Multifractals in dynamical systems

This study considers the multifractal properties of a family of discrete-time dynamical systems from the 2-torus $T^2=\mathbb{R}^2/ \mathbb{Z}^2$ to itself and how uniformly distributed global stable and unstable manifolds can be.
The concepts of fractal and fractal dimensions are explained with examples and Python codes.
Then we review the connection between multifractals and dynamical systems by considering the multifractal formalism (as generalised fractals) and iterated function systems (as a class of dynamical systems), which confirms that dynamical systems can be multifractals.
Then, the classical multifractal formalism, MF-DFA, WTMM, and MFDMA are explained
as methods for addressing multifractality. We regard the dynamical systems as an approach to understanding mixing and the multifractal behavior as an important tool in practical settings to understand the performance of fluid mixing devices.

This research employs the classical multifractal formalism to investigate the multifractal behavior of perturbed maps in dynamical systems, specifically the perturbed CG map as (a single parameter $(p)$ transformation) and the doubly non-monotonic perturbed map as (a two-parameters $(p, q)$ transformation), where we utilise Python data science to demonstrate our results. Then, as a real-world example, we studied the multifractal behavior of the pulsed Dipole system, which displays chaotic advection generated by a pulsed source-sink pairs system, and compared the dynamical behavior to that of the perturbed maps.

The ergodicity of the perturbed maps, which is the range of the parameters where the unstable and stable manifolds eventually cover the whole torus, was investigated.
Then, the distribution of the line-segments on the torus governed by the (forward/inverse) perturbed maps was tested as well as the streamlines generated by the Dipole system.
Then the following multifractal functions are computed: the generalised fractal dimension $D_q$, which is determined by the scaling function $\tau (q)$ and the partition function $Z_q(\delta)$, the q-representation (for which we display $f(q)$ versus $D_q$ curves), and the monotonic function $\alpha(q)$ and the singularity multifractal spectrum $f(\alpha)$, where we also represented the curves of $D_q$, the $q$-representation, and $f(\alpha)$.

Furthermore, we provided two-dimensional line-graphs to illustrate the physical interpretations of the theoretical multifractal functions of the perturbed CG map. In addition, the 3-dimensional surfaces were given for the doubly non-monotonic perturbed map and the pulsed dipole system. These functions are the cumulative curve length, the fractal dimension $D_0$, the exponent $\alpha_0$, the dispersion $\Delta(\alpha)$, the clustering coefficient $\Delta f$, and the skew parameter s. The results from this study suggest that the distribution of the line-segments of the manifold governed by the map affects the map's density, dispersion, and clustering.

The study addressed the cancellation exponents as a result of sign-alternation provides a quantitative characterisation of the singularity inherent in the extreme tendency of the orientation of fast dynamo magnetic fields to oscillate rapidly in space. We demonstrated the distribution of positive and negative line segments and the related dimension spectra governed by the doubly non-monotonic perturbed map, where the perturbed CG map and the CG map are part of it.