In applied mathematics, the Kaplan–Yorke conjecture concerns the

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

of an

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...

, using

Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectory, trajectories. Quantitatively, two trajectories in phase sp ...

s. By arranging the Lyapunov exponents in order from largest to smallest

\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n

, let ''j'' be the largest index for which :

\sum_^j \lambda_i \geqslant 0

and :

\sum_^ \lambda_i < 0.

Then the conjecture is that the dimension of the attractor is :

D=j+\frac.

This idea is used for the definition of the

Lyapunov dimension In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of paper ...

Examples

Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to estimate the

fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...

and the

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...

of the corresponding attractor. * The

Hénon map In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaos theory, chaotic behavior. The Hénon map takes ...

with parameters ''a'' = 1.4 and ''b'' = 0.3 has the ordered Lyapunov exponents

\lambda_1=0.603

and

\lambda_2=-2.34

. In this case, we find ''j'' = 1 and the dimension formula reduces to ::

D=j+\frac=1+\frac=1.26.

* The

Lorenz system The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the ...

shows chaotic behavior at the parameter values

\sigma=16

\rho=45.92

and

\beta=4.0

. The resulting Lyapunov exponents are . Noting that ''j'' = 2, we find ::

D=2+\frac=2.07.

References

{{DEFAULTSORT:Kaplan-Yorke conjecture Dimension Dynamical systems Limit sets Conjectures