality is valid for all numbers d and d_{0}, satisfying 1 ≤ d_{0} < d ≤ m and d_{0} ∈ ℕ. Now, we replace d_{0} by n and consider an arbitrary d with

n + \frac{1}{|\overline{μ}_{n+1}|} max_{j=1,... ,n}(μ_{1} + … + μ_{j})_{+} < d < m.  (3.9)

Then (3.8) results in

(2 Cε\overline{σ}_{n+1}(p)\sqrt{d + 1})^{d}n_{K}(2 Cε\overline{σ}_{n+1}(p)\sqrt{d + 1})

≤ n_{K}(ε)2^{2d}C^{d}ε^{d}(d + 1)^{d/2} max_{1≤l≤n} \overline{ω}_{l}(p)\overline{ω}_{n+1}(p)^{\frac{d−l}{n+1}} .

Setting α = 2 C\overline{σ}_{n+1}(p)\sqrt{d + 1} and ϑ = 2^{2d}C^{d}(d + 1)^{d/2} max_{1≤l≤n} \overline{ω}_{l}(p)\overline{ω}_{n+1}(p)^{\frac{d−l}{n+1}} we obtain

the inequality (αε)^{d}n_{K}(αε) ≤ ϑε^{d}n_{K}(ε).

As in the proof of Theorem 3.1 in a next step we shall estimate α and ϑ. Due to the definition of the Lyapunov exponents and condition (3.9) for d we get for all j = 1, …, n

\begin{matrix}lim_{p→∞} \frac{1}{p} ln(\overline{ω}_{j}(p)\overline{σ}_{n+1}(p)^{d−j}) = μ_{1} + … + μ_{j} + (d − j)\overline{μ}_{n+1} \\ = μ_{1} + … + μ_{j} − (d − j)|\overline{μ}_{n+1}| ≤ (μ_{1} + … + μ_{j})_{+} − (d − j)|\overline{μ}_{n+1}| < 0 . \end{matrix}

It follows that there exists a p_{0} such that for all p ≥ p_{0} there is an a > 0 with

\frac{1}{p} ln(\overline{ω}_{j}(p)\overline{σ}_{n+1}(p)^{d−j}) < −a, i.e., \overline{ω}_{j}(p)\overline{σ}_{n+1}(p)^{d−j} < exp(−pa) → 0 for p → ∞.

The same arguments based on lim sup_{p→∞} \frac{1}{p} ln \overline{σ}_{n+1}(p) = \overline{μ}_{n+1} < 0 lead to the convergence

\overline{σ}_{n+1}(p) → 0 for p → ∞. From both we get α, ϑ → 0 for p → ∞. As in the proof of Theorem 3.1 now dim_{F} K ≤ d follows for any

d > n + \frac{max_{j=1,... ,n}(μ_{1} + … + μ_{j})_{+}}{|\overline{μ}_{n+1}|} .

This completes the proof. ■

Remark 3.3 Theorem 3.3 gives an upper fractal dimension bound equivalent to the well-known Kaplan-Yorke formula [9]. Under the same assumptions as in Theorem 3.3 the Hausdorff dimension can be estimated (see [4]) by

dim_{H} K ≤ n + \frac{|μ_{1} + … + μ_{n}|}{|μ_{n+1}|}.

Theorem 3.4 Let (M,g) be an m-dimensional Riemannian C^{3}-manifold, φ : M → M a C^{1}-map and K ⊂ M a compact φ-invariant set, i.e. φ(K) = K. Assume the global Lyapunov exponents of φ satisfy the conditions −∞ < \underline{μ}_{m} < 0 and μ_{1} + … + μ_{m} < 0. Then the fractal dimension of K can be estimated by

dim_{F} K ≤ m − \frac{|μ_{1} + … + μ_{m}|}{|\underline{μ}_{m}|}.

Proof. The proof of Theorem 3.2 in [1] gives that only

sup_{u∈K} ω_{m}(d_{u}φ^{p}) [ inf_{u∈K} σ_{m}(d_{u}φ^{p})]^{d−m} → 0 for p → +∞  (3.10)

remains to be shown. Let d be an arbitrary number satisfying

d > m − \frac{|μ_{1} + … + μ_{m}|}{|\underline{μ}_{m}|} .

By the definition of the Lyapunov exponents it follows that

lim_{p→∞} \frac{1}{p} ln \{sup_{u∈K} ω_{m}(d_{u}φ^{p}) [ inf_{u∈K} σ_{m}(d_{u}φ^{p})]^{d−m}\} = μ_{1} + … + μ_{m} + (d − m)\underline{μ}_{m} < 0.

This leads directly to (3.10). We can therefore avoid condition (3.7) of Theorem 3.2 and continue the proof as in [1]. Since the only condition on d was (3.10), it follows immediately that dim_{F} K ≤ d. ■

4 Vector fields on Riemannian manifolds

In many cases differential equations occur as vector fields on manifolds. However, even if a differential equation is originally given in ℝ^{m} with standard metric it may be useful to consider it in this space interpretated as Riemannian manifold with a special metric tensor. An other advantage of the interpretation on Riemannian manifolds is, for example, that non-compact invariant sets of systems with periodic right hand-side become compact on the factor-manifold, which is essential for dimension estimates. Moreover, we have a free choice of the smooth Riemannian metric which does not change the dimension and turns out to be a powerful tool to improve various dimension estimates.

4.1 General Fractal dimension estimates for flow-invariant sets

The upper bounds for the fractal dimension given by Theorem 3.1 and 3.2 shall now be used in order to estimate the fractal dimension of attractors of differential equations. Let (M, g) be an m-dimensional Riemannian C^{3}-manifold, U ⊂ M an open subset, f : U → T M a C^{1}-vector field and

u̇ = f(u)  (4.1)

the associated differential equation in U. The differential equation generates a flow {φ^{t}} : U → U which we assume to be defined for all times t ∈ ℝ. In this section we will present a number of fractal dimension estimates of flow-invariant sets which are based on the estimates for maps. It is well-known ([13]) that for arbitrary u ∈ D the differential d_{u}φ^{t} for a solution φ^{(.)}(u) of (4.1) satisfies the variational equation

\frac{Dy}{dt} = ∇f(φ^{t}(u))y  (4.2)

with initial condition d_{u}φ^{t}|_{t=0} = id_{T_{u}M}. The absolute derivative on the left hand-side is taken along the curve φ^{t}(u) as a function of t in direction of the vector field f.

Consider for any u ∈ U the covariant derivative as linear operator ∇f(u) : T_{u}M → T_{u}M in the Euclidean space (T_{u}M, ⟨·, ·⟩_{T_{u}M}) and the adjoint ∇f(u)^{∗}. Let λ_{1}(u) ≥ … ≥ λ_{m}(u) be the eigenvalues of the symmetric part of the covariant derivative, i.e., of S∇f(u) = \frac{1}{2}(∇f(u) + ∇f(u)^{∗}), counted with respect to their multiplicity. It follows that for any u ∈ U we have divf(u) = tr∇f(u) = λ_{1}(u) + … + λ_{m}(u), where divf is the divergence of f with respect to the metric g.

For any k = 1, 2, …, m we introduce the k-partial divergence of f in u given by div_{k}f(u) = λ_{1}(u) + … + λ_{k}(u). For arbitrary d ∈ (0, m], written as d = d_{0} + d_{1} with d_{0} ∈ {0, 1, ., m − 1} and d_{1} ∈ (0, 1], we define the d-partial divergence of f by div_{d}f(u) = div_{d_{0}}f(u) + d_{1}λ_{d_{0+1}}(u). Introduce also the abbreviations

Λ_{i,K}(f) = max_{u∈K} div_{i}f(u) (i = 1, 2, …, m).

As above for maps we use here the notations σ_{i}(t, u) = σ_{i}(d_{u}φ^{t})(i = 1, …, m) for the singular values of d_{u}φ^{t} and ω_{i}(t, u) = ω_{i}(d_{u}φ^{t}) (i = 1, …, m) for the singular value function.

The following result from [13] is crucial for further considerations.

Lemma 4.1 For any u ∈ U and k ∈ {1, …, m} the inequalities

σ_{1}(t, u) … σ_{k}(t, u) ≤ exp ∫_{0}^{t}(λ_{1}(φ^{τ}(u)) + … + λ_{k}(φ^{τ}(u)))dτ and

σ_{m−k+1}(t, u) … σ_{m}(t, u) ≥ exp ∫_{0}^{t}(λ_{m−k+1}(φ^{τ}(u)) + … + λ_{m}(φ^{τ}(u)))dτ .

hold for all t > 0. In particular, for k = m and t > 0, the equation

ω_{m}(t, u) = exp ∫_{0}^{t} divf(φ^{τ}(u))dτ

is fulfilled.

Thanks to this lemma one obtains immediately from Theorems 3.1 and 3.2 the two following theorems.

Theorem 4.1 Let K ⊂ U be a compact set on the m-dimensional C^{3}-manifold M and invariant with respect to the flow of (4.1). Suppose that for k ∈ {1, …, m − 1} the inequalities

Λ_{k+1,K}(f) < 0 and \frac{(Λ_{j,K}(f))_{+}}{−Λ_{k+1,K}(f)} < 1 − \frac{j}{k + 1} (j = 1, …, k)

are satisfied. Then dim_{F} K ≤ max_{j=1,... ,k} [j + (k + 1)\frac{(Λ_{j,K}(f))_{+}}{−Λ_{k+1,K}(f)}] .

Proof. Consider an arbitrary d satisfying

max_{j=1,... ,k} [j + (k + 1)\frac{(Λ_{j,K}(f))_{+}}{−Λ_{k+1,K}(f)}] < d ≤ k + 1.

With d_{0} = k this leads to the inequalities

Λ_{j,K}(f) + \frac{d − j}{d_{0} + 1}Λ_{d_{0}+1,K}(f) < 0 for j = 1, …, d_{0}.

For t > 0 small enough we obtain

sup_{u∈K} ∫_{0}^{t} div_{j}f(φ^{τ}(u))dτ + \frac{d − j}{d_{0} + 1} sup_{u∈K} ∫_{0}^{t} div_{d_{0}+1}f(φ^{τ}(u))dτ < 0.  (4.3)

With Lemma 4.1 it follows for small t > 0 that

ω_{j,K}(φ^{t}) [ω_{d_{0}+1,K}(φ^{t})]^{\frac{d−j}{d_{0}+1}} < 1.  (4.4)

Note that due to (4.4) d_{0} fulfils the conditions of Theorem 3.1 with φ^{t} instead of φ. Applying this theorem we get dim_{F} K ≤ d. Since d was chosen arbitrarily but greater than the left hand-side of (4.3) the claim follows. ■

For a given compact set K ⊂ M and vector field (4.1), we define

Θ_{K}(f) = min_{u∈K} divf(u).

Theorem 4.2 Let K ⊂ U be a compact flow-invariant set of (4.1) on the C^{3}-smooth m-dimensional Riemannian manifold (M, g). Suppose that

Θ_{K}(f) − Λ_{m−1,K}(f) ≤ \frac{1}{m}Λ_{m,K}(f) < 0 .

Then dim_{F} K ≤ m − \frac{Λ_{m,K}(f)}{Θ_{K}(f) − Λ_{m−1,K}(f)} .

Proof. The proof is similar to that of Theorem 4.1. Consider an arbitrary d satisfying

m − \frac{Λ_{m,K}(f)}{ΘK(f) − Λ_{m−1,K}(f)} < d < m

which implies

Λ_{m,K}(f) + (m − d)(Λ_{m−1,K}(f) − Θ_{K}(f)) < 0.

For t > 0 small enough we obtain

sup_{u∈K} ∫_{0}^{t} divf(φ^{τ}(u))dτ

+ (m − d) (sup_{u∈K} ∫_{0}^{t} div_{m−1}f(φ^{τ}(u))dτ − inf_{u∈K} ∫_{0}^{t} divf(φ^{τ}(u))dτ) < 0. (4.5)

With Lemma 4.1 it follows that for small t > 0

ω_{m,K}(φ^{t})ρ_{K}(φ^{t})^{d−m} < 1.

Since divf(φ^{τ}(u)) > −∞ for τ ∈ [0, t] and u is contained in the compact set K we have ω_{m}(φ^{t}(u)) ≠ 0 and σ_{m}(φ^{t}(u)) ≠ 0 for u ∈ K and t > 0. Thus, the assumption ρ_{K}(φ^{t}) > 0 of Theorem 3.2 is fulfilled. Applying this theorem we get dim_{F} K ≤ d. Since d was chosen arbitrarily but greater than the left hand-side of (4.5) the latter one proves the result. ■

Remark 4.1 The proofs of both Theorems 4.1 and 4.2 obviously show that the only conditions needed to be satisfied are (4.3) and (4.5), respectively. This leads in many cases to better fractal dimension estimates. But one should pay attention to the fact that these conditions are essentially based on estimates for the Λ_{i}(t, u) using certain information about the flow {φ^{t}}.

Remark 4.2 Theorems 4.1 and 4.2 can be applied without any change to non-au-tonomous differential equations of the form u̇ = f(t, u), too. All steps can be done analogously.

4.2 Introduction of a new metric by frequency domain conditions

The fact that the fractal dimension of compact sets on smooth manifolds is independent of the choice of the Riemannian metric turns out to be beneficial. It allows to alter the metric tensor which, indeed, can be used to improve dimension estimates essentially. In the remaining part of this paper we adjust the metric by introducing frequency domain conditions. Similar conditions appear for the first time in stability investigations of feedback systems and stem from the well-known Yakubovich-Kalman theorem ([8, 16]). In [10] and [13] similar techniques were applied to estimate the Hausdorff dimension. For a detailed theory of these topics we refer also to [14].

To verify the conditions of Theorems 4.1 and 4.2 we want to estimate the values Λ_{i,K}(f) from above. If we introduce a continuous function ϑ(u) on M satisfying λ_{i}(u) ≥ −ϑ(u) for all i = 1, …, m we get such an estimate from

div_{i}f(u) = divf(u) − (λ_{i+1}(u) + … + λ_{m}(u))

≤ divf(u) + (m − i)ϑ(u) for i = 1, …, m − 1.

Taking the supremum we obtain

Λ_{i,K}(f) ≤ sup_{u∈K} (divf(u) + (m − i)ϑ(u)) for i = 1, …, m − 1.  (4.6)

For differential equations in ℝ^{m} the following lemma provides a criterion for such a lower bound −ϑ(u) of the eigenvalues. Suppose that on the open set U ⊂ ℝ^{m} the C^{1}-vector field

ẋ = f(x)  (4.7)

is given, which generates on U the flow {φ^{t}}. Assume also that a compact set K ⊂ M is invariant with respect to the flow {φ^{t}}.

Lemma 4.2 Consider the differential equation (4.7) and suppose that ϑ : U → ℝ is a continuous function. Let u ∈ U be arbitrary and suppose also that for a constant real m × m-matrix G = G^{T} > 0 the inequality

x^{T} [f′(u)^{T} G + Gf′(u) + 2ϑ(u)G] x ≥ 0  (4.8)

holds for all x ∈ ℝ^{m}. Then the eigenvalues λ_{i}(u) of the symmetric part \frac{1}{2}(f′(u)+(f′(u)^{∗}) of the Jacobi matrix of (4.7), determined with respect to G as the metric tensor on U, satisfy the inequalities

λ_{i}(u) ≥ −ϑ(u) for i = 1, …, m.

Proof. The matrix of the symmetrized covariant derivative of the vector field f in u is given by

S(u) = \frac{1}{2}[G^{−1}f′(u)^{T} G + f′(u)].  (4.9)

It follows that the eigenvalues λ_{i}(u) of S(u) are the solutions of

det[GS(u) − λG] = 0 .

For the eigenvectors x_{i}(u) corresponding to λ_{i}(u) we have

x_{i}(u)^{T} [GS(u) − λ_{i}(u)G]x_{i}(u) = 0.

With (4.9) the last relation can be written as

\frac{1}{2}x_{i}(u)^{T} [Gf′(u) + f′(u)^{T} G] x_{i}(u) = λ_{i}(u)x_{i}(u)^{T} Gx_{i}(u).

Setting x = x_{i}(u) in (4.8) we obtain now

\frac{1}{2}x_{i}(u)^{T} [Gf′(u) + f′(u)^{T} G] x_{i}(u) ≥ −ϑ(u)x_{i}(u)^{T} Gx_{i}(u) (i = 1, …, m).

Since G is positive definite this implies λ_{i}(u) ≥ −ϑ(u) for i = 1, …, m. ■

Any differential equation (4.7) can be written as feedback control system

ẋ = P x + Qϕ(σ),  σ = R^{T} x,  (4.10)

where P, Q and R are constant m × m-matrices and ϕ(·) is a C^{1}-vector function. Recall that the pair (P, Q) is said to be controllable if rank(Q, P Q, …, P^{m−1}Q) = m, and the pair (P, R) is said to be observable if rank(R, P^{T} R, …, (P^{T})^{m−1}R) = m. Denote by χ(z) = R^{T} (zI − P)^{−1}Q the transfer matrix function of the linear part of (4.10), defined for all z ∈ ℂ with det(zI − P) ≠ 0.

Theorem 4.3 Consider the differential equation (4.7), given as feedback control system (4.10). Suppose that there exist positive numbers λ and γ such that the following conditions are satisfied:

(i) \frac{1}{2}η^{T} [φ′(R^{T} x)^{T} + φ′(R^{T} x)] η ≤ λ∥η∥^{2} for all η ∈ ℝ^{m} and x ∈ U;

(ii) The pairs (P, Q) and (P, R) are controllable and observable, respectively;

(iii) All eigenvalues of P + γ I have positive real part;

(iv) Reχ(iω − γ) − λχ^{∗}(iω − γ)χ(iω − γ) ≥ 0 for all ω ∈ ℝ.

Then the following statements hold:

(a) If k ∈ {1, …, m − 1}, sup_{u∈K} divf(u) + (m − k − 1)γ < 0 and

\frac{j}{k + 1} < \frac{[sup_{u∈K} divf(u) + (m − j)γ]_{+}}{sup_{u∈K} divf(u) + (m − k − 1)γ} + 1 for j = 1, …,k, then

dim_{F} K ≤ max_{j=1,... ,k} [j + (k + 1)\frac{(sup_{u∈K} divf(u) + (m − j)γ)_{+}}{sup_{u∈K} divf(u) + (m − k − 1)γ} ] .

(b) If m [ inf_{u∈K} divf(u) − sup_{u∈K} divf(u) − γ] ≤ sup_{u∈K} divf(u) then

dim_{F} K ≤ m − \frac{sup_{u∈K} divf(u)}{sup_{u∈K} divf(u) − inf_{u∈K} divf(u) + γ} .

Proof. The conditions (i) − (iv) of the theorem provide the existence of a constant real m×m-matrix G = G^{T} > 0 satisfying the inequality v^{T} [f′(x)^{T} G+Gf′(x)+2γG]v ≥ 0 for arbitrary x ∈ U and for every vector v ∈ ℝ^{m}. Setting ϑ(u) = γ the last exactly agrees with condition (4.8) of Lemma 4.2., which guarantees that the eigenvalues λ_{i}(u), i = 1, …, m, of S∇f(u) fulfil the inequality λ_{i}(u) ≥ −γ.

The dimension estimates in (i) and (ii) now follow immediately from both Theorems 4.1 and 4.2 using relation (4.6). ■

Remark 4.3 The Hausdorff dimension estimate

dim_{H} K ≤ m − \frac{sup_{u∈K} |divf(u)|}{γ}

is given in [2] under exactly the same conditions as in Theorem 4.3. If the divergence of the vector field is constant we get in (b) the same estimate for the fractal dimension.

4.3 A fractal dimension estimate for the Lorenz attractor

In this section we use some approximation results from [11] on the Lorenz system which are carried out there to estimate the Hausdorff dimension of its attractor. The Lorenz system is defined in ℝ^{3} by the differential equation

\begin{matrix}ẋ = −dx + dy , \\ ẏ = rx − y − xz , \\ ż = −bz + xy , \end{matrix}  (4.11)

where d, b and r are positive parameters. Numerical calculations show that for parameter values d = 10, b = \frac{8}{3} and r = 28 system (4.11) has a strange attractor characterized also by a non-integer Hausdorff dimension. For parameter values r ≤ 1 and 2d < b system (4.11) is globally stable, i.e., any solution converges to an equilibrium for t → +∞ ([12]).

One can show (see [2, 12]) that any attractor of (4.11) is located in the set

C = {(x, y, z) : x ∈ [x_{0}, x_{1}], y^{2} + (z − r)^{2} ≤ l^{2}r^{2}, z ≥ 0},

where x_{0} and x_{1} are certain numbers and

l = \{ \begin{matrix}1 & for & b ≤ 2, \\ \frac{b}{2\sqrt{b − 1}} & for & b ≥ 2. \end{matrix}

We write now the system (4.11) in feedback form (4.10) with the matrices

P = [ \begin{matrix}−d & d_{1} & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −b \end{matrix} ] , Q = [ \begin{matrix}−κ_{3} & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ] , R = [ \begin{matrix}κ_{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ]

and the smooth function ϕ(u, v, w) = \frac{1}{κ_{2}} (−κ_{1}v, (w − r)u, −uv), where κ_{1} ≥ 0, κ_{2} >

0, κ_{3} > 0 are suitable parameters and d_{1} = d − \frac{κ_{1}κ_{3}}{κ_{2}} .

The calculation of the maximum eigenvalue λ_{max} of the matrix

\frac{1}{2} [\frac{∂ϕ^{T}}{∂σ} (R^{T} x) + \frac{∂ϕ}{∂σ}(R^{T} x)]

gives

λ_{max} = \frac{1}{2κ_{2}} \sqrt{y^{2} + (z − r)^{2} + 2κ_{1}r + κ_{1}^{2} − 2κ_{1}z.}

Using the inclusion K ⊂ C we can estimate this value by

λ_{max} ≤ \frac{1}{2κ_{2}} \sqrt{l^{2}r^{2} + 2κ_{1}r + κ_{1}^{2}} = λ′.

For λ = λ′ the assumptions of Theorem 4.3. are satisfied if the inequalities

\begin{matrix}d + λ′κ_{2}κ_{3} − γ < 0, b + λ′ − γ < 0 and \\ 4κ_{3}(d + λ′κ_{2}κ_{3} − γ)(1 − λ′ − γ) − κ_{2} (d − \frac{κ_{1}κ_{3}}{κ_{2}} )^{2} > 0 \end{matrix}  (4.12)

hold. Let now γ′ be the maximum real root of the quadratic equation

4κ_{3}(d + λ′κ_{2}κ_{3} − γ)(1 − λ′ − γ) − κ_{2} (d − \frac{κ_{1}κ_{3}}{κ_{2}} )^{2} = 0 .

(If there is no real solution we define γ′ = 0.) Setting now

γ = inf_{κ_{1}≥0,κ_{2}>0,κ_{3}>0} max{d + λ′κ_{2}κ_{3}, b + λ′, γ′}  (4.13)

the above inequalities (4.12) hold. Since for the right hand-side f of (4.11) divf ≡ −d − b − 1 Theorem 4.3 (b) leads immediately to the fractal dimension estimate

dim_{F} K ≤ 3 − \frac{d + b + 1}{γ}

for the considered compact set K.

Choosing now the above mentioned parameters d = 10, b = \frac{8}{3} and r = 28 we obtain the infimum in (4.13) for κ_{1} = 24, κ_{2} = 1.3, κ_{3} = 0.45. This leads to the estimate dim_{F} K ≤ 2.405. Note, that numerical calculations of the Hausdorff dimension of K for these parameters give dim_{H} K ≈ 2.05.

Finally note, that other available dimension estimates of the Lorenz attractor are dim_{H} K ≤ 2.666 ([14]), dim_{H} K ≤ 2.538 ([15]), dim_{F} K ≤ 2.408 ([7]) and dim_{H} K ≤ 2.405 ([11]).

Literatur

[1] V. A. Boichenko, A. Franz, G. A. Leonov and V. Reitmann: Hausdorff and fractal dimension estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre Anwendungen, 17 (1), 207-223, 1998.

[2] A. V. Boichenko and G. A. Leonov: Frequency-domain estimates of the Hausdorff dimension of attractors of nonlinear systems. Differential’nye Uravneniya, 26 (4), 555-563, 1990. (Russian)

[3] P. Constantin, C. Foias and R. Temam: Attractors representing turbulent flows. Amer. Math. Soc. Memoirs, vol. 53, 314, 1985.

[4] P. Constantin and C. Foias: Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations. Comm. Pure. Appl. Math., 38, 1-27, 1985.

[5] P. Constantin and C. Foias: Navier-Stokes Equations. Chicago Lectures in Mathematics, Chicago, 1988.

[6] A. Douady et J. Oesterlé: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A 290, 1135-1138, 1980.

[7] A. Eden, C. Foias and R. Temam: Local and global Lyapunov exponents. J. Dynam. Diff. Equations, 3, 133-177, 1991.

[8] R. E. Kalman: Lyapunov functions for the problem of Lur’e in automatic control. Proc. National Acad. Sci. USA, 49 (2), 201-205, 1963.

[9] J. Kaplan and J. Yorke: Chaotic behaviour of multidimensional difference equations. Lect. Notes in Mathematics, vol. 730, 204-227,1979.

[10] G. A. Leonov: On estimates of the Hausdorff dimension of attractors. Vestnik Leningrad gos. univ., Ser. 1, (15), 41-44, 1991. (Russian)

[11] G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova: Frequency-Domain Methods for Nonlinear Analysis. World Scientific, Singapore-New Jersey-London-Hong Kong, 1996.

[12] G. A. Leonov und V. Reitmann: Attraktoreingrenzung für nichtlineare Systeme. Teubner-Texte zur Mathematik, Bd. 97, Teubner-Verlag, Leipzig, 1986.

[13] A. Noack and V. Reitmann: Hausdorff dimension estimates for invariant sets of time-dependent vector fields. Zeitschrift für Analysis und ihre Anwendungen, 15 (2), 457-473, 1996.

[14] R. A. Smith: Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proc. Roy. Soc. Edinburgh, 104A, 235-259, 1986.

[15] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York-Berlin, 1988.

[16] V. A. Yakubovich: The solution of some matrix inequalities which appear in the automatic control theory. Doklady Akad. Nauk SSSR, 143 (6), 1304-1307, 1962. (Russian).