Nonlinear differential equations and dynamical systems非线性微分方程和动态系统

书籍目录

1  Introduction　1.1  Definitions and notation　1.2  Existence and uniqueness　1.3  Gronwall's inequality2  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  Exercises3  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  Exercises4  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  Exercises5  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  Exercises6  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  Exercises8  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  Exercises9  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  Exercises10 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 Exercises11 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 Exercises12 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

作者简介

"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)

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