May 4, 1998

(Revised May 27, 2005)

J. Kaplan and J. A. Yorke [*Springer Lecture Notes in Mathematics***730**,
204 (1979)] have conjectured that the dimension of a strange attractor
can be approximated from the spectrum of Lyapunov exponents. Such
a dimension has been called the Kaplan-Yorke (or Lyapunov) dimension,
and
it has been shown that this dimension is close to other dimensions such
as the box-counting, information, and correlation dimensions for
typical
strange attractors.

For a system with *N* variables, there are *N* Lyapunov
exponents.
The sum of these exponents is the average rate at which a cluster of
initial
conditions expands in *N*-dimensional hypervolume:

d*V*/dt / *V* = l_{1}
+
l_{2} + l_{3}
+ ... + l_{N}

For a conservative (Hamiltonian) system, this quantity is zero (by
Liouville's
theorem). For a dissipative system, the quantity is negative and
there exists an attractor for the dynamics towards which initial
conditions
in the basin of attraction are drawn. If the system is chaotic,
at
least one of the Lyapunov exponents must be positive, and a strange
attractor
will exist. Following the usual convention of ordering the
Lyapunov
exponents from the largest (most positive) to the smallest (most
negative),
we conclude that l_{1} must be
positive
for a chaotic system. Systems with more than one positive
Lyapunov
exponent are called "hyperchaotic."

If we let *S*(*D*) represent the sum of the exponents
from
1 to *D* where *D* __<__ *N*, then it is
evident that
for a strange attractor, there is some maximum integer *D* = *j*
for which *S* is positive and an integer *j* + 1 for which *S*
is negative. The attractor must then have a fractal dimension
that
lies between *j* and *j* + 1. The essence of the
Kaplan-Yorke
conjecture is simply to interpolate the function *S*(*D*)
and
evaluate the value of *D* for which *S* = 0. That is
to
say, we seek the hypothetical fractional dimension in which there is
neither
expansion nor contraction. Using a linear interpolation, this
value
is

*D*_{KY} = *j* - *S*(*j*) / l_{j+1}

Since *S*(*j*) is positive and l_{j+1}
is negative, it follows that *D*_{KY} > *j. *Numerical
evaluation of *D*_{KY} can be problematic, for example
with
an attractor that is a 2-Torus or nearly so. In such a case the
first
two exponents are very small, and numerical errors can lead to
calculated
values almost anywhere between 1 and 2.

This and other errors can be reduced by using a polynomial
interpolation
rather than a linear one. For example, suppose we have a system
with
*N*
= 3. It is natural in such a case to consider fitting *S*(*D*)
to a parabola, with the result:

*D*_{KY} = {l_{2} +
3l_{3} + [9l_{2}^{2}
+ 6l_{2}l_{3}
- 8l_{1}l_{3}
+ 8l_{1}l_{2}
+ l_{3}^{2}]^{1/2}}
/ 2(l_{3} - l_{2})

If the system consists of ordinary differential equations and is
known
to be chaotic, then l_{2} must
equal
zero, and expression above simplifies to:

*D*_{KY} = 1.5 + 0.5[1 - 8l_{1}/l_{3}]^{1/2}

As an example, the Lorenz attractor has Lyapunov exponents (0.906, 0, -14.572), for which the standard Kaplan-Yorke formula gives 2.062. By comparison, the quadratic interpolation gives 2.112, which is a bit higher than the dimension calculated by other methods.

A good project would be to test how this prediction compares with
the
standard Kaplan-Yorke formula for attractors whose dimensions can be
continuously
varied from 2 to 3. Good candidates for such studies are chaotic
Hamiltonian systems such as the standard (Chirikov) map, the Ueda
attractor,
or Sprott Case A to which a small variable
dissipation is added. It it likely that the Kaplan-Yorke formula
will be a good approximation for strange attractors with dimensions
near
an integer, but that the quadratic modification will make a significant
difference when the dimension is close to an odd half integer. (This
study has now been done, and the results are published.)

Ref: J. C. Sprott, *Chaos and Time-Series
Analysis*
(Oxford University Press, 2003), pp.121-122.

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