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Differential equations, Nonlinear --- Differentiable dynamical systems

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The study of non-linear dynamical systems nowadays is an intricate mixture of analysis, geometry, algebra and measure theory and this book takes all aspects into account. Presenting the contents of its authors' graduate courses in non-linear dynamical systems, this volume aims at researchers who wish to be acquainted with the more theoretical and fundamental subjects in non-linear dynamics and is designed to link the popular literature with research papers and monographs. All of the subjects covered in this book are extensively dealt with and presented in a pedagogic

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

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Differential geometry. Global analysis --- Differentiable dynamical systems --- Point mappings (Mathematics) --- Dynamique différentiable --- 515.16 --- Equations, Recurrent --- Mappings, Point (Mathematics) --- Recurrence relations in functional differential equations --- Recurrent equations --- Functional differential equations --- Mappings (Mathematics) --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Topology of manifolds --- Differentiable dynamical systems. --- Point mappings (Mathematics). --- 515.16 Topology of manifolds --- Dynamique différentiable

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Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. Most results are obtained in the infinite-dimensional setting of Banach spaces. Furthermore, the linear variational equations are always assumed to possess a nonuniform exponential behavior, given either by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. The presentation is self-contained and has unified character. The volume contributes towards a rigorous mathematical foundation of the theory in the infinite-dimension setting, and may lead to further developments in the field. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.

Lyapunov stability --- Differential equations --- Calculus --- Applied Mathematics --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Lyapunov stability. --- Differential equations. --- 517.91 Differential equations --- Liapunov stability --- Ljapunov stability --- Mathematics. --- Dynamics. --- Ergodic theory. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Control theory --- Stability --- Differential Equations. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this book, two different approaches are developed which are based on special definitions of local attractivity and repulsivity. It is shown that these notions lead to nonautonomous Morse decompositions, which are useful to describe the global asymptotic behavior of systems on compact phase spaces. Furthermore, methods from the qualitative theory for linear and nonlinear systems are derived, and nonautonomous counterparts of the classical one-dimensional autonomous bifurcation patterns are developed.

Differentiable dynamical systems. --- Differential equations, Linear. --- Bifurcation theory. --- Dynamique différentiable --- Equations différentielles linéaires --- Théorie de la bifurcation --- Electronic books. -- local. --- Differentiable dynamical systems --- Differential equations, Linear --- Bifurcation theory --- Mathematics --- Mathematical Theory --- Calculus --- Physical Sciences & Mathematics --- Linear differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Mathematics. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- 517.91 Differential equations --- Differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Differential equations, Nonlinear --- Stability --- Linear systems --- Global analysis (Mathematics) --- Topological dynamics --- Numerical solutions --- Differential Equations.