Author(s): D.K. Arrowsmith, C.M. Place
Publisher: Chapman and Hall
Year: 1982
Language: English
Pages: 261
Cover......Page 1
Title page......Page 2
Preface page......Page 4
1.1.1 Existence and uniqueness......Page 10
1.1.2 Geometrical representation......Page 12
1.2.1 Solution curves and the phase portrait......Page 16
1.2.2 Phase portraits and dynamics......Page 21
1.3 Autonomous systems in the plane......Page 23
1.4 Construction of phase portraits in the plane......Page 27
1.4.1 Use of calculus......Page 28
1.4.2 Isoclines......Page 32
1.5 Flows and evolution......Page 34
Exercises......Page 39
2.1 Linear changes of variable......Page 48
2.2 Similarity types for 2 x 2 real matrices......Page 51
2.3.1 Simple canonical systems......Page 57
2.4.1 Phase portrait of a simple Iinear system......Page 61
2.4.2 Types of canonical system and qualitative equivalence......Page 65
2.5 The evolution operator......Page 67
2.6 Affine systems......Page 70
2.7.1 Three-dimensional systems......Page 73
2.7.2 Four-dimensional systems......Page 76
2.7.3 n-Dimensional systems......Page 78
Exercises......Page 79
3.1 Local and global behaviour......Page 88
3.2 Linearization at a fixed point......Page 91
3.3 The linearization theorem......Page 94
3.4 Non-simple fixed points......Page 100
3.5 Stability of fixed points......Page 102
3.6.1 Ordinary points......Page 106
3.6.2 Global phase portraits......Page 109
3.7 First integrals......Page 110
3.8 Limit cycles......Page 115
3.9 Poincaré-Bendixson theory......Page 118
Exercises......Page 122
4.1.1 A mechanical oscillator......Page 128
4.1.2 Electrical circuits......Page 134
4.1.3 Economics......Page 138
4.1.4 Coupled oscillators......Page 140
4.2 Affine models......Page 144
4.2.2 Resonance......Page 146
4.3.1 Competing species......Page 149
4.3.2 Volterra-Lotka equations......Page 153
4.3.3 The Holling-Tanner model......Page 156
4.4.1 Van der Pol oscillator......Page 160
4.4.2 lumps and regularization......Page 164
4.5.1 The jump assumption and piecewise models......Page 168
4.5.2 A limit cycle from linear equations......Page 171
Exercises......Page 175
5.1 The Liénard equation......Page 188
5.2 Regularization and some economic models......Page 193
5.3 The Zeeman models of heartbeat and nerve impulse......Page 199
5.4.1 Theory......Page 207
5.4.2 A model of animal conflict......Page 213
5.5.1 Some simple examples......Page 219
5.5.2 The Hopf bifurcation......Page 221
5.5.3 An application of the Hopf bifurcation theorem......Page 223
5.6 A mathematical mode] of tumor growth......Page 225
5.6.1 Construction of the model......Page 226
5.6.2 An analysis of the dynamics......Page 227
Exercises......Page 235
Bibliography......Page 239
Hints to exercises......Page 241
Index......Page 258
