The fractal dimension of an attractor can be defined by the following construction. Let b(ε) be the minimum number of balls of diameter ε needed to cover the attractor. The fractal dimension is defined by the limit,

(10.19)

As the length scale ε is reduced, the number of balls required to cover the attractor increases. As b(ε) is a positive integer, its logarithm is positive. The term lnε is negative as soon as the length scale ε is less than one (in the appropriate units), the dimension is a positive real quantity.

To obtain the information dimension we suppose an observer makes an isolated measurement of the coarse grained probability distribution function p_i. Coarse graining implies a length scale ε for the observation, and an associated number of cells N(ε). The discrete entropy S(ε) as a function of the length scale is given by

(10.20)

Notice that S(ε) is positive as for each i, -p_ilnp_i is positive. The information dimension D_I is then defined by

(10.21)

This dimension is a property of any distribution function as nothing in the definition is specific to attractors, or to some underlying dynamics.

If all the N(ε) elements have the same probability then S(ε)=lnN(ε). Further if b(ε) is a minimal covering, then a smaller covering can be formed by removing the overlapping parts of circles so that ln b(ε)≥lnN(ε)=S(ε). It is then straightforward to see that the fractal dimension is an upper bound on the information dimension. (We will generalize this result later.) From a computational point of view it is easier to tabulate the steady state distribution function and calculate D_I, rather than to attempt to identify the attractor and construct a covering to calculate D_F.

The correlation dimension D_C introduced by Grassberger and Procaccia (1983) is a scaling relation on the correlation function C(ε) where

(10.22)

Here is the Heavyside step function. C(ε) is the correlation integral which counts the number of pairs of points whose distance of separation |Γ_i-Γ_j| is less than ε. The correlation dimension is

(10.23)

It has been argued that the correlation dimension can be calculated numerically, more easily and more reliably than either the information dimension or the fractal dimension.

In a series of papers by Grassberger, Hentschel, Procaccia and Halsey et. al. have been shown that the concept of dimension can be generalized further. They introduce a generating function D_q which provides an infinite spectrum of dimensions depending upon the value of a parameter q. We will show that all previous dimensions are related to special values of q. Again we begin with a discrete probability distribution p_i taken at a course graining length ε. By averaging powers of the p_is over all boxes, the generalized dimension D_q is obtained

(10.24)

There are formal similarities between the D_q and the free energy per particle F_β in the thermodynamic limit,

(10.25)

where E_i are the energy levels in the system, N is the number of particles and β=(k_BT)^-1 is the inverse temperature. The analogy is not a strict one as the probability of state i is exp(-βE_i) rather than simply exp(-E_i) as implied above. Also the probabilities p_i are normalized, while neither exp(-βE_i) nor exp(-E_i) are normalized. This is crucial in statistical mechanics as if normalized probabilities are inserted into equation (10.25) in place of exp(-βE_i), the free energy F_β is trivially zero.

It straightforward to see that D_q gives each of the previously defined dimensions. For q=0, q^q_i=1 for all values of i, so that

(10.26)

This is the fractal or Hausdorff dimension equation (10.19).

For q=1 consider the limit

(10.27)

Substituting this limit into the expression for D_q gives

(10.28)

This is simply the information dimension. For q=2 it is easy to show that the generalized dimension is the correlation dimension.

The generalized dimension D_q is a non-increasing function of q. To show this we consider the generalized mean M(t) of the set of positive quantities {a₁,...,a_n}, where p_i is the probability of observing a_i. The generalized mean is defined to be

(10.29)

This reduces to the familiar special cases; M(1) is the arithmetic mean and the limit as t→0 is the geometric mean. It is not difficult to show that if a_i=p_i(ε), where the p_i(ε) are a set of discrete probabilities calculated using a length scale of ε, then the generalized dimension in equation (10.24) is related to the generalized mean by

(10.30)

Using a theorem concerning generalized means, namely if t<s then M(t)≤ M(s) (Hardy, Littlewood and Pólya (1934), page 26) it follows that if s>t then D_s≤D_t.

If we consider the quadratic map for μ=4, the distribution of the iterates shown in Figure 10.5, is characterized by the two singularities at x=0 and x=1. For μ=3.65, the distribution of iterates, shown in Figure 10.4, has approximately ten peaks which also appear to be singularities. It is common to find a probability distribution on the attractor which consist of sets of singularities with differing fractional power law strengths. This distribution of singularities can be calculated from the generalized dimension D_q. To illustrate the connection between the generalized dimensions D_q and the singularities of the distribution function, we consider a one-dimensional system whose underlying distribution function is

for . (10.31)

First note that, despite the fact that ρ(x) is singular, ρ(x) is integrable on the interval 0≤ x≤ 1 and it is correctly normalized. The generalized dimension D_q is defined in terms of discrete probabilities so we divide the interval into bins of length ε - [0,ε) is bin 0, [ε,2ε) is bin 1, etc.. The probability of bin 0 is given by

(10.32)

and in general the probability of bin i is given by

(10.33)

where x_i=iε. As (x_i+iε)^½ is analytic for i≠0, we can expand this term to obtain

(10.34)

So for i=0, p_i~ε^½ but for all nonzero values of i, p_i~ε. To construct D_q we need to calculate

(10.35)

We can replace the last sum in this equation by an integral,

(10.36)

where . Combining this result with that for i=0 we obtain

(10.37)

The distribution function ρ(x) in equation (10.31) gives rise to singularities in the discrete probabilities p_i

If the discrete probabilities scale with exponent α_i, so that and

(10.38)

then α_i can take on a range of values corresponding to different regions of the underlying probability distribution. In particular, if the system is divided into pieces of size ε, then the number of times α_i takes on a value between α' and α'+dα' will be of the form

(10.39)

where f(α') is a continuous function. The exponent f(α') reflects the differing dimensions of the sets whose singularity strength is α'. Thus fractal probability distributions can be modeled by interwoven set of singularities of strength α, each characterized by its own dimension f(α).

In order to determine the function f(α) for a given distribution function, we must relate it to observable properties, in particular we relate f(α) to the generalized dimensions D_q. As q is varied, different subsets associated with different scaling indices become dominant. Using equation (10.39) we obtain

(10.40)

Since ε is very small, the integral will be dominated by the value of α' which makes the exponent qα'-f(α') smallest, provided that ρ(α') is nonzero. The condition for an extremum is

and (10.41)

If α(q) is the value of α' which minimizes qα'-f(α') then f'(α(q))=q and f''(α(q))<0. If we approximate the integral in equation (10.40) by its maximum value, and substitute this into equation (10.24) then

(10.42)

so that

(10.43)

Thus if we know f(α), and the spectrum of α values we can find D_q. Alternatively, given D_q we can find α(q), since f'(α)=q implies that

(10.44)

and knowing α(q), f(α(q)) can be obtained.

Grassberger and Procaccia (1983) and Eckmann and Procaccia (1986) have shown that it is possible to define a range of scaling indices for the dynamical properties of chaotic systems. Suppose that phase space is partitioned into boxes of size ε, and that a measured trajectory X(t) is in the basin of attraction. The state of the system is measured at intervals of time τ. Let p(i₁,i₂,...,i_n) be the joint probability that X(t=τ) is in box i₁, X(t=2τ) is in box i₂,..., and X(t=nτ) is in box i_n. The generalized entropies K_q are defined by

(10.45)

where the sum is over all possible sequences i₁,i₂,...,i_n. As before the most interesting K_q for experimental applications are the low order ones. The limit q→0, K_q=K is the Kolmogorov or metric entropy, whereas K₂ has been suggested as a useful lower bound on the metric entropy. For a regular dynamical system K=0, and for a random signal K=∞. In general for a chaotic system K is finite, and related to the inverse predictability time and to the sum of the positive Lyapunov exponents. The Legendre transform of (q-1)K_q, that is g(Λ), is the analogue of singularity structure quantities f(α) introduced in the last section (see Jensen, Kadanoff and Procaccia, 1987 for more details).

In §3.4 we introduced the concept of Lyapunov exponents as a quantitative measure of the mixing properties of a system. Here we will develop these ideas further, but first we review the methods which can be used to calculate the Lyapunov exponents. The standard method of calculating Lyapunov exponents for dynamical systems is due to Benettin et. al. (1976) and Shimada and Hagashima (1979). They linearize the equations of motion and study the time evolution of a set of orthogonal vectors. To avoid problems with rapidly growing vector lengths they periodically renormalize the vectors using a Gram-Schmidt procedure. This allows one vector to follow the fastest growing direction in phase space, and the second to follow the next fastest direction, while remaining orthogonal first vector, etc. The Lyapunov exponents are given by the average rates of growth of each of the vectors.

A new method of calculating Lyapunov exponents has been developed by Hoover and Posch (1985) and extended to multiple exponents by Morriss (1988) and Posch and Hoover (1988). It uses Gauss' principle of least constraint to fix the length of each tangent vector, and to maintain the orthogonality of the set of tangent vectors. The two extensions of the method differ in the vector character of the constraint forces - the Posch-Hoover method uses orthogonal forces, while the Morriss method uses non-orthogonal constraint forces. In earlier chapters we have used Gauss' principle to change from one ensemble to another. This application of Gauss' principle to the calculation of Lyapunov exponents exactly parallels this situation. In the Benettin method one monitors the divergence of a pair of trajectories, with periodic rescaling. In the Gaussian scheme we monitor the force required to keep two trajectories a fixed distance apart in phase space.

The rate of exponential growth of a vector δx(t) is given by the largest Lyapunov exponent. The rate of growth of a surface element δσ(t)=δx₁(t)xδx₂(t) is given by the sum of the two largest Lyapunov exponents. In general the exponential rate of growth of a k-volume element is determined by the sum of the largest k Lyapunov exponents λ₁+....+λ_k. This sum may be positive implying growth of the k-volume element, or negative implying shrinkage of the k-volume element.

A calculation of the Lyapunov spectrum gives as many Lyapunov exponents as phase space dimensions. All of the previous characterizations of chaos that we have considered, have led to a single scalar measure of the dimension of the attractor. From a knowledge of the complete spectrum of Lyapunov exponents Kaplan and Yorke (1979) have conjectured that the effective dimension of an attractor is given by that value of k for which the k-volume element neither grows nor decays. This requires some generalization of the idea of a k-dimensional volume element as the result is almost always non-integer. The Kaplan and Yorke conjecture is that the Lyapunov dimension can be calculated from

(10.46)

where n is the largest integer for which ∑ⁿ_i=1λ_i>0.

Essentially the Kaplan-Yorke conjecture corresponds to plotting the sum of Lyapunov exponents ∑ⁿ_i=1λ_i versus n, and the dimension is estimated by finding where the curve intercepts the n-axis by linear interpolation.

There is a second postulated relation between Lyapunov exponents and dimension due to Mori (1980).

(10.47)

where m₀ and m⁺ are the number of zero and positive exponents respectively, and λ^± is the mean value of the positive or negative exponents (depending upon the superscript). Farmer (1982) gives a modified form of the Mori dimension which is found to give integer dimensions for systems of an infinite number of degrees of freedom.