Author : Peter W. Bates
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 11,17 MB
Release : 1998
Category : Mathematics
ISBN : 0821808680
Extends the theory for normally hyperbolic invariant manifolds to infinite dimensional dynamical systems in a Banach space, thereby providing tools for the study of PDE's and other infinite dimensional equations of evolution. In the process, the authors establish the existence of center-unstable and center-stable manifolds in a neighborhood of the unperturbed compact manifold. No index. Annotation copyrighted by Book News, Inc., Portland, OR
Author : Peter W. Bates
Publisher :
Page : 145 pages
File Size : 41,88 MB
Release : 2014-09-11
Category : MATHEMATICS
ISBN : 9781470402341
Since the early 1970s, mathematicians have tried to extend the work of N. Fenichel and M. Hirsch, C. Pugh and M. Shub, to give conditions under which invariant manifolds for semi-flows persist under perturbation of the semiflow. This work provides natural conditions and establishes the desired theorem. The technique is geometric in nature, and in addition to rigorous proofs, an informal outline of the approach is given with useful illustrations.
Author : Kenji Nakanishi
Publisher : European Mathematical Society
Page : 264 pages
File Size : 10,55 MB
Release : 2011
Category : Hamiltonian systems
ISBN : 9783037190951
The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein-Gordon and Schrodinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. The authors' entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors. The proofs rely on an interplay between the variational structure of the ground states and the nonlinear hyperbolic dynamics near these states. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein-Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.
Author : Jaap Eldering
Publisher : Springer Science & Business Media
Page : 197 pages
File Size : 46,56 MB
Release : 2013-08-17
Category : Mathematics
ISBN : 9462390037
This monograph treats normally hyperbolic invariant manifolds, with a focus on noncompactness. These objects generalize hyperbolic fixed points and are ubiquitous in dynamical systems. First, normally hyperbolic invariant manifolds and their relation to hyperbolic fixed points and center manifolds, as well as, overviews of history and methods of proofs are presented. Furthermore, issues (such as uniformity and bounded geometry) arising due to noncompactness are discussed in great detail with examples. The main new result shown is a proof of persistence for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. This extends well-known results by Fenichel and Hirsch, Pugh and Shub, and is complementary to noncompactness results in Banach spaces by Bates, Lu and Zeng. Along the way, some new results in bounded geometry are obtained and a framework is developed to analyze ODEs in a differential geometric context. Finally, the main result is extended to time and parameter dependent systems and overflowing invariant manifolds.
Author : Stephen Shing-Toung Yau
Publisher : American Mathematical Soc.
Page : 596 pages
File Size : 48,99 MB
Release : 2001
Category : Mathematics
ISBN : 0821826522
The International Congress of Mathematicians was an historical event that was held at the Morningside Center of Mathematics of the Chinese Academy of Sciences (Beijing). It was the first occasion where Chinese mathematicians from all over the world gathered to present their research. The Morningside Mathematics lectures were given by R. Borcherds, J. Coates, R. Graham, and D. Stroock. Other distinguished speakers included J.-P. Bourguignon, J. Jöst, M. Taylor, and S. L. Lee. Topics covered in the volume include algebra and representation theory, algebraic geometry, number theory and automorphic forms, Riemannian geometry and geometric analysis, mathematical physics, topology, complex analysis and complex geometry, computational mathematics, and combinatorics. Titles in this series are copublished with International Press, Cambridge, MA.
Author : Mickaël D. Chekroun
Publisher : Springer
Page : 136 pages
File Size : 28,11 MB
Release : 2014-12-20
Category : Mathematics
ISBN : 331912496X
This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
Author : Donald G. Babbitt
Publisher : American Mathematical Soc.
Page : 762 pages
File Size : 43,22 MB
Release : 2000-05-05
Category : Mathematics
ISBN : 9780821896709
This second volume of Featured Reviews makes available special detailed reviews of some of the most important mathematical articles and books published from 1997 through 1999. Also included are excellent reviews of several classic books and articles published prior to 1970. Among those reviews, for example, are the following: Homological Algebra by Henri Cartan and Samuel Eilenberg, reviewed by G. Hochschild; Faisceaux algebriques coherents by Jean-Pierre Serre, reviewed by C. Chevalley; and On the Theory of General Partial Differential Operators by Lars Hormander, reviewed by J. L. Lions. In particular, those seeking information on current developments outside their own area of expertise will find the volume very useful. By identifying some of the best publications, papers, and books that have had or are expected to have a significant impact in applied and pure mathematics, this volume will serve as a comprehensive guide to important new research across all fields covered by MR.
Author : Huaizhong Zhao
Publisher : World Scientific
Page : 458 pages
File Size : 29,81 MB
Release : 2012
Category : Mathematics
ISBN : 9814360910
The volume is dedicated to Professor David Elworthy to celebrate his fundamental contribution and exceptional influence on stochastic analysis and related fields. Stochastic analysis has been profoundly developed as a vital fundamental research area in mathematics in recent decades. It has been discovered to have intrinsic connections with many other areas of mathematics such as partial differential equations, functional analysis, topology, differential geometry, dynamical systems, etc. Mathematicians developed many mathematical tools in stochastic analysis to understand and model random phenomena in physics, biology, finance, fluid, environment science, etc. This volume contains 12 comprehensive review/new articles written by world leading researchers (by invitation) and their collaborators. It covers stochastic analysis on manifolds, rough paths, Dirichlet forms, stochastic partial differential equations, stochastic dynamical systems, infinite dimensional analysis, stochastic flows, quantum stochastic analysis and stochastic Hamilton Jacobi theory. Articles contain cutting edge research methodology, results and ideas in relevant fields. They are of interest to research mathematicians and postgraduate students in stochastic analysis, probability, partial differential equations, dynamical systems, mathematical physics, as well as to physicists, financial mathematicians, engineers, etc.
Author : Robert A. Meyers
Publisher : Springer Science & Business Media
Page : 1885 pages
File Size : 15,11 MB
Release : 2011-10-05
Category : Mathematics
ISBN : 1461418054
Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
Author : Àlex Haro
Publisher : Springer
Page : 280 pages
File Size : 17,38 MB
Release : 2016-04-18
Category : Mathematics
ISBN : 3319296620
This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
