Statistical self-similarity in Lozi and Hénon's strange attractors

Abstract

Non-classical characterizations of strange attractors assign occupancy times and numbers to each of the hypercubes that cover them, local information, and generate statistical distributions over such quantities, global information, at the expense of losing all the local ones. The confrontation between local and global information would allow statistical quantification, important for the characterization of the self-similarity of the involved attractor: correspondence between global and local properties are expected as manifestations of the self-similarity that characterizes that fractal set (statistical self-similarity). Hence, the importance of clarifying what the local distributions would be. The core of this work is the presentation of statistical distributions of the mentioned times and numbers, and quantities derived from them, linked to each hypercube of a set that corresponds to a small fraction of their total, that confirm the correspondence between local and global statistical properties.

Keywords:

• Lozi strange attractor
• Hénon strange attractor
• multifractality
• chaotic advection
• statistical self-similarity
• power laws

Mathematics Subject Classifications:

• 37M22
• 37M05

Acknowledgments

Thanks to Valdir Barbosa Bezerra, Full Professor at the Federal University of Paraí ba, Brazil, for his continuous encouragement and guidance at crucial times and to René Pierre Lozi, Professor Emeritus at Université Côte d'Azur, France, for his suggestions and encouragement to publish in this journal.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 From this point on, the subscript indexes n and t will respectively designate occupancy numbers and occupancy times.

2 From this point forward the superscript index (q) will refer to quantities linked to some hypercube, as opposed to the quantities associated with the corresponding attractor taken as a whole.