Dynamical Systems: Theories and Applications

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface......Page 4
Table of Contents......Page 6
 1.2 Poincaré map technique......Page 10
 1.3 Smale horseshoe......Page 12
 1.4 Symbolic dynamics......Page 16
 1.5 Strange attractors......Page 19
 1.6 Basins of attraction......Page 22
 1.7 Density, robustness and persistence of chaos......Page 25
 1.8 Entropies of chaotic attractors......Page 32
 1.9 Period 3 implies chaos......Page 41
 1.10 The Snap-back repeller and the Li-Chen-Marotto theorem......Page 46
 1.11 Shilnikov criterion for the existence of chaos......Page 47
 2.2 Definition of Human Immunodeficiency Virus (HIV)......Page 55
 2.3 Modelling the Human Immunodeficiency Virus (HIV)......Page 57
 2.4 Dynamics of sexual transmission of the Human Immunodeficiency Virus......Page 58
 2.5 The effects of variable infectivity on the HIV dynamics......Page 62
 2.6 The CD4+ Lymphocyte dynamics in HIV infection......Page 65
 2.7 The viral dynamics of a highly pathogenic Simian/Human Immunodeficiency Virus......Page 69
 2.8 The effects of morphine on Simian Immunodeficiency Virus Dynamics......Page 74
 2.9 The dynamics of the HIV therapy system......Page 76
 2.10 Dynamics of urbanization......Page 80
 3.2 Chaos in one-dimensional piecewise smooth maps......Page 87
 3.3 Chaos in one-dimensional singular maps......Page 92
 3.4 Chaos in 2-D piecewise smooth maps......Page 95
 4.1 Introduction......Page 105
 4.2 Chaos in neural networks models......Page 106
 4.3 Robust chaos in discrete time neural networks......Page 107
  4.3.1 Robust chaos in 1-D piecewise-smooth neural networks......Page 110
  4.3.2 Fragile chaos (blocks with smooth activation function)......Page 111
  4.3.3 Robust chaos (blocks with non-smooth activation function)......Page 114
  4.3.4 Robust chaos in the electroencephalogram model......Page 118
  4.3.5 Robust chaos in Diluted circulant networks......Page 122
  4.3.6 Robust chaos in non-smooth neural networks......Page 123
 4.4 The importance of robust chaos in mathematics and some open problems......Page 124
 5.2 Lyapunov exponents of the discrete hyperchaotic double scroll map......Page 126
 5.3 Lyapunov exponents for a class of 2-D piecewise linear mappings......Page 130
 5.4 Lyapunov exponents of a family of 2-D discrete mappings with separate variables......Page 133
 5.5 Lyapunov exponents of a discontinuous piecewise linear mapping of the plane governed by a simple switching law......Page 137
 5.6 Lyapunov exponents of a modified map-based BVP model......Page 144
 6.1 Introduction......Page 151
 6.2 Compound synchronization of different chaotic systems......Page 152
 6.3 Synchronization of 3-D continuous-time quadratic systems using a universal non-linear control law......Page 160
 6.4 Co-existence of certain types of synchronization and its inverse......Page 164
 6.5 Synchronization of 4-D continuous-time quadratic systems using a universal non-linear control law......Page 173
 6.6 Quasi-synchronization of systems with different dimensions......Page 178
 6.7 Chaotification of 3-D linear continuous-time systems using the signum function feedback......Page 182
 6.8 Chaos control problem of a 3-D cancer model with structured uncertainties......Page 190
 6.9 Controlling homoclinic chaotic attractor......Page 191
 6.10 Robustification of 2-D piecewise smooth mappings......Page 195
 6.11 Chaotifying stable n-D linear maps via the controller of any bounded function......Page 201
 7.2 Boundedness of certain forms of 3-D quadratic continuous-time systems......Page 206
 7.3 Bounded jerky dynamics......Page 211
  7.3.1 Boundedness of general forms of jerky dynamics......Page 214
  7.3.2 Examples of bounded jerky chaos......Page 223
  7.3.3 Appendix A......Page 225
 7.4 Bounded hyperjerky dynamics......Page 229
 7.5 Boundedness of the generalized 4-D hyperchaotic model containing Lorenz-Stenflo and Lorenz-Haken systems......Page 233
  7.5.1 Estimating the bounds for the Lorenz-Haken system......Page 237
  7.5.2 Estimating the bounds for the Lorenz-Stenflo system......Page 238
 7.6 Boundedness of 2-D Hénon-like mapping......Page 242
 7.7 Examples of fully bounded chaotic attractors......Page 248
 8.1 Introduction......Page 251
 8.2 Direct Lyapunov stability for ordinary differential equations......Page 252
 8.3 Exponential stability of non-linear time-varying......Page 260
 8.4 Lasalle’s Invariance Principle......Page 268
 8.5 Direct Lyapunov-type stability for fractional-like systems......Page 270
 8.6 Construction of globally asymptotically stable n-D discrete mappings......Page 276
 8.7 Construction of superstable n-D mappings......Page 280
 8.8 Examples of globally superstable 1-D quadratic mappings......Page 284
 8.9 Construction of globally superstable 3-D quadratic mappings......Page 289
 8.10 Hyperbolicity of dynamical systems......Page 294
 8.11 Consequences of uniform hyperbolicity......Page 306
  8.11.1 Classification of singular-hyperbolic attracting sets......Page 307
  8.12.1 The concept of structural stability......Page 311
  8.12.2 Conditions for structural stability......Page 312
  8.12.3 The Jordan normal form J1......Page 315
  8.12.5 The Jordan normal form J3......Page 318
  8.12.7 The Jordan normal form J5......Page 319
  8.12.8 The Jordan normal form J6......Page 320
 8.13 Construction of globally asymptotically stable partial differential systems......Page 321
 8.14 Construction of globally stable system of delayed differential equations......Page 331
 8.15 Stabilization by the Jurdjevic-Quinn method......Page 338
  8.15.1 The minimization problem......Page 339
  8.15.2 The inverse optimization problem......Page 340
  8.15.3 Input-to-state stability......Page 341
 9.2 Transformation of 3-D dynamical systems to jerk form......Page 345
 9.3 Transformation of 3-D dynamical systems to rational and cubic jerks forms......Page 349
 9.4 Transformation of 4-D dynamical systems to hyperjerk form......Page 352
  9.4.1 The expression of the transformation between (9.45) and (9.61)-(9.62)......Page 358
  9.4.2 Examples of 4-D hyperjerky dynamics......Page 361
 9.5 Examples of crackle and top dynamics......Page 368
References......Page 370
Index......Page 398