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The Computer Algebra and Differential Equations meeting held in France in June 1992 (CADE-92) was the third of a series of biennial workshops devoted to recent developments in computer algebra systems. This book contains selected papers from that meeting. Three main topics are discussed. The first of these is the theory of D-modules. This offers an excellent way to effectively handle linear systems of partial differential equations. The second topic concerns the theoretical aspects of dynamical systems, with an introduction to Ecalle theory and perturbation analysis applied to differential equations and other nonlinear systems. The final topic is the theory of normal forms. Here recent improvements in the theory and computation of normal forms are discussed.
D-modules --- Differentiable dynamical systems --- Normal forms (Mathematics) --- Algebra --- Differential equations --- 517.91 Differential equations --- Forms, Normal (Mathematics) --- Mathematics --- Modules (Algebra) --- Data processing --- 517.91 --- #KVIV:BB --- 517.982.4 --- 517.982.4 Theory of generalized functions (distributions) --- Theory of generalized functions (distributions) --- Mathematical analysis --- Data processing&delete& --- Congresses --- Numerical solutions&delete& --- Algèbre --- Dynamique différentiable --- Equations différentielles --- Informatique --- Congrès --- Ordered algebraic structures --- Numerical solutions --- D-modules - Congresses --- Differentiable dynamical systems - Congresses --- Normal forms (Mathematics) - Congresses --- Algebra - Data processing - Congresses --- Differential equations - Congresses
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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
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An extension of different lectures given by the authors, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems provides the reader with a comprehensive overview of these topics. Starting with the simplest bifurcation problems arising for ordinary differential equations in one- and two-dimensions, this book describes several tools from the theory of infinite dimensional dynamical systems, allowing the reader to treat more complicated bifurcation problems, such as bifurcations arising in partial differential equations. Attention is restricted to the study of local bifurcations with a focus upon the center manifold reduction and the normal form theory; two methods that have been widely used during the last decades. Through use of step-by-step examples and exercises, a number of possible applications are illustrated, and allow the less familiar reader to use this reduction method by checking some clear assumptions. Written by recognised experts in the field of center manifold and normal form theory this book provides a much-needed graduate level text on bifurcation theory, center manifolds and normal form theory. It will appeal to graduate students and researchers working in dynamical system theory.
Bifurcation theory. --- Normal forms (Mathematics) --- Forms, Normal (Mathematics) --- Mathematics. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Statistical physics. --- Dynamical Systems and Ergodic Theory. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Applications of Mathematics. --- Nonlinear Dynamics. --- Mathematics --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Differentiable dynamical systems. --- Differential Equations. --- Differential equations, partial. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Math --- Science --- Partial differential equations --- 517.91 Differential equations --- Topological manifolds. --- Physics --- Mathematical statistics --- Engineering --- Engineering analysis --- Mathematical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Statistical methods
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