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Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems / by Mariana Haragus, Gérard Iooss
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Universitext | SpringerLink BücherPublisher: London : Springer London, 2011Description: Online-Ressource (XIV, 325p. 34 illus, digital)ISBN:- 9780857291127
- 515.39
- 515.48
- QA313
Contents:
Summary: Elementary Bifurcations -- Center Manifolds -- Normal Forms -- Reversible Bifurcations -- Applications -- Appendix.Summary: 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.PPN: PPN: 165061439XPackage identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface""; ""Contents""; ""Elementary Bifurcations""; ""Bifurcations in Dimension 1""; ""Saddle-Node Bifurcation""; ""Pitchfork Bifurcation""; ""Bifurcations in Dimension 2""; ""Hopf Bifurcation""; ""Example: Homogeneous Brusselator""; ""Hopf Bifurcation with SO(2) Symmetry""; ""Steady Bifurcation with O(2) Symmetry""; ""Center Manifolds""; ""Notations""; ""Local Center Manifolds""; ""Hypotheses""; ""Main Result""; ""Checking Hypothesis 2.7""; ""Examples""; ""Particular Cases and Extensions""; ""Parameter-Dependent Center Manifolds""; ""Nonautonomous Center Manifolds"" ; ""Symmetries and Reversibility""""Empty Unstable Spectrum""; ""Further Examples and Exercises""; ""A Fourth Order ODE""; ""Burgers Model""; ""Swift--Hohenberg Equation""; ""Brusselator Model ""; ""Elliptic PDE in a Strip""; ""Normal Forms""; ""Main Theorem""; ""Proof of Theorem 1.2""; ""Examples in Dimension 2: i, 02""; ""Examples in Dimension 3: 0(i), 03""; ""Examples in Dimension 4: (i1)(i2), (i)2, 02(i), 0202""; ""Parameter-Dependent Normal Forms""; ""Main Result""; ""Linear Normal Forms""; ""Derivation of the Parameter-Dependent Normal Form""; ""Example: 02 Normal Form with Parameters"" ; ""Symmetries and Reversibility""""Equivariant Vector Fields""; ""Reversible Vector Fields""; ""Example: van der Pol System""; ""Normal Forms for Reduced Systems on Center Manifolds""; ""Computation of Center Manifolds and Normal Forms""; ""Example 1: Hopf Bifurcation""; ""Example 2: Hopf Bifurcations with Symmetries""; ""Example 3: Takens--Bogdanov Bifurcation""; ""Example 4: (i1)(i2) bifurcation""; ""Further Normal Forms""; ""Time-Periodic Normal Forms""; ""Example: Periodically Forced Hopf Bifurcation""; ""Normal Forms for Analytic Vector Fields""; ""Reversible Bifurcations"" ; ""Dimension 2""""Reversible Takens--Bogdanov Bifurcation 02+""; ""Reversible Takens--Bogdanov Bifurcation 02-""; ""Dimension 3""; ""Reversible 03+ Bifurcation""; ""Reversible 03- Bifurcation (Elements)""; ""Reversible 002 Bifurcation (Elements)""; ""Reversible 0(i) Bifurcation (Elements)""; ""Dimension 4""; ""Reversible 02+(i) Bifurcation""; ""Reversible 02-(i ) Bifurcation (Elements)""; ""Reversible (i )2 Bifurcation (1-1 resonance)""; ""Reversible (i 1)(i 2) Bifurcation (Elements)""; ""Reversible 04+ Bifurcation (Elements)""; ""Reversible 0202 Bifurcation with SO(2) Symmetry"" ; ""Applications""""Hydrodynamic Instabilities""; ""Hydrodynamic Problem""; ""Couette--Taylor Problem""; ""Bénard--Rayleigh Convection Problem""; ""Existence of Traveling Waves""; ""Gravity-Capillary Water-Waves""; ""Almost-Planar Waves in Reaction-Diffusion Systems""; ""Waves in Lattices""; ""Appendix""; ""Elements of Functional Analysis""; ""Bounded and Closed Operators""; ""Resolvent and Spectrum""; ""Compact Operators and Operators with Compact Resolvent""; ""Adjoint Operator""; ""Fredholm Operators""; ""Basic Sobolev Spaces""; ""Center Manifolds"" ; ""Proof of Theorem 2.9 (Center Manifolds)""
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