This book presents an introduction to chaos in dynamical system and fundamental theories( ordinary differential equations. Considering the fact that many readers have unclear concepts about chaotic dynamical systems and some mathematical concepts are also abstract, this book mainly focuses on the explanation of the systems'basic concepts,with detailed examples and diagrams. It also attaches great importance to the calculational and analytic methods of the chaos systems and the application of their theories. It will be of interest to advanced undergraduates in mathematics and graduate students in engineering taking courses in dynamical systems, nonlinear dynamics,nonlinear systems as well aschaos.

CHAPTER 1 Computational Techniques of Linear Differential Equation 1

1.1Basic concepts 1

1.2 First order linear differential equation

      1.2.1Separable equation 4

      1.2.2 Linear equation7

      1.2.3 Exact equations and integrating factors 10

      1.2.4 Direction fields 13

1.3 Second order differential equation 16

      1.3.1 Homogeneous linear equation 16

      1.3.2Nonhomogeneous linear equation 17

1.4 First order differential equations 19

      1.4.1 Basic theories of the first order DEs 19

      1.4.2 Homogeneous linear DEs with constant coefficients 20

      1.4.3 Nonhomogeneous linear DEs with constant coefficients 28 C2\1.5 Three special methods 31

      1.5.1 Laplace transform method 31

      1.5.2 Power series method 37

      1.5.3 Fourier series method 39

1.6 Numerical solution of differential equations 40

CHAPTER2 Qualitative Analysis of Planar Differential Equations 43

2.1 Flow and manifold 43

      2.1.1 Flow 43

      2.1.2 Maniflod 46

2.2 Planar linear systems 49

2.3 Linearization of nonlinear systems 62

      2.3.1 Singularities analysis of nonlinear systems 62

      2.3.2 Stability of singularities 71

2.4 Periodic solutions of nonlinear systems 74

      2.4.1 Orbit and limit set 74

      2.4.2 Periodic orbit and limit cycle 77

2.5 Conservative system and dissipative system 87

      2.5.1 Hamiltonian system 87

      2.5.2 Dissipative Systems 98

HAPTER 3 Calculation and Analysis of Chaotic Systems 101

3.1 Attractor, Lyapunov exponent 101

      3.1.1 Attractor 101

      3.1.2 Lyapunov exponent 106

3.2 Center manifolds 109

      3.2.1 Eigenspaces and manifolds 109

      3.2.2 Center manifolds116

3.3 Hopf bifurcation 120

      3.3.1 Andronov-Hopf bifurcation 120

      3.3.2 Hopf bifurcation ofLorenz-like system 121

3.4Dimension reduction analysis 132

      3.4.1 Invariant algebraic surface 132

      3.4.2 Invariant algebraic surface ofT system 136

3.5 Infinity analysis 146

      3.5.1 Poincare compactification on R2 146

      3.5.2 Poincare compactification on R3 161

3.6 Melnikov method 170

CHAPTER 4 Control and Synchronization of Chaotic Systems 185

4.1 Feedback control 185

      4.1.1 Feedback control ofTsystem 185

      4.1.2 Differential fedback control of Jerk system 190

4.2 Backstepping Control 194

      4.2.1 Backstepping for strict feedback systems 194

      4.2.2 Adaptive backstepping control of electromechanical system 197

      4.2.3 Adaptive backstepping control of T system 204

4.3 Periodic parametric perturbation control 208

      4.3.1 Periodic parametric perturbation system 208

      4.3.2 Melnikov homoclinic orbits analysis 211

      4.3.3 Melnikov periodic orbits analysis 214

      4.3.4 Numerical experiments 220

4.4 Generalized synchronization 230

      4.4.1 Preliminary 230

      4.4.2 GS of fractional unified chaotic system 233

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