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Many dynamical systems are described by differential equations that can be separated into one part, containing linear terms with constant coefficients, and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic oscillators in the absence of perturbations. This book is devoted primarily to applied asymptotic methods in nonlinear oscillations which are associated with the names of N. M. Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. The advantages of the present methods are their simplicity, especially for computing higher approximations, and their applicability to a large class of quasi-linear problems. In this book, we confine ourselves basi cally to the scheme proposed by Krylov, Bogoliubov as stated in the monographs [6,211. We use these methods, and also develop and improve them for solving new problems and new classes of nonlinear differential equations. Although these methods have many applications in Mechanics, Physics and Technique, we will illustrate them only with examples which clearly show their strength and which are themselves of great interest. A certain amount of more advanced material has also been included, making the book suitable for a senior elective or a beginning graduate course on nonlinear oscillations.
Many dynamical systems are described by differential equations that can be separated into one part, containing linear terms with constant coefficients, and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic oscillators in the absence of perturbations. This book is devoted primarily to applied asymptotic methods in nonlinear oscillations which are associated with the names of N. M. Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. The advantages of the present methods are their simplicity, especially for computing higher approximations, and their applicability to a large class of quasi-linear problems. In this book, we confine ourselves basi cally to the scheme proposed by Krylov, Bogoliubov as stated in the monographs [6,211. We use these methods, and also develop and improve them for solving new problems and new classes of nonlinear differential equations. Although these methods have many applications in Mechanics, Physics and Technique, we will illustrate them only with examples which clearly show their strength and which are themselves of great interest. A certain amount of more advanced material has also been included, making the book suitable for a senior elective or a beginning graduate course on nonlinear oscillations.
This text maps out the modern theory of non-linear oscillations. The material is presented in a non-traditional manner and emphasises the new results of the theory - obtained partially by the author, who is one of the leading experts in the area. Among the topics are: synchronization and chaotization of self-oscillatory systems and the influence of weak random vibration on modification of characteristics and behaviour of the non-linear systems.
In various fields of science, notably in physics and biology, one is con fronted with periodic phenomena having a remarkable temporal structure: it is as if certain systems are periodically reset in an initial state. A paper of Van der Pol in the Philosophical Magazine of 1926 started up the investigation of this highly nonlinear type of oscillation for which Van der Pol coined the name "relaxation oscillation". The study of relaxation oscillations requires a mathematical analysis which differs strongly from the well-known theory of almost linear oscillations. In this monograph the method of matched asymptotic expansions is employed to approximate the periodic orbit of a relaxation oscillator. As an introduction, in chapter 2 the asymptotic analysis of Van der Pol's equation is carried out in all detail. The problem exhibits all features characteristic for a relaxation oscillation. From this case study one may learn how to handle other or more generally formulated relaxation oscillations. In the survey special attention is given to biological and chemical relaxation oscillators. In chapter 2 a general definition of a relaxation oscillation is formulated.
This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae. The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entirenonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary. The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and appliedmathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects. The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is knownas the Courant point of view!! --Percy Deift, Courant Institute, New York Peter D. Miller is an associate professor of mathematics at the University of Michigan at Ann Arbor. He earned a Ph.D. in Applied Mathematics from the University of Arizona and has held positions at the Australian NationalUniversity (Canberra) and Monash University (Melbourne). His current research interests lie in singular limits for integrable systems.
This book is the frrst to focus on mechanical aspects of fibrous and layered composite material with curved structure. By mechanical aspects we mean statics, vibration, stability loss, elastic and fracture problems. By curved structures we mean that the reinforcing layers or fibres are not straight: they have some initial curvature, bending or distortion. This curvature may occur as a result of design, or as a consequence of some technological process. During the last two decades, we and our students have investigated problems relating to curved composites intensively. These investigations have allowed us to study stresses and strains in regions of a composite which are small compared to the curvature wavelength. These new, accurate, techniques were developed in the framework of continuum theories for piecewise homogeneous bodies. We use the exact equations of elasticity or viscoelasticity for anisotropic bodies, and consider linear and non-linear problems in the framework of this continuum theory as well as in the framework of the piecewise homogeneous model. For the latter the method of solution of related problems is proposed. We have focussed our attention on self-balanced stresses which arise from the curvature, but have provided sufficient information for the study of other effects. We assume that the reader is familiar with the theory of elasticity for anisotropic bodies, with partial differential equations and integral transformations, and also with the Finite Element Method.
This textbook on models and modeling in mechanics introduces a new unifying approach to applied mechanics: through the concept of the open scheme, a step-by-step approach to modeling evolves. The unifying approach enables a very large scope on relatively few pages: the book treats theories of mass points and rigid bodies, continuum models of solids and fluids, as well as traditional engineering mechanics of beams, cables, pipe flow and wave propagation.
This book presents a unified hierarchical formulation of theories for three-dimensional continua, two-dimensional shells, one-dimensional rods, and zero-dimensional points. It allows readers with varying backgrounds easy access to fundamental understanding of these powerful Cosserat theories.