Scaling Laws in Dynamical Systems: Nonlinear Physical Science
Autor Edson Denis Leonelen Limba Engleză Hardback – 27 aug 2021
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.
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.
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Specificații
ISBN-13: 9789811635434
ISBN-10: 9811635439
Pagini: 276
Ilustrații: XXVII, 247 p. 109 illus., 79 illus. in color.
Dimensiuni: 160 x 241 x 21 mm
Greutate: 0.53 kg
Ediția:1st ed. 2021
Editura: Springer
Colecția Nonlinear Physical Science
Seria Nonlinear Physical Science
Locul publicării:Singapore, Singapore
ISBN-10: 9811635439
Pagini: 276
Ilustrații: XXVII, 247 p. 109 illus., 79 illus. in color.
Dimensiuni: 160 x 241 x 21 mm
Greutate: 0.53 kg
Ediția:1st ed. 2021
Editura: Springer
Colecția Nonlinear Physical Science
Seria Nonlinear Physical Science
Locul publicării:Singapore, Singapore
Cuprins
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.
Recenzii
“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)
Notă biografică
Dr. Vinícius Barros is a physicist with a profound understanding and expertise in the area of statistical mechanics, dynamical systems and information geometry. He earned his Ph.D. in Applied Physics from São Paulo State University "Júlio de Mesquita Filho" (UNESP) in 2023, with a focus on topics intrinsically linked to information geometry and dynamical systems, including bifurcations. His solid academic background also includes a Master's degree (2018) and a Bachelor's degree in Physics (2016), both from UNESP. Dr. Barros's expertise encompasses several crucial areas for the geometric theory of bifurcations, such as dynamical systems, chaos theory, Fisher information geometry, differential geometry, and the Fisher and Rao metrics, as well as scalar curvature. His research has focused on applying tools from statistical physics and information geometry to gain deep insights into dynamical systems, with a particular interest in understanding and characterizing bifurcations through a geometric lens. Dr. João Peres Vieira earned a bachelor’s degree in Mathematics from the São Carlos Federal University in 1984, a Master’s in Mathematics from the São Paulo University in 1988, and a Ph.D. in Mathematics from the same institution in 1995. In 2012, he achieved his Habilitation in Mathematics from São Paulo State University "Júlio de Mesquita Filho", where he currently serves as an Associate Professor. With extensive expertise in Mathematics, Dr. Vieira specializes in algebraic topology and dynamical systems. His primary research interests focus on fixed points, coincidence theory, and their applications within topological dynamics, offering important insights into the behaviour and structure of complex dynamical systems. His contributions reflect a deep commitment to advancing the understanding of both theoretical and applied aspects of these mathematical fields. Dr. Edson Denis Leonel is a Professor of Physics at São Paulo State University (UNESP), Rio Claro, Brazil. He has been working on scaling investigations since his Ph.D. in 2003, where he conducted the first study of scaling behavior in the chaotic sea of the Fermi-Ulam model. His research group has developed a variety of approaches and formalisms to investigate and characterize scaling properties across a wide range of systems, including one-dimensional mappings, ordinary differential equations, cellular automata, meme propagation, and time-dependent billiards. His group has investigated different types of transitions using scaling investigations, including but not limited to: (i) the transition from integrability to non-integrability; (ii) the transition from limited to unlimited diffusion; and (iii) the production and suppression of Fermi acceleration - the latter involving the analytical solution of the diffusion equation. Professor Leonel and his collaborators have published more than 190 scientific papers in respected international journals, including Physical Review Letters, Physics Reports, and many others. He is the author of Scaling Laws in Dynamical Systems (Springer & Higher Education Press, 2021), and Dynamical Phase Transitions in Chaotic Systems (Springer & Higher Education Press, 2023), Foundations of Statistical Mechanics (Generis, 2025) as well as two books in Portuguese: one on Statistical Mechanics (Blucher, 2015) and another on Nonlinear Dynamics (Blucher, 2019).
Textul de pe ultima copertă
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.
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.
Caracteristici
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