Almost periodicity, chaos, and asymptotic equivalence

Preface......Page 7
 References......Page 11
Contents......Page 13
 1.1 Almost Periodicity......Page 18
 1.2 Chaos......Page 22
 1.3 Piecewise Constant Argument......Page 30
 1.4 Asymptotic Behavior of Solutions......Page 39
 References......Page 50
  2.1.1 Piecewise Continuous Functions and Their Points of Discontinuity......Page 59
  2.2.1 Equivalent Integral Equations......Page 60
  2.2.2 The Gronwall–Bellman Lemma for Piecewise Continuous Functions......Page 62
  2.3.2 Basics of Periodic Systems......Page 65
  2.3.3 An Example......Page 66
  2.4.1 Linear Homogeneous Systems......Page 68
  2.4.2 Examples......Page 70
  2.4.3 The Adjoint System......Page 74
  2.4.4 Linear Exponentially Dichotomous Systems......Page 75
  2.5.1 The Solution of Cauchy Problem......Page 77
  2.5.2 An Example......Page 78
  2.5.3 The Bounded Solution......Page 80
 References......Page 83
  3.1.1 Almost Periodic Sequences......Page 84
  3.1.3 Examples......Page 86
 3.2 Discontinuous Almost Periodic Functions......Page 88
  3.2.1 Examples......Page 89
  3.2.2 The Basic Properties of Almost Periodic Functions......Page 90
  3.2.3 Invariance of Discontinuous Almost Periodicity......Page 91
 3.3 Note......Page 98
 References......Page 99
 4.1 Fundamental Lemmas......Page 100
 4.2 Linear Non-homogeneous Almost Periodic Systems......Page 102
 4.3 Quasilinear Almost Periodic Impulsive Systems......Page 104
 4.4 Systems with Periodic Homogeneous Parts......Page 106
 4.5 Note......Page 113
 References......Page 114
5 Bohr and Bochner Discontinuities......Page 117
 5.1 Bohr Discontinuous Almost Periodic Functions......Page 118
 5.2 Bochner Criterion......Page 123
 5.3 Amerio and Favard Theorems......Page 126
 5.4 Mukhamadiev Theorem......Page 128
 5.5 Recurrent and Limit Almost Periodic Solutions of Non-autonomous Impulsive Systems......Page 132
 References......Page 134
 6.1 Introduction and Preliminaries......Page 136
 6.2 Exponential Dichotomy......Page 139
 6.3 The Non-homogeneous Linear System......Page 141
 6.4 Periodic Solutions......Page 146
 6.5 Almost Periodic Solutions......Page 148
 References......Page 153
 7.1 Introduction and Preliminaries......Page 155
 7.2 Retarded Functional Differential Equations with Retarded Constancy of Argument......Page 159
 7.3 Retarded Functional Differential Equations with Alternate Constancy of Argument......Page 162
 7.4 Quasilinear Systems: Preliminaries......Page 164
 7.5 Bounded Solutions of Quasilinear Systems......Page 168
  7.5.1 Periodic Solutions......Page 172
  7.5.2 Almost Periodic Solutions......Page 174
  7.5.3 Examples......Page 180
 7.6 Further Investigations......Page 183
 References......Page 186
 8.1 Introduction......Page 188
 8.2 Preliminaries......Page 192
 8.3 Existence and Uniqueness......Page 194
 8.4 Bounded Solutions......Page 196
 8.5 Almost Periodic Solutions......Page 201
 8.6 An Example......Page 205
 References......Page 207
 9.1 Introduction......Page 212
 9.2 Description of the DETC......Page 213
 9.3 ψ-Substitution......Page 215
 9.4 Reduction of the DETC (DETS) to Impulsive Differential Equations......Page 216
  9.5.1 Homogeneous Linear Systems......Page 218
  9.5.2 Non-homogeneous Linear Systems......Page 220
  9.5.3 Linear Systems with Constant Coefficients......Page 221
  9.6.1 Description of Periodical DETC......Page 223
  9.6.2 Floquet Theory......Page 225
  9.6.3 Massera Theorem......Page 227
 9.7 Almost Periodic Solutions of Quasilinear Systems......Page 228
 9.8 Note......Page 231
 References......Page 232
 10.1 Introduction......Page 234
 10.3 Chaos Through a Cascade of Almost Periodic Solutions......Page 237
  10.3.2 Cascade of Almost Periodic Solutions......Page 239
 10.4 Li-Yorke Chaos with Infinitely Many Almost PeriodicMotions......Page 240
 10.5 An Example......Page 242
 10.6 Control of Tori......Page 247
 10.7 Note......Page 249
 References......Page 250
 11.1 Introduction and Preliminaries......Page 254
 11.2 The Model......Page 256
 11.3 Bounded Solutions......Page 258
 11.4 Li-Yorke Chaos......Page 260
 11.5 An Example......Page 267
 References......Page 272
 12.1 Introduction......Page 275
  12.2.1 Exponential Decaying Input Currents (EDICs)......Page 279
  12.2.2 Rectangular Input Currents (RICs)......Page 280
  12.2.3 The External Input, Lij(t), of the Cell Cij......Page 281
  12.3.1 Almost Periodicity of EDICs and RICs......Page 283
  12.3.2 Almost Periodic Solutions......Page 285
 12.4 The Li-Yorke Chaos......Page 289
  12.4.1 The Li-Yorke Chaos in SICNNs......Page 290
  12.4.2 An Example......Page 296
 12.5 The Network with EDICs......Page 297
 12.6 A Chaos Extension in SICNNs......Page 299
 12.7 Note......Page 305
 References......Page 306
 13.1 Introduction and Preliminaries......Page 318
 13.2 Asymptotic Equivalence of Linear Systems and Asymptotically Almost Periodic Solutions......Page 321
 13.3 Asymptotic Equivalence of Linear and Quasilinear Systems......Page 324
 13.4 Bi-asymptotic Equivalence and Almost Periodic Solutions of Linear Systems......Page 327
 13.5 Note......Page 328
 References......Page 329
 14.1 On Asymptotic Equivalence of Impulsive Linear Homogeneous Differential Systems......Page 331
  14.1.1 Introduction......Page 332
  14.1.2 Ordinary Differential Equations......Page 333
  14.1.3 Delay Differential Equation......Page 334
  14.1.4 Example......Page 336
  14.2.1 Introduction and Preliminaries......Page 337
  14.2.2 Main Result......Page 338
 14.3 Asymptotic Equivalence for EPCAG......Page 340
  14.3.1 Introduction and Preliminaries......Page 341
  14.3.2 Main Results......Page 342
  14.4.1 Introduction and Preliminaries......Page 348
  14.4.2 Asymptotic Equilibriums......Page 350
  14.4.3 Asymptotic Equivalence......Page 357
 References......Page 361
Bibliography......Page 363
Index......Page 365