Bifurcation and stability in nonlinear dynamical systems

Preface......Page 7
Contents......Page 9
 1.1 Continuous Dynamical Systems......Page 12
 1.2 Equilibriums and Stability......Page 15
  1.3.1 Hyperbolic Stability on Eigenvectors......Page 29
  1.3.2 Spiral Stability on an Invariant Eigenplane......Page 41
  1.3.3 Spiral Stability Based on the Fourier Series Base......Page 51
 1.4 Spiral Stability in Second-Order Nonlinear Systems......Page 55
 1.5 Lyapunov Functions and Stability......Page 59
 References......Page 68
 2.1 Bifurcations......Page 69
 2.2 Hyperbolic Bifurcations on Eigenvectors......Page 70
 2.3 Hopf Bifurcation on an Eigenvector Plane......Page 79
 2.4 Hopf Bifurcation Based on the Fourier Series Base......Page 85
 2.5 Hopf Bifurcations in Second-Order Nonlinear Systems......Page 90
 Reference......Page 95
  3.1.1 Stability and Singularity......Page 96
  3.1.2 Bifurcations......Page 106
   3.1.3a Saddle-Node-Appearing Bifurcation......Page 113
   3.1.3b Saddle-Node-Switching Bifurcation......Page 116
   3.1.3c Pitchfork-Switching/Appearing Bifurcation......Page 117
 3.2 2-Dimensional Nonlinear Systems......Page 118
  3.2.1 Stability and Singularity......Page 119
  3.2.2 Hopf Bifurcation......Page 126
 References......Page 131
 4.1 System Classifications......Page 132
 4.2 Equilibrium Stability......Page 136
 4.3 One-Equilibrium Systems......Page 139
 4.4 Two-Equilibrium Systems......Page 140
 4.5 Three-Equilibrium Systems......Page 148
 Reference......Page 157
 5.1 Linear Systems......Page 158
 5.2 Quadratic Nonlinear Systems......Page 160
 5.3 Cubic Nonlinear Systems......Page 173
 5.4 Quartic Nonlinear Systems......Page 195
 Reference......Page 238
 6.1 Global Stability and Bifurcations......Page 239
  6.2.1 Appearing Bifurcation......Page 256
  6.2.2 Switching Bifurcations......Page 263
  6.2.3 Switching and Appearing Bifurcations......Page 268
  6.3.1 Appearing Bifurcations......Page 273
  6.3.2 Switching Bifurcations......Page 284
  6.3.3 Appearing and Switching Bifurcations......Page 289
 Reference......Page 296
 7.1 Global Stability and Bifurcations......Page 297
  7.2.1 Appearing Bifurcations......Page 314
  7.2.2 Switching Bifurcations......Page 327
  7.2.3 Switching and Appearing Bifurcations......Page 332
  7.3.1 Higher Order Equilibrium Bifurcations......Page 339
  7.3.2 Switching Bifurcations......Page 359
  7.3.3 Switching and Appearing Bifurcations......Page 364
 Reference......Page 371
 8.1 Equilibrium Computations......Page 372
 8.2 Normal Forms......Page 380
 8.3 Infinite-Equilibrium Systems......Page 392
  8.3.1 One-Infinite-Equilibrium Systems......Page 393
  8.3.2 Two-Infinite-Equilibrium Systems......Page 395
  8.3.3 Higher Order Infinite-Equilibrium Systems......Page 400
 8.4 Network-Infinite-Equilibrium Systems......Page 403
  8.4.1 A Network-Infinite-Equilibrium System......Page 405
  8.4.2 Circular Infinite-Equilibrium Systems......Page 407
 Reference......Page 415
Index......Page 416