Complex hamiltonian dynamics

Cover......Page 1
Complex Hamiltonian Dynamics......Page 4
Foreword......Page 8
Preface......Page 10
Acknowledgments......Page 16
Contents......Page 18
Acronyms......Page 22
1.1 Preamble......Page 25
1.2 Lyapunov Stability of Dynamical Systems......Page 27
1.3 Hamiltonian Dynamical Systems......Page 31
1.4 Complex Hamiltonian Dynamics......Page 34
2.1 The Case of N=1 Degree of Freedom......Page 37
2.2 The Case of N=2 Degrees of Freedom......Page 43
2.2.1 Coordinate Transformations and Solution by Quadratures......Page 44
2.2.2 Integrability and Solvability of the Equations of Motion......Page 48
2.3 Non-autonomous One Degree of Freedom Hamiltonian Systems......Page 54
2.3.1 The Duffing Oscillator with Quadratic Nonlinearity......Page 55
2.3.2 The Duffing Oscillator with Cubic Nonlinearity......Page 58
Exercises......Page 61
Appendix B: Example algorithms for the computation of the SALI and GALI chaos indicators......Page 63
3.1.1 Equilibrium Points......Page 64
3.1.2 Periodic Orbits......Page 67
3.2 Linear Stability Analysis......Page 69
3.2.1 An Analytical Criterion for ``Weak'' Chaos......Page 74
3.3.1 Lyapunov Spectra and Their Convergence......Page 76
3.3.1.1 Lyapunov Spectra and the Thermodynamic Limit......Page 78
3.4 Distinguishing Order from Chaos......Page 79
3.4.1 The SALI Method......Page 81
3.4.2 The GALI Method......Page 82
Problems......Page 84
4.1 Normal Modes of Linear One-Dimensional Hamiltonian Lattices......Page 86
4.2 Nonlinear Normal Modes (NNMs) and the Problemof Continuation......Page 88
4.3.1 NNMs as One-Dimensional Bushes......Page 90
4.4 A Group Theoretical Study of Bushes......Page 93
4.4.1 Subgroups of the Parent Group and Bushesof NNMs......Page 95
4.4.2 Bushes in Modal Space and Stability Analysis......Page 97
4.5.1 Bushes of NNMs for a Square Molecule......Page 103
4.5.2 Bushes of NNMs for a Simple Octahedral Molecule......Page 107
Problems......Page 111
5.1 Variational Equations and Tangent Map......Page 113
5.2 The SALI Method......Page 116
5.3 The GALI Method......Page 124
5.3.1.1 Exponential Decay of GALI for Chaotic Orbits......Page 126
5.3.1.2 The Evaluation of GALI for Ordered Orbits......Page 129
5.3.2.1 Low-Dimensional Hamiltonian Systems......Page 133
5.3.2.2 High-Dimensional Hamiltonian Systems......Page 139
5.3.2.3 Symplectic Maps......Page 142
5.3.2.4 Motion on Low-Dimensional Tori......Page 144
Exercises......Page 148
A MAPLE Algorithm for the Computation of the GALI......Page 150
Exercises......Page 152
Problems......Page 153
6.1.1 Historical Remarks......Page 155
6.1.2 The Concept of q-breathers......Page 158
6.1.3 The Concept of q-tori......Page 160
6.2 Existence and Stability of q-tori......Page 161
6.2.1 Construction of q-tori by Poincaré-Linstedt Series......Page 162
6.2.2 Profile of the Energy Localization......Page 169
6.3 A Numerical Study of FPU Trajectories......Page 174
6.3.1 Long Time Stability near q-tori......Page 177
6.4 Diffusion and the Breakdown of FPU Recurrences......Page 179
6.4.1 Two Stages of the Diffusion Process......Page 180
6.4.2 Rate of Diffusion of Energy to the Tail Modes......Page 182
6.4.3 Time Interval for Energy Equipartition......Page 184
Problems......Page 186
7.1 Introduction and Historical Remarks......Page 187
7.1.1 Localization in Configuration Space......Page 188
7.2 Discrete Breathers and Homoclinic Dynamics......Page 190
7.2.1 How to Construct Homoclinic Orbits......Page 193
7.3 A Method for Constructing Discrete Breathers......Page 196
7.3.1 Stabilizing Discrete Breathers by a Control Method......Page 197
7.4.1 Anderson Localization in Disordered Linear Media......Page 202
7.4.2.1 Two Basic Models......Page 205
7.4.2.2 Regimes of Wave Packet Spreading......Page 207
7.4.2.3 Numerical Results......Page 208
Exercises......Page 210
Problems......Page 211
8.1 From Deterministic Dynamics to Statistical Mechanics......Page 213
8.1.1 Nonextensive Statistical Mechanics and q-Gaussian pdfs......Page 215
8.2 Statistical Distributions of Chaotic QSS and Their Computation......Page 217
8.3 FPU -Mode Under Periodic Boundary Conditions......Page 219
8.3.1 Chaotic Breathers and the FPU -Mode......Page 222
8.4 FPU SPO1 and SPO2 Modes Under Fixed Boundary Conditions......Page 225
8.5 q-Gaussian Distributions for a Small Microplasma System......Page 230
8.6 Chaotic Quasi-stationary States in Area-Preserving Maps......Page 234
8.6.1 Time-Evolving Statistics of pdfs in Area-Preserving Maps......Page 235
8.6.2 The Perturbed MacMillan Map......Page 236
8.6.2.1 The =0.9, =1.6 Class of Examples......Page 237
8.6.2.2 The =1.2, =1.6 Class of Examples......Page 239
Problems......Page 241
9.1 Conclusions......Page 243
9.2.1 Singularity Analysis: Where Mathematics Meets Physics......Page 246
9.2.2 Nonlinear Normal Modes, Quasiperiodicityand Localization......Page 248
9.2.3 Diffusion, Quasi-stationary States and Complex Statistics......Page 251
9.3.1 Anomalous Heat Conduction and Control of Heat Flow......Page 253
9.3.2 Complex Soliton Dynamics in Nonlinear PhotonicStructures......Page 254
9.3.3 Kinetic Theory of Hamiltonian Systems and Applications to Plasma Physics......Page 257
References......Page 261
Index......Page 275