Improved Kaplan-Yorke Dimension

J. Kaplan and J. A. Yorke [Springer Lecture Notes in Mathematics730, 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:

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 l₁ 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

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:

If the system consists of ordinary differential equations and is known to be chaotic, then l₂ must equal zero, and expression above simplifies to:

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.