Analytic Theory of Global Bifurcation
Boris Buffoni, John Toland
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
Power series, Symmetric bilinear form, Fredholm operator, Multilinear form, Diagram (category theory), Differentiable manifold, Linear space (geometry), Ordinary differential equation, Open mapping theorem (complex analysis), Boundary value problem, Morse theory, Complex analysis, Taylor's theorem, Linear subspace, Froude number, Linear algebra, Bifurcation theory, Analytic function, Connected space, Linear function, Union (set theory), Proportionality (mathematics), Solution set, Polynomial, Special case, Existential quantification, Derivative, Coordinate system, Pseudo-differential operator, Operator (physics), Differentiable function, Linear map, Partial differential equation, Singularity theory, Theorem, Implicit function theorem, Complex number, Homeomorphism, Metric space, Sturm–Liouville theory, Banach space, Elliptic integral, Linearity, Bounded operator, Functional analysis, Linear independence, Prime number, Hilbert space, Skew-symmetric matrix, Parametrization, Codimension, Permutation group, Equation, Resultant, Cartesian coordinate system, Eigenvalues and eigenvectors, Vector space, Algebraic equation, Bounded set (topological vector space), Permutation, Euler equations (fluid dynamics), Linearization, Taylor series, Dimension (vector space), Analytic manifold, Dimension, Complex conjugate, Bifurcation diagram, Puiseux series, Variable (mathematics)