According to the articles, we use fourier transform to compute power spetra density(psd) in spectral analysis ,
Sm = psd = lim {2|Ym|^2 /( N * delta) }, m=1,2,3,..., N / 2
psd is a function of frequency ( Fm = m / N),
In the logarithmic graph, Fm is expressed in terms of Sm.
Sm ~ Fm ^(- Beta) ---> log Sm ~ log Fm^(- Beta) ---> Beta (spectral exponent) = - log Sm / log Fm
the relation of spectral exponent with fractal dimension :
Beta = 5 - 2 D ----> D = (Beta - 5) / 2
for example : 0 < Beta < 1 ---> 0 < 5 - 2 D < 1--> 2 < D < 2.5
Answer range for fractal dimension(FD) : 2 < D < 2.5
I want to know what is the direct impact of the fractal dimension on the analysis of the Rossler system?
In other words, How can I connect this answer range(FD) to the analytical solution of Rossler system?
The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system.
See also https://web.math.pmf.unizg.hr/Dynsys_workshop_2018/zupanovic.pdf
Dear Muhammad Ali
Thanks for your response and the file,
Your answer leads me to a better understanding.
Article The dimension of chaotic attractors
Since the days of Cantor, sets in Euclidian that are "space filling" have been studied and there are all sorts of examples, e.g., subsets of the unit interval that have the same cardinality as the unit interval but measure zero which was generalized to subsets with the same cardinality as the unit interval but any arbitrary measure between 0 and 1. Dynamical systems can have very complex behavior and generate very complex phase curves in the ambient space. However, there are numerous examples of space filling curves where it is not obvious what classical dimension means, e.g. Smale Horseshoe.
Hence such concepts of Fractal dimension, Box dimension and Hausdorff dimension were developed. For the system mentions there has been a lot of work
https://iopscience.iop.org/article/10.1088/1742-6596/1003/1/012099/pdf
The system has even been proposed for a cryptographic application.
https://eprint.iacr.org/2009/447.pdf
Fractal geometry is closely tied to ergodic theory. See
https://www.ams.org/books/cbms/120/cbms120-endmatter.pdf
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