Mathematical Software - Chaotic System - Attractor Dimension

Attractor's Dimension

Embed and Correlation Dimension

The algorithm of Grassberger and Procaccia (Phys. Rev. Lett. 50, 346-9) allows to find both the embed dimension and the correlation dimension in a single computations. When an attractor is embed in growing dimensions starting from 1 then its correlation dimension d_G grows as well up to some critical value. The correlation dimension is computed from the correlation function

	N
C(R) = lim_{N → ∞} [ 1 / N²	∑	H(R - \|x_i - x_j\|) ]
	i,j=1

where R is a correlation sphere radius and H denotes the Heaviside's function. |x_i - x_j| is a distance between two points from the attractor.
In the optfinderML user should set ranges of the sphere radius and parameters for the attractor reconstruction via the Time Delay Technique i.e. the assumed value of the time delay and a final value of the embed dimension up to which program will compute the correlation dimensions.

chaos, chaotic systems, dynamical systems, correlation dimension, ilona kosinska, kosinska, optfinderML, taketechease

On the basis of C(R) function the correlation dimension is computed from the log-log plot of C(R) function as

d(log₁₀(C(R))) / d(log₁₀(R))

at small R. The d_G with the growing embed dimension reaches its limit value. Further increasing of embed dimensions D does not change the correlation dimension (see figure below).

Obtained values of the d_G correlation dimension are as follows:

d_G = 0.754 for D = 1
d_G = 1.061 for D = 2
d_G = 1.019 for D = 3.

In the optfinderML, uncertainty of regression can be shown in the graph. In all presented cases, computed error lines are too close to regression lines and they are not seen as different from the central regression lines. Thus for better figure's clarity error lines are not plotted in the picture above.

Chaotic System - References

^{^} G. L. Baker, J. P. Gollub, Chaotic dynamics: an introduction, Cambridge University Press, 1996
^{^} Vadim S. Anishchenko et al., Nonlinear Dynamics of Chaotic and Stochastic Systems, Springer-Verlag, 2007
^{^} Boris P. Bezruchko and Dmitry A. Smirnov, Extracting Knowledge From Time Series, Springer-Verlag, 2010

Machine Learning - OptFinderML

Package for machine learning - OptFinderML.