Leseprobe
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Analytic Theory of Global Bifurcation
Boris Buffoni, John Toland
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence.
This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory.
Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.
Kundenbewertungen
Complex conjugate, Algebraic equation, Elliptic integral, Prime number, Functional analysis, Dimension, Homeomorphism, Union (set theory), Derivative, Dimension (vector space), Special case, Polynomial, Symmetric bilinear form, Linear subspace, Linear function, Taylor's theorem, Theorem, Skew-symmetric matrix, Permutation group, Complex analysis, Euler equations (fluid dynamics), Equation, Taylor series, Banach space, Diagram (category theory), Linear algebra, Open mapping theorem (complex analysis), Parametrization, Existential quantification, Bifurcation theory, Metric space, Permutation, Linear map, Linearization, Multilinear form, Morse theory, Analytic function, Pseudo-differential operator, Boundary value problem, Partial differential equation, Bounded set (topological vector space), Resultant, Fredholm operator, Linearity, Froude number, Differentiable function, Operator (physics), Linear independence, Implicit function theorem, Sturm–Liouville theory, Coordinate system, Vector space, Bounded operator, Bifurcation diagram, Proportionality (mathematics), Differentiable manifold, Analytic manifold, Hilbert space, Variable (mathematics), Complex number, Ordinary differential equation, Connected space, Power series, Singularity theory, Puiseux series, Codimension, Solution set, Linear space (geometry), Eigenvalues and eigenvectors, Cartesian coordinate system