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The idea of modeling the behavior of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems.   "Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters.  For each topic the theoretical framework is carefully explained using examples and applications.   The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises.  This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences.   Herb Kunze is a Professor in the Department of Mathematics and Statistics, University of Guelph.  Davide La Torre is an Associate Professor in the Department of Economics, Business and Statistics, University of Milan.   Franklin Mendivil is a Professor in the Department of Mathematics and Statistics, Acadia University. Edward R. Vrscay is a Professor in the Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo.  A major focus of their research is fractals and their applications.

Mathematical analysis. --- Numerical analysis. --- Numerical analysis --- Mathematical analysis --- Fractals --- Civil & Environmental Engineering --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Operations Research --- 517.1 Mathematical analysis --- Mathematics. --- Approximation theory. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Mathematical models. --- Physics. --- Dynamical Systems and Ergodic Theory. --- Mathematical Methods in Physics. --- Approximations and Expansions. --- Mathematical Modeling and Industrial Mathematics. --- Ordinary Differential Equations.

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Over the last two decades, the dimension theory of dynamical systems has progressively developed into an independent and extremely active field of research. The main aim of this volume is to offer a unified, self-contained introduction to the interplay of these three main areas of research: ergodic theory, hyperbolic dynamics, and dimension theory. It starts with the basic notions of the first two topics and ends with a sufficiently high-level introduction to the third. Furthermore, it includes an introduction to the thermodynamic formalism, which is an important tool in dimension theory. The volume is primarily intended for graduate students interested in dynamical systems, as well as researchers in other areas who wish to learn about ergodic theory, thermodynamic formalism, or dimension theory of hyperbolic dynamics at an intermediate level in a sufficiently detailed manner. In particular, it can be used as a basis for graduate courses on any of these three subjects. The text can also be used for self-study: it is self-contained, and with the exception of some well-known basic facts from other areas, all statements include detailed proofs.

Differentiable dynamical systems. --- Differential equations, Hyperbolic. --- Dimension theory (Topology). --- Ergodic theory. --- Differentiable dynamical systems --- Ergodic theory --- Differential equations, Hyperbolic --- Dimension theory (Topology) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Hyperbolic differential equations --- Ergodic transformations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Mathematics. --- Dynamics. --- Dynamical Systems and Ergodic Theory. --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Topology --- Differential equations, Partial --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics

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The guiding light of this monograph is a question easy to understand but difficult to answer: {What is the shape of the universe? In other words, how do we measure the shortest distance between two points of the physical space? Should we follow a straight line, as on a flat table, fly along a circle, as between Paris and New York, or take some other path, and if so, what would that path look like? If you accept that the model proposed here, which assumes a gravitational law extended to a universe of constant curvature, is a good approximation of the physical reality (and I will later outline a few arguments in this direction), then we can answer the above question for distances comparable to those of our solar system. More precisely, this monograph provides a mathematical proof that, for distances of the order of 10 AU, space is Euclidean. This result is, of course, not surprising for such small cosmic scales. Physicists take the flatness of space for granted in regions of that size. But it is good to finally have a mathematical confirmation in this sense. Our main goals, however, are mathematical. We will shed some light on the dynamics of N point masses that move in spaces of non-zero constant curvature according to an attraction law that naturally extends classical Newtonian gravitation beyond the flat (Euclidean) space. This extension is given by the cotangent potential, proposed by the German mathematician Ernest Schering in 1870. He was the first to obtain this analytic expression of a law suggested decades earlier for a 2-body problem in hyperbolic space by Janos Bolyai and, independently, by Nikolai Lobachevsky. As Newton's idea of gravitation was to introduce a force inversely proportional to the area of a sphere the same radius as the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition using the hyperbolic distance in hyperbolic space. The recent generalization we gave to the cotangent potential to any number N of bodies, led to the discovery of some interesting properties. This new research reveals certain connections among at least five branches of mathematics: classical dynamics, non-Euclidean geometry, geometric topology, Lie groups, and the theory of polytopes.

Celestial mechanics. --- Differentiable dynamical systems. --- Differential equations. --- Many-body problem -- Numerical solutions. --- Many-body problem --- Differential equations --- Celestial mechanics --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Numerical solutions --- Many-body problem. --- Equilibrium. --- Balance --- Balance (Physics) --- Balancing (Physics) --- n-body problem --- Problem of many bodies --- Problem of n-bodies --- Mathematics. --- Dynamics. --- Ergodic theory. --- Dynamical Systems and Ergodic Theory. --- Ordinary Differential Equations. --- Mathematics, general. --- Stability --- Statics --- Mechanics, Analytic --- Differential Equations. --- Math --- Science --- 517.91 Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Force and energy --- Mechanics --- Physics

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In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.   The second edition is substantially and formally revised and new material is added. Among this is bifurcation with a two-dimensional kernel with applications, the buckling of the Euler rod, the appearance of Taylor vortices, the singular limit process of the Cahn-Hilliard model, and an application of this method to more complicated nonconvex variational problems.

Bifurcation theory. --- Mathematical analysis. --- Bifurcation theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- 517.1 Mathematical analysis --- Mathematical analysis --- Mathematics. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Mechanics. --- Mechanics, Applied. --- Partial Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Applications of Mathematics. --- Theoretical and Applied Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Engineering --- Engineering analysis --- Partial differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Math --- Science --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Differential equations, partial. --- Differentiable dynamical systems. --- Mechanics, applied. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics

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Dynamical system theory has developed rapidly over the past fifty years. It is a subject upon which the theory of limit cycles has a significant impact for both theoretical advances and practical solutions to problems. Hopf bifurcation from a center or a focus  is integral to the theory of bifurcation of limit cycles, for which normal form theory is a central tool. Although Hopf bifurcation has been studied for more than half a century, and normal form theory for over 100 years, efficient computation in this area is still a challenge with implications for Hilbert’s 16th problem. This book introduces the most recent developments in this field and provides major advances in fundamental theory of limit cycles. Split into two parts, the first focuses on  the study of limit cycles bifurcating from Hopf singularity using normal form theory with later application to Hilbert’s 16th problem, while the second considers near Hamiltonian systems using Melnikov function as the main mathematical tool. Classic topics with new results are presented in a clear and concise manner and are accompanied by the liberal use of illustrations throughout. Containing a wealth of examples and structured algorithms that are treated in detail, a good balance between theoretical and applied topics is demonstrated. By including complete Maple programs within the text, this book also enables the reader to reconstruct the majority of formulas provided, facilitating the use of concrete models for study. Through the adoption of an elementary and practical approach, this book will be of use to graduate mathematics students wishing to study the theory of limit cycles as well as scientists, across a number of disciplines, with an interest in the applications of periodic behavior.

Limit cycles. --- Nonlinear systems. --- Limit cycles --- Bifurcation theory --- Normal forms (Mathematics) --- Nonlinear systems --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Calculus --- Systems, Nonlinear --- Cycles, Limit --- Differential equations --- Limit cycles of differential equations --- Mathematics. --- Approximation theory. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Computer software. --- Statistical physics. --- Dynamical Systems and Ergodic Theory. --- Approximations and Expansions. --- Ordinary Differential Equations. --- Mathematical Software. --- Nonlinear Dynamics. --- Physics --- Mathematical statistics --- Software, Computer --- Computer systems --- 517.91 Differential equations --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Statistical methods --- System theory --- Differentiable dynamical systems --- Differentiable dynamical systems. --- Differential Equations. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics