Scaling Laws in Dynamical Systems

This book discusses many of the common scaling properties observed in some nonlinear dynamical systems mostly described by mappings. The unpredictability of the time evolution of two nearby initial conditions in the phase space together with the exponential divergence from each other as time goes by lead to the concept of chaos. Some of the observables in nonlinear systems exhibit characteristics of scaling invariance being then described via scaling laws.

From the variation of control parameters, physical observables in the phase space may be characterized by using power laws that many times yield into universal behavior. The application of such a formalism has been well accepted in the scientific community of nonlinear dynamics. Therefore I had in mind when writing this book was to bring together few of the research results in nonlinear systems using scaling formalism that could treated either in under-graduation as well as in the post graduation in the several exact programs but no earlier requirements were needed from the students unless the basic physics and mathematics. At the same time, the book must be original enough to contribute to the existing literature but with no excessive superposition of the topics already dealt with in other text books. The majority of the Chapters present a list of exercises. Some of them are analytic and others are numeric with few presenting some degree of computational complexity.

Discusses scaling investigation in nonlinear dynamics Applies the scaling formalism to discuss bifurcations and diffusion Discusses transition from limited to unlimited diffusion that may also be applied to diffusion in energy, which is a hot topic investigated in billiard dynamics. Can be used as textbook and reference book in mathematics, physics, mechanical and control engineering

Auteur

Edson Denis Leonel is Professor of Physics at São Paulo State University, Rio Claro, Brazil. He has been dealing with scaling investigation since his PhD in 2003 where the first scaling investigation in chaotic sea for the Fermi-Ulam model was studied. His research group developed different approaches and formalisms to investigate and characterise the several scaling properties in a diversity types of systems ranging from one-dimensional mappings, passing to ordinary differential equations, cellular automata, in meme propagations and also in the time dependent billiards. There are different types of transition we considered and discussed in these scaling investigations: (I) transition from integrability to non-integrability; (ii) transition from limited to unlimited diffusion and; (iii) production and suppression of Fermi acceleration. The latter approach involves the analytical solution of the diffusion equation. His group and he published more than 150 scientific papers in respected international journals including three papers in Physical Review Letters. He is author of two books in Portuguese language, one dealing with statistical mechanics (2015) and the other one dealing with nonlinear dynamics (2019), both edited by Blucher.

Résumé
"The book is useful for getting a sense of the dynamics typical for these types of maps and the analysis that can be employed to understand them. The exercises allow the interested reader to tinker and explore for themselves. Little technical background is required, and each chapter starts with an abstract and ends with a summary, both of which I found very useful. The writing is clear ... ." (D. J. W. Simpson, SIAM Review, Vol. 65 (1), March, 2023)
"The titles of the chapters are quite informative and they reflect not only the structure of the book but also its content ... . The book provides useful introductory materials, including exercises, for undergraduate and graduate students who wish to have an overview of common scaling properties at their related methodologies in mathematics, physics, mechanical and control engineering." (Vladimir Sobolev, zbMATH 1483.37002, 2022)

Contenu
Introduction.- One-dimensional mappings.- Some dynamical properties for the logistic map.- The logistic-like map.- Introduction to two dimensional mappings.- A Fermi accelerator model.- Dissipation in the Fermi-Ulam model.- Dynamical properties for a bouncer model.- Localization of invariant spanning curves.- Chaotic diffusion in non-dissipative mappings.- Scaling on a dissipative standard mapping.- Introduction to billiards dynamics.- Time dependent billiards.- Suppression of Fermi acceleration in the oval billiard.- A thermodynamic model for time dependent billiards.