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Lattice Dynamical Systems&Shui-Nee Chow 1

1 Spatially Discrete Nonlinear Diffusion Equations 1

1.1 Introduction 1

1.2 Spatially Discrete Models 2

1.3 Gradient Systems 9

1.4 Numerical Methods 13

1.5 Numerical Results 15

2 Dynamics in a Discrete Nagumo Equation-Spatial Chaos 20

2.1 Introduction 20

2.2 Basic Facts 23

2.3 Spatial Chaos 26

2.4 Non-existence of Traveling Waves 31

2.5 Non-existence of Standing Waves 35

3 Pattern Formation and Spatial Chaos in Lattice Dynamical Systems 36

3.1 Introduction 36

3.2 Lattice Dynamical Systems 37

3.3 Bifurcation of Checkerboard and Stripe Patterns 41

3.4 Traveling Waves and Prorogation Failure 43

3.5 Homoclinic Points of ZZD Actions 45

4 Pattern Formation and Spatial Chaos in Spatially Discrete Evolution Equations 46

4.1 Introduction 46

4.2 Existence and Uniqueness of Solutions 47

4.3 Mosaic Solutions：Existence and Stability 53

4.4 Examples of Two-Dimensional Mosaic Solutions 58

5 Spatial Entropy and Spatial Chaos 64

5.1 Spatial Entropy，Spatial Chaos and Pattern Formation 64

5.2 Rigorous Bounds on the Spatial Entropy 72

5.3 One-Dimensional Mosaics 76

6 Synchronization in Lattice Dynamical Systems 80

6.1 Introduction 80

6.2 Statement of Results 81

6.3 Proof of Theorem 6.1 83

6.4 Proof of Theorem 6.2 86

7 Synchronization，Stability and Normal Hyperbolicity 90

7.1 Introduction 90

7.2 Definitions and Results 90

7.3 Synchronization of Fully Coupled Systems 93

7.4 Master-Slave Synchronization and Partially Coupled System 95

References 99

Totally Bounded Cubic Systems in R2&Roberto Conti，Marcello Galeotti 103

1 Finitely Many Singular Points 103

1.1 Totally Bounded Cubic Systems 103

1.2 Singular Points 104

1.3 Annulus of the System 104

1.4 Petals 105

1.5 A Unique Singular Point 105

2 Two Singular Points 106

2.1 Total Boundedness and Reversibility 106

2.2 The Case D＝0 108

2.3 Uniqueness of the Singular Points S，O 110

2.4 Conditions of Total Boundedness 111

2.5 Regular Systems 116

2.6 Limit Lines at S 116

2.7 The Cuspidal Case 117

2.8 One Vertical Limit Line at S 119

2.9 Three Limit Lines at S 124

2.10 Petal at S 127

2.11 The Tangential Limit Point O 127

2.12 O Center 133

2.13 O Poincare Center 133

2.14 O Degenerate Center 136

2.15 O Right Pseudo-center 140

2.16 O Left Pseudo-center 142

2.17 O Poincare Saddle 146

2.18 O Degenerate Saddle 148

2.19 Two Heteroclines with indexO＝1 152

2.20 An Invariant Ellipse 157

2.21 An Invariant Ellipse.Two Heteroclines 159

2.22 An Invariant Ellipse.Infinitely Many Heteroclines 163

2.23 Infinitely Many Heteroclines with No Invariant Ellipse 165

3 Limit Cycles 168

3.1 Limit Cycles of Cubic Totally Bounded Systems 168

References 171

Non-Autonomous Differential Equations&Russell Johnson，Francesca Mantellini 173

1 Introduction 173

2 Basic Methods 179

3 Cantor Spectrum for Quasi-Periodic Schroedinger Operators 193

4 Almost Automorphy in Semilinear Parabolic PDEs 206

5 Positive Solutions of the Scalar Curvature Equation 213

References 223

Traveling Waves in Spatially Discrete Dynamical Systems of Diffusive Type&John Mallet-Paret 231

1 Introduction 231

2 Equilibria，Stability，and Patterns 235

3 Traveling Waves in PDE’s 239

4 Traveling Waves in Lattice Differential Equations：General Features 245

5 An Abstract Approach to Traveling Waves 248

6 Linear Theory of Differential-Difference Equations 252

7 The Global Structure of Traveling Waves in LDE’s 259

8 Pinning 269

9 Crystallographic Pinning 274

10 Stability，Perturbations，and Time-Discretizations 281

11 Exponential Dichotomies for Linear Systems of Mixed Type 286

References 293

Limiting Profiles for Solutions of Differential-Delay Equations&Roger D.Nussbaum 299

1 Background Material：The Fixed Point Index and Measures of Noncompactness 299

2 The Singularly Perturbed Differential-Delay Equation ex’（t）＝-x（t）＋f（x（t-1）） 310

3 The Differential-Delay Equation εx’（t）＝f（x（t），x（t-r（x（t））） 324

4 Ω-Theory for Solutions of εx’（t）＝f（x（t），x（t-r）），r：＝r（x（t）） 332

References 340