Qualitative Theory Of Differential Equations Wei Fen Fang Chʻeng Ting Hsing Li Lun. Chinese

This book provides an introduction to and a comprehensive study of the qualitative theory of ordinary differential equations. It begins with fundamental theorems on existence, uniqueness, and initial conditions, and discusses basic principles in dynamical systems and Poincaré-Bendixson theory. The authors present a careful analysis of solutions near critical points of linear and nonlinear planar systems and discuss indices of planar critical points. A very thorough study of limit cycles is given, including many results on quadratic systems and recent developments in China. Other topics included are: the critical point at infinity, harmonic solutions for periodic differential equations, systems of ordinary differential equations on the torus, and structural stability for systems on two-dimensional manifolds. This books is accessible to graduate students and advanced undergraduates and is also of interest to researchers in this area. Exercises are included at the end of each chapter. Readership: Graduate students and research mathematicians interested in differential equations. Cover Translations of Mathematical Monographs 101 S Title Qualitative Theory of Differential Equations Copyright ©1992 by the American Mathematical Society ISBN 0-8218-4551-9 QA372.W3812 1991 515'.35-dc20 LCCN 91-23961 Contents Preface Translator's note Symbols CHAPTER I Fundamental Theorems §1. Existence and uniqueness of solutions, dependence of solutions on initial conditions and parameters §2. Continuation of solutions §3. General concepts in dynamical systems §4. Dynamical systems on the plane Exercises REFERENCES CHAPTER II Critical Points on the Plane §1. Critical points and regular points §2. Critical points for linear differential equations with constant coefficients §3. Critical points for nonlinear systems §4. Effects of nonlinear terms when the eigenvalues have nonzero real parts §5. Effects of nonlinear terms when the eigenvalues are a pair of pure imaginary eigenvalues. (Tests for center or focus.) Method I. Test for center or focus. Method II. Test for center or focus. §6. * Geometric configurations near critical points Composite interval of the first kind Composite interval of the second kind. §7. * Effect of nonlinear terms for the case of zero eigenvalue(s) Exercises REFERENCES CHAPTER III Indices of Planar Critical Points §1. Rotation number for a continuous vector field [2] §2. Indices for planar critical points §3. Cauchy's index §4. Computation of the index for an isolated critical point of homogeneous equations by rational calculation §5. * The rational calculation of the index of a singular critical point §6. * Bendixson's formula Exercises REFERENCES CHAPTER IV Limit Cycles §1. Existence of limit cycles §2. Successor function. Multiplicity and stability of limit cycles §3. Rotated vector fields §4. The uniqueness of limit cycles §5. Existence of two limit cycles §6. * The number of limit cycles for quadratic systems §7. * Existence of n limit cycles (A) S. P. Diliberto's theorem (B) Construction of a Lienard's equation which has exactly n limit cycles. (C) Sufficient conditions for a type of Lienard's equation to have at least n limit cycles Exercises REFERENCES CHAPTER V Critical Points at Infinity §1. Poincare transformation §2. Global structures of planar systems §3. Analysis of the existence of limit cycles by means of critical points at infinity §4. The sum of indices of critical points for continuous vector fields on the two-dimensional compact surface S2 , P2, and T2 Exercises REFERENCES CHAPTER VI Harmonic Solutions for Two-Dimensional Periodic Systems §1. Preliminaries §2. Linear systems with constant coefficients and periodic forcing §3. Almost linear systems §4. Method of averaging §5. Small perturbations of Duffing's equatio §6. Small amplitude harmonic solutions for high frequency forced oscillations §7. Large amplitude harmonic solutions for high frequency forced oscillations §8. Dissipative systems §9. Duffing's equation with no damping Exercises REFERENCES CHAPTER VII Systems of Ordinary Differential Equations on the Torus §1. Introduction §2. Rotation numbers §3. The limit set §4. Ergodicity §5. An example for the singular case §6. Description of Schweitzer's example §7. On Birkhoff s conjecture Exercises REFERENCES CHAPTER VIII Structural Stability §1. Structural stability for systems of ordinary differential equations on a planar disk §2. Structural stability for systems of ordinary differential equations on two-dimensional manifolds (A) Systems of ordinary differential equations on two-dimensional manifolds, and Peixoto's structural stability theorem (B) On the elimination of nontrivial minimal se (C) More approximation lemma (D) Proofs of the structural stability theorem and density theorem Exercises REFERENCES Back Cover Gives an introduction to and a comprehensive study of the qualitative theory of ordinary differential equations. This title begins with fundamental theorems on existence, uniqueness, and initial conditions, and discusses basic principles in dynamical systems and Poincare-Bendixson theory. Each chapter includes exercises. Zhang Zhi-fen ... [et Al. ; Translated From The Chinese By Anthony Wing-kwok Leung]. Translation Of: Wei Fen Fang Chʻeng Ting Hsing Li Lun. Includes Bibliographical References And Index.