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This book introduces the reader to the study of Hamiltonian systems, focusing on the stability of autonomous and periodic systems and expanding to topics that are usually not covered by the canonical literature in the field. It emerged from lectures and seminars given at the Federal University of Pernambuco, Brazil, known as one of the leading research centers in the theory of Hamiltonian dynamics.This book starts with a brief review of some results of linear algebra and advanced calculus, followed by the basic theory of Hamiltonian systems. The study of normal forms of Hamiltonian systems is covered by Ch.3, while Chapters 4 and 5 treat the normalization of Hamiltonian matrices. Stability in non-linear and linear systems are topics in Chapters 6 and 7. This work finishes with a study of parametric resonance in Ch. 8. All the background needed is presented, from the Hamiltonian formulation of the laws of motion to the application of the Krein-Gelfand-Lidskii theory of stronglystable systems.With a clear, self-contained exposition, this work is a valuable help to advanced undergraduate and graduate students, and to mathematicians and physicists doing research on this topic.
This book is a revised and updated version, including a substantial portion of new material, of our text Perturbation Methods in Applied Mathematics (Springer Verlag, 1981).
Materials science is an area of growing research as composite materials become widely used in such areas as civil engineering, electrotechnics, and the aerospace industry.
This book examines the main theorems in bifurcation theory. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces and shows how to apply the theory.
The revised edition of this systematic investigation of convection in systems comprised of liquid layers with deformatable interfaces offers new material on flows in ultra thin films. The text also reflects progress into dynamics of complex fluids.
This text gives a common treatment to three areas of application of global analysis to mathematical physics: the geometry of manifolds applied to classical mechanics; stochastic differential geometry used in quantum and statistical mechanics; and infinites-dimensional differential geometry.
This book develops methods for describing random dynamical systems, and it illustrats how the methods can be used in a variety of applications. Appeals to researchers and graduate students who require tools to investigate stochastic systems.
This book presents rigorous treatment of boundary value problems in nonlinear theory of shallow shells. The consideration of the problems is carried out using methods of nonlinear functional analysis.
This monograph provides in-depth analyses of vortex dominated flows via matched and multiscale asymptotics, and demonstrates how insight gained through these analyses can be exploited in the construction of robust, efficient, and accurate numerical techniques.
An up-to-date and unified treatment of bifurcation theory for variational inequalities in reflexive spaces and the use of the theory in a variety of applications, such as: obstacle problems from elasticity theory, unilateral problems;
This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level.
In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equation, as well as for the existence and persistence of fibrations of these invariant manifolds.
Enlarged, updated, and extensively revised, this second edition illuminates specific problems of nonlinear elasticity, emphasizing the role of nonlinear material response. Subsequent chapters cover tensors, three-dimensional continuum mechanics, three-dimensional elasticity , general theories of rods and shells, and dynamical problems.
By including classical results as well as recent developments in the field of hydrodynamic stability and transition, the book can be used as a textbook for an introductory, graduate-level course in stability theory or for a special-topics fluids course.
Presents a mathematical analysis of hysteretic phenomena, where two complementary viewpoints are taken. This work studies scalar rate independent hysteresis in a general setting; and discusses the connections between the occurrence of hysteresis and physical mechanisms like energy dissipation and phase transitions.
This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations: from modeling issues to well-posedness results and the coupled models of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD).
This work aims to introduce students to active areas of research in mathematical physics in a rather direct way, thus minimizing the use of abstract mathematics. The book's main features are geometric methods in spectral analysis, semi-classical analysis of resonance and other topics.
Providing a solid basis in dynamical systems theory, and explicit procedures for application, this is a must-have for students of advanced mathematics. 'I know of no other book that so clearly explains the basic phenomena of bifurcation theory.' - Math Reviews
This is a monograph on the emerging branch of mathematical biophysics combining asymptotic analysis with numerical and stochastic methods to analyze partial differential equations arising in biological and physical sciences.In more detail, the book presents the analytic methods and tools for approximating solutions of mixed boundary value problems, with particular emphasis on the narrow escape problem. Informed throughout by real-world applications, the book includes topics such as the Fokker-Planck equation, boundary layer analysis, WKB approximation, applications of spectral theory, as well as recent results in narrow escape theory. Numerical and stochastic aspects, including mean first passage time and extreme statistics, are discussed in detail and relevant applications are presented in parallel with the theory.Including background on the classical asymptotic theory of differential equations, this book is written for scientists of various backgrounds interested in deriving solutions to real-world problems from first principles.
This book provides an authoritative introduction to the rapidly growing field of chemical reaction network theory.
This is the first comprehensive book on information geometry, written by the founder of the field. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry.
A study of topology and geometry, beginning with a comprehensible account of the extraordinary and rather mysterious impact of mathematical physics, and especially gauge theory, on the study of the geometry and topology of manifolds.
This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes the fundamental ideas of the past two decades of research, carefully balancing theory and application.
The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces.
This book introduces the area of inverse problems. It examines basic notions and difficulties encountered with ill-posed problems and presents two special nonlinear inverse problems in detail: the inverse spectral problem and the inverse scattering problem.
This treatment of pull-back attractors for non-autonomous Dynamical systems emphasizes the infinite-dimensional variety but also analyzes those that are finite. As a graduate primer, it covers everything from basic definitions to cutting-edge results.
This book covers adaptive mesh generation and moving mesh methods for solving time-dependent PDEs. It gives a general description of the components of moving mesh methods as well as examples of their application for a number of nontrivial physical problems.
This mathematically rigorous survey of the special theory of relativity also details the physical significance of that mathematics. In addition to kinematics, particle dynamics and electromagnetic fields it treats subjects usually bypassed at elementary level.
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