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Chapter 1 Linear Discrete Systems and Stability 1

1.1 Basic iterative solutions 1

1.2 Linear discrete systems with distinct eigenvalues 3

1.3 Linear discrete systems with repeated eigenvalues 8

1.4 Stability and boundary 16

1.5 Lower-dimensional discrete systems 31

1.5.1 One-dimensional systems 31

1.5.2 Planar discrete linear systems 32

1.5.3 Three-dimensional discrete systems 46

Reference 62

Chapter 2 Stability,Bifurcation and Routes to Chaos 63

2.1 Discrete dynamical systems 63

2.2 Fixed points and stability 65

2.3 Bifurcation and stability switching 80

2.3.1 Stability and switching 87

2.3.2 Bifurcations 111

2.4 Routes to chaos 124

2.4.1 One-dimensional maps 124

2.4.2 Two-dimensional maps 130

References 131

Chapter 3 Fractality and Complete Dynamics 133

3.1 Multifractals in 1-D iterative maps 133

3.1.1 Similar structures in period doubling 134

3.1.2 Fractality of chaos via period doubling bifurcations 139

3.1.3 An example 141

3.2 Bouncing ball dynamics 146

3.2.1 Periodic motions 148

3.2.2 Stability and bifurcations 151

3.2.3 Numerical illustrations 158

3.3 Positive and negative dynamics of discrete systems 162

3.4 Complete dynamics of Henon map 170

References 175

Chapter 4 Switching Systems with Transports 177

4.1 Continuous subsystems 177

4.2 Switching systems 179

4.3 Measuring functions and stability 186

4.4 Mappings and periodic flows 205

4.5 Linear switching systems 211

4.5.1 Vibrations with piecewise forces 214

4.5.2 Vector fields switching 222

References 227

Chapter 5 Mapping Dynamics and Fragmentation 229

5.1 Discontinuous dynamical systems 229

5.2 G-functions to boundaries 232

5.3 Mapping dynamics 235

5.4 A semi-active suspension system 241

5.4.1 Analytical dynamics 243

5.4.2 Illustrations 248

5.5 Grazing singular sets and fragmentation 257

5.6 Fragmentized strange attractors 269

References 281

Index 283