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Il est inhabituel qu’une idée mathématique se diffuse dans la société. C’est pourtant le cas avec la théorie du chaos, popularisée grâce à l’effet papillon, selon lequel le battement d’ailes d’un papillon au Brésil pourrait provoquer une tornade au Texas. Depuis Galilée et Newton, la physique et les mathématiques sont traversées par cette question du déterminisme. Si la science semble d’abord en état de tout prédire, elle doit reconnaître, vers la fin du XIXe siècle, l’infinie complexité du monde et l’impossibilité pratique de prévoir le futur en fonction du présent. Étienne Ghys restitue ici le cheminement de grands scientifiques et nous fait toucher du doigt, toujours de manière très accessible, la portée conceptuelle de l’abstraction mathématique. (4e de couverture)

Chaotic behavior in systems

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ePDF and ePUB available Open Access under CC-BY-NC-ND licence. Building on research in public health, social epidemiology and the social determinants of health, this book presents complexity theory as an alternative basis for an outcome-oriented public management praxis.

Chaotic behavior in systems. --- Municipal services --- Management. --- Municipal services within corporate limits --- Public services --- Municipal government --- Public utilities --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory

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This textbook introduces the language and the techniques of the theory of dynamical systems of finite dimension for an audience of physicists, engineers, and mathematicians at the beginning of graduation. Author addresses geometric, measure, and computational aspects of the theory of dynamical systems. Some freedom is used in the more formal aspects, using only proofs when there is an algorithmic advantage or because a result is simple and powerful. The first part is an introductory course on dynamical systems theory. It can be taught at the master's level during one semester, not requiring specialized mathematical training. In the second part, the author describes some applications of the theory of dynamical systems. Topics often appear in modern dynamical systems and complexity theories, such as singular perturbation theory, delayed equations, cellular automata, fractal sets, maps of the complex plane, and stochastic iterations of function systems are briefly explored for advanced students. The author also explores applications in mechanics, electromagnetism, celestial mechanics, nonlinear control theory, and macroeconomy. A set of problems consolidating the knowledge of the different subjects, including more elaborated exercises, are provided for all chapters.

System theory. --- Dynamical systems. --- Mathematical physics. --- Complex Systems. --- Dynamical Systems. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Mathematics --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Systems, Theory of --- Systems science --- Science --- Philosophy --- Chaotic behavior in systems. --- Dynamics. --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory

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This book discusses some scaling properties and characterizes two-phase transitions for chaotic dynamics in nonlinear systems described by mappings. The chaotic dynamics is determined by the unpredictability of the time evolution of two very close initial conditions in the phase space. It yields in an exponential divergence from each other as time passes. The chaotic diffusion is investigated, leading to a scaling invariance, a characteristic of a continuous phase transition. Two different types of transitions are considered in the book. One of them considers a transition from integrability to non-integrability observed in a two-dimensional, nonlinear, and area-preserving mapping, hence a conservative dynamics, in the variables action and angle. The other transition considers too the dynamics given by the use of nonlinear mappings and describes a suppression of the unlimited chaotic diffusion for a dissipative standard mapping and an equivalent transition in the suppression of Fermi acceleration in time-dependent billiards. This book allows the readers to understand some of the applicability of scaling theory to phase transitions and other critical dynamics commonly observed in nonlinear systems. That includes a transition from integrability to non-integrability and a transition from limited to unlimited diffusion, and that may also be applied to diffusion in energy, hence in Fermi acceleration. The latter is a hot topic investigated in billiard dynamics that led to many important publications in the last few years. It is a good reference book for senior- or graduate-level students or researchers in dynamical systems and control engineering, mathematics, physics, mechanical and electrical engineering.

Chaotic behavior in systems. --- Phase transformations (Statistical physics) --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory --- Dynamical systems. --- Mathematical analysis. --- Condensed matter. --- Dynamical Systems. --- Scale Invariance. --- Phase Transitions and Multiphase Systems. --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids --- 517.1 Mathematical analysis --- Mathematical analysis --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics