Mutual singularities of overlapping attractor and repeller.

Area of Science:

• Dynamical Systems Theory
• Fractal Geometry
• Chaos Theory

Background:

• Chaotic dynamical systems often exhibit complex fractal attractors and repellers.
• Understanding the interplay between attractors and repellers is crucial for characterizing system dynamics.
• Existing methods may struggle with non-orthogonal stable and unstable directions.

Purpose of the Study:

• To apply relative dimensions and mutual singularities for characterizing fractal properties of overlapping attractors and repellers.
• To analyze these properties in analytically solvable and numerically explored chaotic maps.
• To assess the validity of orthogonality assumptions in fractal analysis.

Main Methods:

• Utilized concepts of relative dimensions and mutual singularities.
• Analyzed a generalized baker's map analytically.
• Numerically explored the Anosov-Möbius and Chirikov-Möbius maps on a two-dimensional torus.
• Calculated relative Rényi and Kullback-Leibler dimensions and mutual singularity spectra.

Main Results:

• Successfully characterized fractal properties of overlapping attractors and repellers.
• Demonstrated applicability to both analytical and numerical examples.
• Showed that relative dimensions and singularity spectra can be approximated even with non-orthogonal directions.
• Validated the utility of orthogonality assumptions for approximation.

Conclusions:

• Relative dimensions and mutual singularities provide effective tools for analyzing fractal properties in chaotic systems.
• The methods are robust and offer good approximations even when orthogonality assumptions are not strictly met.
• This work advances the understanding of fractal structures in complex dynamical systems.