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Lattice Dynamical Systems&Shui-Nee Chow1

1 Spatially Discrete Nonlinear Diffusion Equations1

1.1 Introduction1

1.2 Spatially Discrete Models2

1.3 Gradient Systems9

1.4 Numerical Methods13

1.5 Numerical Results15

2 Dynamics in a Discrete Nagumo Equation-Spatial Chaos20

2.1 Introduction20

2.2 Basic Facts23

2.3 Spatial Chaos26

2.4 Non-existence of Traveling Waves31

2.5 Non-existence of Standing Waves35

3 Pattern Formation and Spatial Chaos in Lattice Dynamical Systems36

3.1 Introduction36

3.2 Lattice Dynamical Systems37

3.3 Bifurcation of Checkerboard and Stripe Patterns41

3.4 Traveling Waves and Prorogation Failure43

3.5 Homoclinic Points of ZZD Actions45

4 Pattern Formation and Spatial Chaos in Spatially Discrete Evolution Equations46

4.1 Introduction46

4.2 Existence and Uniqueness of Solutions47

4.3 Mosaic Solutions：Existence and Stability53

4.4 Examples of Two-Dimensional Mosaic Solutions58

5 Spatial Entropy and Spatial Chaos64

5.1 Spatial Entropy，Spatial Chaos and Pattern Formation64

5.2 Rigorous Bounds on the Spatial Entropy72

5.3 One-Dimensional Mosaics76

6 Synchronization in Lattice Dynamical Systems80

6.1 Introduction80

6.2 Statement of Results81

6.3 Proof of Theorem 6.183

6.4 Proof of Theorem 6.286

7 Synchronization，Stability and Normal Hyperbolicity90

7.1 Introduction90

7.2 Definitions and Results90

7.3 Synchronization of Fully Coupled Systems93

7.4 Master-Slave Synchronization and Partially Coupled System95

References99

Totally Bounded Cubic Systems in R2&Roberto Conti，Marcello Galeotti103

1 Finitely Many Singular Points103

1.1 Totally Bounded Cubic Systems103

1.2 Singular Points104

1.3 Annulus of the System104

1.4 Petals105

1.5 A Unique Singular Point105

2 Two Singular Points106

2.1 Total Boundedness and Reversibility106

2.2 The Case D＝0108

2.3 Uniqueness of the Singular Points S，O110

2.4 Conditions of Total Boundedness111

2.5 Regular Systems116

2.6 Limit Lines at S116

2.7 The Cuspidal Case117

2.8 One Vertical Limit Line at S119

2.9 Three Limit Lines at S124

2.10 Petal at S127

2.11 The Tangential Limit Point O127

2.12 O Center133

2.13 O Poincare Center133

2.14 O Degenerate Center136

2.15 O Right Pseudo-center140

2.16 O Left Pseudo-center142

2.17 O Poincare Saddle146

2.18 O Degenerate Saddle148

2.19 Two Heteroclines with indexO＝1152

2.20 An Invariant Ellipse157

2.21 An Invariant Ellipse.Two Heteroclines159

2.22 An Invariant Ellipse.Infinitely Many Heteroclines163

2.23 Infinitely Many Heteroclines with No Invariant Ellipse165

3 Limit Cycles168

3.1 Limit Cycles of Cubic Totally Bounded Systems168

References171

Non-Autonomous Differential Equations&Russell Johnson，Francesca Mantellini173

1 Introduction173

2 Basic Methods179

3 Cantor Spectrum for Quasi-Periodic Schroedinger Operators193

4 Almost Automorphy in Semilinear Parabolic PDEs206

5 Positive Solutions of the Scalar Curvature Equation213

References223

Traveling Waves in Spatially Discrete Dynamical Systems of Diffusive Type&John Mallet-Paret231

1 Introduction231

2 Equilibria，Stability，and Patterns235

3 Traveling Waves in PDE’s239

4 Traveling Waves in Lattice Differential Equations：General Features245

5 An Abstract Approach to Traveling Waves248

6 Linear Theory of Differential-Difference Equations252

7 The Global Structure of Traveling Waves in LDE’s259

8 Pinning269

9 Crystallographic Pinning274

10 Stability，Perturbations，and Time-Discretizations281

11 Exponential Dichotomies for Linear Systems of Mixed Type286

References293

Limiting Profiles for Solutions of Differential-Delay Equations&Roger D.Nussbaum299

1 Background Material：The Fixed Point Index and Measures of Noncompactness299

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

References340