Approaches to the qualitative theory of ordinary differential equations : dynamical systems and nonlinear oscillations

This book is an ideal text for advanced undergraduate students and graduate students with an interest in the qualitative theory of ordinary differential equations and dynamical systems. Elementary knowledge is emphasized by the detailed discussions on the fundamental theorems of the Cauchy problem, fixed-point theorems (especially the twist theorems), the principal idea of dynamical systems, the nonlinear oscillation of Duffing's equation, and some special analyses of particular differential equations. It also contains the latest research by the author as an integral part of the book.

Preface	p. V
Chapter 1 Cauchy Problem	p. 1
1.1 Fundamental Theorems	p. 1
1.2 Method of Euler Polygons	p. 15
1.3 Local Behavior of Integral Curves	p. 20
1.4 Peano Phenomenon	p. 24
1.5 Convergence Theorem on Difference Methods	p. 33
Chapter 2 Global Behavior of Solution	p. 47
2.1 Global Existence of Solution	p. 47
2.2 Predictability of Solution	p. 58
2.3 Liapunov Stability	p. 61
2.4 Liapunov Unstability	p. 71
Chapter 3 Autonomous Systems	p. 73
3.1 Phase Portrait	p. 73
3.2 Orbital Box	p. 76
3.3 Types of Orbits	p. 77
3.4 Singular Points	p. 79
3.5 General Property of Singular Points	p. 86
3.6 Closed Orbit	p. 87
3.7 Invariant Torus	p. 91
3.8 Limit-Point Set	p. 95
3.9 Poincare-Bendixson Theorem	p. 97
Chapter 4 Non-Autonomous Systems	p. 101
4.1 General Systems	p. 101
4.2 Conservative Systems	p. 104
4.3 Dissipative Systems	p. 106
4.4 Planar Periodic Systems	p. 115
4.5 Invariant Continuum	p. 119
Chapter 5 Dynamical Systems	p. 123
5.1 The Originality	p. 123
5.2 Recurrence	p. 128
5.3 Quasi-Minimal Set	p. 132
5.4 Minimal Set	p. 134
5.5 Almost Periodic Motion	p. 144
Chapter 6 Fixed-Point Theorems	p. 155
6.1 Poincare Index	p. 155
6.2 Vector Fields on Closed Surfaces	p. 165
6.3 Spatial Vector Fields	p. 172
6.4 Fixed-Point Theorems of Brouwer Type	p. 176
Chapter 7 Bend-Twist Theorem	p. 181
7.1 Generalized Poincare-Birkhoff Twist Theorem	p. 181
7.2 Analytic Bend-Twist Theorem	p. 184
7.3 Analytic Poincare-Birkhoff Twist Theorem	p. 189
7.4 Application of the Bend-Twist Theorem	p. 191
Chapter 8 Chaotic Motions	p. 199
8.1 Definition of Chaotic Motion	p. 199
8.2 Chaotic Quasi-Minimal Set	p. 201
8.3 Sufficient Conditions for Chaotic Sets	p. 202
8.4 Chaotic Closed Surfaces	p. 205
8.5 Applications	p. 209
Chapter 9 Perturbation Method	p. 217
9.1 Nonlinear Differential Equation of Second Order	p. 217
9.2 Method of Averaging	p. 225
9.3 High Frequency Forced Oscillations	p. 230
Chapter 10 Duffing Equations of Second Order	p. 241
10.1 Periodic Oscillations	p. 241
10.2 Time-Map	p. 250
10.3 Duffing Equation of Super-Linear Type	p. 261
10.4 Duffing Equation of Sub-Linear Type	p. 275
10.5 Duffing Equation of Semi-Linear Type	p. 288
Chapter 11 Some Special Problems	p. 313
11.1 Reeb's Problem	p. 313
11.2 Birkhoff's Conjecture	p. 319
11.3 Morse's Conjecture	p. 326
11.4 Kolmogorov's Problem	p. 331
11.5 Brillouin Focusing System	p. 345
11.6 A Retarded Equation	p. 355
11.7 Periodic Lotka-Volterra System	p. 365
Bibliography	p. 377

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