Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields | springerprofessional.de

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1983 | Buch

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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

verfasst von: John Guckenheimer, Philip Holmes

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #Book Review - Engineering Societies Library, New York#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #American Mathematical Monthly#2

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction: Differential Equations and Dynamical Systems

Abstract

In this introductory chapter we review some basic topics in the theory of ordinary differential equations from the viewpoint of the global geometrical approach which we develop in this book. After recalling the basic existence and uniqueness theorems, we consider the linear, homogeneous, constant coefficient system and then introduce nonlinear and time-dependent systems and concepts such as the Poincaré map and structural stability. We then review some of the better-known results on two-dimensional autonomous systems and end with a statement and sketch of the proof of Peixoto’s theorem, an important result which summarizes much of our knowledge of two-dimensional flows.

John Guckenheimer, Philip Holmes

Chapter 2. An Introduction to Chaos: Four Examples

Abstract

In this chapter we introduce four nonlinear systems which possess fascinating properties and which are still improperly understood. We have chosen two periodically forced single degree of freedom oscillators, a three-dimensional autonomous differential equation, and a two-dimensional map. The oscillators of van der Pol [1927] and Duffing [1918] originally arose as models in electric circuit theory and solid mechanics, respectively, while the Lorenz equations (Lorenz [1963]) represent a truncation of the nonlinear partial differential equations governing convection in fluids. Finally, our map models a simple repeated impact problem (Holmes [1982]) and, in a slightly different form, resonance problems in atomic physics (Chirikov [1979], Greene [1980]). In fact the conservative, area preserving version of this map has been studied intensively as a canonical example of the transition to stochasticity and chaos in Hamiltonian systems. The range of applications for the models outlined here should suggest the pervasive importance of nonlinear systems.

John Guckenheimer, Philip Holmes

Chapter 3. Local Bifurcations

Abstract

In this chapter, we study the local bifurcations of vector fields and maps. As we have seen, systems of physical interest typically have parameters which appear in the defining systems of equations. As these parameters are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcations and the parameter values are called bifurcation values. To the extent possible, we develop in this chapter and Chapters 6 and 7, a systematic theory which describes and permits the analysis of the typical bifurcations one encounters. We pay careful attention to the examples introduced in Chapter 2 and use these to illustrate the theory that we present.

John Guckenheimer, Philip Holmes

Chapter 4. Averaging and Perturbation from a Geometric Viewpoint

Abstract

In this chapter we describe some classical methods of analysis which are particularly applicable to problems in nonlinear oscillations. While these methods might be familiar to the reader who has studied nonlinear mechanics and perturbation theory, the present geometrical approach and the stress on obtaining approximations to Poincaré maps will probably be less familiar.

John Guckenheimer, Philip Holmes

Chapter 5. Hyperbolic Sets, Symbolic Dynamics, and Strange Attractors

Abstract

The solutions of ordinary differential equations can have an erratic time dependence which appears in some ways to be random. We have seen several such examples in Chapter 2. The present chapter is devoted to a discussion of simple, geometrically defined systems in which such chaotic motion occurs. We shall describe both the irregular character of individual solutions and the complicated geometric structures associated with their limiting behavior. The principal technique which we use is called symbolic dynamics and the general approach to the questions we adopt is referred to as dynamical systems theory. We shall not develop this theory systematically but will state some of its major results and provide a brief guide to its literature. Our strategy in solving specific problems will generally involve the use of numerical or perturbation methods, such as those of Chapter 4, to establish the existence of interesting geometrical structure in appropriate Poincaré maps, followed by the use of the methods of this chapter.

John Guckenheimer, Philip Holmes

Chapter 6. Global Bifurcations

Abstract

In Chapter 3 we dealt with the local bifurcation properties of equilibrium points and periodic orbits. The theory developed there relied upon coordinate transformations which bring general systems into normal forms, from which dynamical information can be deduced from the Taylor series of a vector field or map at a single point.

John Guckenheimer, Philip Holmes

Chapter 7. Local Codimension Two Bifurcations of Flows

Abstract

In this chapter we discuss bifurcations from equilibria which have multiple degeneracies. We start with the analogues of the sadble-node and Hopf bifurcations with the same linear part, but additional degeneracy in the nonlinear terms of the Taylor expansion expressed in normal form. The theory here is complete, at least for the first few cases, and is essentially obtained by unfolding degenerate singularities of functions, since in each case we can reduce to a noe-dimensional flow.

John Guckenheimer, Philip Holmes

Backmatter

Titel: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
verfasst von: John Guckenheimer
Philip Holmes
Copyright-Jahr: 1983
Verlag: Springer New York
Electronic ISBN: 978-1-4612-1140-2
Print ISBN: 978-1-4612-7020-1
DOI: https://doi.org/10.1007/978-1-4612-1140-2

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