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.

A comprehensive and unified introduction to describing and understanding complex interacting systems.

The author describes and teaches the art of discovering scaling laws, starting from dimensional analysis and physical similarity, which are here given a modern treatment. He demonstrates the concepts of intermediate asymptotics and the renormalisation group as natural consequences of self-similarity and shows how and when these notions and tools can be used to tackle the task at hand, and when they cannot. Based on courses taught to undergraduate and graduate students, the book can also be used for self-study by biologists, chemists, astronomers, engineers and geoscientists.

During a century, from the Van der Waals mean field description (1874) of gases to the introduction of renormalization group (RG techniques 1970), thermodynamics and statistical physics were just unable to account for the incredible universality which was observed in numerous critical phenomena. The great success of RG techniques is not only to solve perfectly this challenge of critical behaviour in thermal transitions but to introduce extremely useful tools in a wide field of daily situations where a system exhibits scale invariance. The introduction of scaling, scale invariance and universality concepts has been a significant turn in modern physics and more generally in natural sciences. Since then, a new "physics of scaling laws and critical exponents", rooted in scaling approaches, allows quantitative descriptions of numerous phenomena, ranging from phase transitions to earthquakes, polymer conformations, heartbeat rhythm, diffusion, interface growth and roughening, DNA sequence, dynamical systems, chaos and turbulence. The chapters are jointly written by an experimentalist and a theorist. This book aims at a pedagogical overview, offering to the students and researchers a thorough conceptual background and a simple account of a wide range of applications. It presents a complete tour of both the formal advances and experimental results associated with the notion of scaling, in physics, chemistry and biology.

The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and example-driven. The first part on ODEs fits even a lower undergraduate level, while the most advanced multiphysics fluid mechanics examples target the graduate level. The scientific literature is full of scaled models, but in most of the cases, the scales are just stated without thorough mathematical reasoning. This book explains how the scales are found mathematically. This book will be a valuable read for anyone doing numerical simulations based on ordinary or partial differential equations.

Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.

This Volume constitutes the Proceedings of the IUTAM Symposium on 'Scaling Laws in Ice Mechanics and Ice Dynamics', held in Fairbanks, Alaska from 13th to 16th of June 2000. Ice mechanics deals with essentially intact ice: in this discipline, descriptions of the motion and deformation of Arctic/ Antarctic and river/lake ice call for the development of physically based constitutive and fracture models over an enormous range in scale: 0.01 m - 10 km. Ice dynamics, on the other hand, deals with the movement of broken ice: descriptions of an aggregate of ice floes call for accurate modeling of momentum transfer through the sea/ice system, again over an enormous range in scale: 1 km (floe scale) - 500 km (basin scale). For ice mechanics, the emphasis on lab-scale (0.01 - 0.5 m) research con trasts with applications at the scale of order 1 km (ice-structure interaction, icebreaking); many important upscaling questions remain to be explored.