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This book provides an introduction to discrete dynamical systems -- a framework of analysis commonly used in the fields of biology, demography, ecology, economics, engineering, finance, and physics. The book characterizes the fundamental factors that govern the qualitative and quantitative trajectories of a variety of deterministic, discrete dynamical systems, providing solution methods for systems that can be solved analytically and methods of qualitative analysis for systems that do not permit or necessitate an explicit solution. The analysis focuses initially on the characterization of the factors the govern the evolution of state variables in the elementary context of one-dimensional, first-order, linear, autonomous systems. The fundamental insights about the forces that affect the evolution of these elementary systems are subsequently generalized, and the determinants of the trajectory of multi-dimensional, nonlinear, higher-order, non-autonomous dynamical systems are established.

Differentiable dynamical systems. --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Differentiable dynamical systems

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Based on the subjects from the Clay Mathematics Institute/Mathematical Sciences Research Institute Workshop titled 'Recent Progress in Dynamics' in September and October 2004, this volume contains surveys and research articles by leading experts in several areas of dynamical systems that have experienced substantial progress. One of the major surveys is on symplectic geometry, which is closely related to classical mechanics and an exciting addition to modern geometry. The survey on local rigidity of group actions gives a broad and up-to-date account of another flourishing subject. Other papers cover hyperbolic, parabolic, and symbolic dynamics as well as ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this book fascinating. The book also includes a fifty-page commented problem list that takes the reader beyond the areas covered by the surveys, to inspire and guide further research.

Differentiable dynamical systems. --- Ergodic theory. --- Geometry. --- Mathematics --- Euclid's Elements --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics

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This textbook contains the lecture series originally delivered at the "Advanced Course on Limit Cycles of Differential Equations" in the Centre de Recerca Matemàtica Barcelona in 2006. The topics covered are the center-focus problem for polynomial vector fields, and the application of abelian integrals to limit cycle bifurcations. Both topics are related to Hilbert's sixteenth problem. In particular, the book will be of interest to students and researchers working in the qualitative theory of dynamical systems.

Mathematics. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Differential Equations. --- Differentiable dynamical systems. --- 517.91 Differential equations --- Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Limit cycles. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics

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Brings together two research areas: synchronization in coupled nonlinear systems and complex networks, and study conditions under which a complex network of dynamical systems synchronizes. This book introduces ideas from systems and graph theory, and linear algebra and synergy between them that are necessary to derive synchronization conditions.

Differentiable dynamical systems. --- System analysis. --- Nonlinear theories. --- Synchronization. --- Synchronism --- Time measurements --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Network analysis --- Network science --- Network theory --- Systems analysis --- System theory --- Mathematical optimization --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Differentiable dynamical systems --- System Analysis --- Nonlinear theories --- Synchronization

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This book is drawn from across many active fields of mathematics and physics, and has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors f this book explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and the tools to answer them. In reading this book, the reader will learn how to research a topic and how to understand statistical mechanics treatments of fluid dynamics. Of particular interest should be the application of Monte Carlo methods to problems like dispersal of points on the sphere, the phase transitions of in viscid fluid flows in models that increasingly approach the conditions of actual planetary atmospheres, and the treatment of negative absolute temperatures and the effects these extremely high-energy states have on fluid flows. Special attention is given to spherical models as well. This book is intended for the upper-level undergraduate or the beginning graduate level courses of mathematics and physics. It will also be of interest to readers interested in statistical mechanics methods applied to fluid mechanics problems. Readers will gain an understanding of how to synthesize new mathematics by applying familiar tools in new ways, and develop new tools to fit particular applications.

Fluid mechanics. --- Statistical mechanics. --- Mathematical physics. --- Physical mathematics --- Physics --- Mechanics --- Mechanics, Analytic --- Quantum statistics --- Statistical physics --- Thermodynamics --- Hydromechanics --- Continuum mechanics --- Mathematics --- Mathematics. --- Differentiable dynamical systems. --- Statistical physics. --- Applications of Mathematics. --- Dynamical Systems and Ergodic Theory. --- Complex Systems. --- Fluid- and Aerodynamics. --- Statistical Physics and Dynamical Systems. --- Mathematical statistics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Math --- Science --- Statistical methods