Bifurcation and Chaos in Engineering Book

Ch. 1	Dynamical Systems, Ordinary Differential Equations and Stability of Motion	1
1.1	Concepts of Dynamical Systems	1
1.2	Ordinary Differential Equations	5
1.3	Properties of Flow	14
1.4	Limit Point Sets	17
1.5	Liapunov Stability of Motion	23
1.6	Poincare-Bendixson Theorem and its Applications	29
Ch. 2	Calculation of Flows	35
2.1	Divergence of Flows	35
2.2	Linear Autonomous Systems and Linear Flows and the Calculation of Flows about the IVP	38
2.3	Hyperbolic Operator (or Generality)	47
2.4	Non-linear Differential Equations and the Calculation of their Flows	55
2.5	Stable Manifold Theorem	60
Ch. 3	Discrete Dynamical Systems	66
3.1	Discrete Dynamical Systems and Linear Maps	66
3.2	Non-linear Maps and the Stable Manifold Theorem	68
3.3	Classification of Generic Systems	71
3.4	Stability of Maps and Poincare Mapping	73
3.5	Structural Stability Theorem	76
Ch. 4	Liapunov-Schmidt Reduction	84
4.1	Basic Concepts of Bifurcation	84
4.2	Classification of Bifurcations of Planar Vector Fields	88
4.3	The Implicit Function Theorem	91
4.4	Liapunov-Schmidt Reduction	93
4.5	Methods of Singularity	102
4.6	Simple Bifurcations	119
4.7	Bifurcation Solution of the 1/2 Subharmonic Resonance Case of Non-linear Parametrically Excited Vibration Systems	127
4.8	Hopf Bifurcation Analyzed by Liapunov-Schmidt Reduction	143
Ch. 5	Centre Manifold Theorem and Normal Form of Vector Fields	154
5.1	Centre Manifold Theorem	154
5.2	Saddle-Node Bifurcation	166
5.3	Normal Form of Vector Fields	169
Ch. 6	Hopf Bifurcation	176
6.1	Hopf Bifurcation Theorem	176
6.2	Complex Normal Form of the Hopf Bifurcation	179
6.3	Normal Form of the Hopf Bifurcation in Real Numbers	182
6.4	Hopf Bifurcation with Parameters	185
6.5	Calculating Formula for the Hopf Bifurcation Solution	192
6.6	Stability of the Hopf Bifurcation Solution	194
6.7	Effective Method for Computing the Hopf Bifurcation Solution Coefficients	198
6.8	Bifurcation Problem Involving Double Zero Eigenvalues	203
Ch. 7	Application of the Averaging Method in Bifurcation Theory	230
7.1	Standard Equation	230
7.2	Averaging Method and Poincare Maps	237
7.3	The Geometric Description of the Averaging Method	241
7.4	An Example of the Averaging Method - the Duffing Equation	248
7.5	The Averaging Method and Local Bifurcation	255
7.6	The Averaging Method, Hamiltonian Systems and Global Behaviour	261
Ch. 8	Brief Introduction to Chaos	265
8.1	What is Chaos?	265
8.2	Some Examples of Chaos	268
8.3	A Brief Introduction to the Analytical Method of Chaotic Study	273
8.4	The Hamiltonian System	289
8.5	Some Statistical Characteristics	303
8.6	Conclusions	305
Ch. 9	Construction of Chaotic Regions	311
9.1	Incremental Harmonic Balance Method (IHB Method)	312
9.2	The Newtonian Algorithm	317
9.3	Number of Harmonic Terms	318
9.4	Stability Characteristics	318
9.5	Transition Sets in Physical Parametric Space	319
9.6	Example of the Duffing Equation with Multi-Harmonic Excitation	320
Ch. 10	Computational Methods	341
10.1	Normal Form Theory	341
10.2	Symplectic Integration and Geometric Non-Linear Finite Element Method	359
10.3	Construction of the Invariant Torus	375
Ch. 11	Non-linear Structural Dynamics	399
11.1	Bifurcations in Solid Mechanics	399
11.2	Non-Linear Dynamics of an Unbalanced Rotating Shaft	406
11.3	Galloping Vibration Analysis for an Elastic Structure	421
11.4	Other Applications of Bifurcation Theory	431
	References	436
	Index	451