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

Dear Naseem Ali

Thanks for your response.