Nonlinear differential equations and dynamical systems非线性微分方程和动态系统
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作者： Ferdinand Verhulst著
出 版 社： 广东教育出版社
出版时间： 2006-3-1字数：版次： 1页数： 303印刷时间： 2006/03/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540609346包装： 平装编辑推荐
作者简介：
Ferdinand Verhulst was born in Amsterdam, The Netherlands, in 1939.
He graduated at the University of Amsterdam in Astrophysics and Mathematics. A period of five years at the Technological University of Delft, started his interest in technological problems, resulting in various cooperations with engineers. His other interests include the methods and applications of asymptotic analysis, nonlinear oscillations and wave theory.
He holds a chair of dynamical systems at the department of mathematics at the University of Utrecht.
Among his other interests are a publishing company, Epsilon Uitgaven, that he founded in 1985, and the relation between dynamical systems and psychoanalysis.
内容简介
"A good book for a nice price!" (Monatshefte für Mathematik)
"... for lecture courses that cover the classical theory of nonlinear differential equations associated with Poincaré and Lyapunov and introduce the student to the ideas of bifurcation theory and chaos this is an ideal text ..." (Mathematika)
"The pedagogical style is excellent, consisting typically of an insightful overview followed by theorems, illustrative examples and exercises." (Choice)
目录
1 Introduction
1.1 Definitions and notation
1.2 Existence and uniqueness
1.3 Gronwall's inequality
2 Autonomous equations
2.1 Phase-space, orbits
2.2 Critical points and linearisation
2.3 Periodic solutions
2.4 First integrals and integral manifolds
2.5 Evolution of a volume element, Liouville's theorem
2.6 Exercises
3 Critical points
3.1 Two-dimensional linear systems
3.2 Remarks on three-dimensional linear systems
3.3 Critical points of nonlinear equations
3.4 Exercises
4 Periodic solutions
4.1 Bendixson's criterion
4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem
4.3 The Poincare-Bendixson theorem
4.4 Applications of the Poincar6-Bendixson theorem
4.5 Periodic solutions inRn
4.6 Exercises
5 Introduction to the theory of stability
5.1 Simple examples
5.2 Stability of equilibrium solutions
5.3 Stability of periodic solutions
5.4 Linearisation
5.5 Exercises
6 Linear Equations
6.1 Equations with constant coefficients
6.2 Equations with coefficients which have a limit
6.3 Equations with periodic coefficients
6.4 Exercises
Stability by linearisation
7.1 Asymptotic stability of the trivial solution
7.2 Instability of the trivial solution
7.3 Stability of periodic solutions of autonomous equations
7.4 Exercises
8 Stability analysis by the direct method
8.1 Introduction
8.2 Lyapunov functions
8.3 Hamiltonian systems and systems with first integrals
8.4 Applications and examples
8.5 Exercises
9 Introduction to perturbation theory
9.1 Background and elementary examples
9.2 Basic material
9.3 Naive expansion
9.4 The Poincare expansion theorem
9.5 Exercises
10 The Poincare-Lindstedt method
10.1 Periodic solutions of autonomous second-order equations
10.2 Approximation of periodic solutions on arbitrary long time-scales
10.3 Periodic solutions of equations with forcing terms
10.4 The existence of periodic solutions
10.5 Exercises
11 The method of averaging
11.1 Introduction
11.2 The Lagrange standard form
11.3 Averaging in the periodic case
11.4 Averaging in the general case
11.5 Adiabatic invariants
11.6 Averaging over one angle, resonance manifolds
11.7 Averaging over more than one angle, an introduction
11.8 Periodic solutions
11.9 Exercises
12 Relaxation Oscillations
13 Bifurcation Theory
14 Chaos
15 Hamiltonian systems
Appendix 1 The Morse lemma
Appendix 2 Linear periodic equations with a small parameter
Appendix 3 Trigonometric formulas and averages
Appendix 4 A sketch of Cotton's proof of the stable and unstable manifold theorem
Appendix 5 Bifurcations of self-excited oscillations
Appendix 6 Normal forms of Hamiltonian systems near equilibria
Answers and hints to the exercises
References
Index
