Perspectives of nonlinear dynamics.

Contents of Volume\0......Page 7
Preface\0......Page 15
Acknowledgements\0......Page 18
6.1 Some origins of maps in R^2\0......Page 19
6.2 Rotation and winding numbers: Maps and flows; knots, algebraic constants of the motion\0......Page 27
6.3 The Cartwright-Littlewood, Levinson and Levi analyses\0......Page 34
6.4 Some abstract nonconservative maps in R^2\0......Page 40
6.5 The standard map; twist maps\0......Page 51
6.6 \'Near-integrable\' systems\0......Page 59
6.7 The breakup of KAM curves\0......Page 75
6.8 Physical regularity in mathematical chaos\0......Page 83
6.9 Chirikov\'s resonance-overlap criterion: A periodically kicked oscillator example\0......Page 85
6.10 The numerical Poincare map and discontinuous dynamics: Henon\'s integration method\0......Page 90
6.11 The Henon-Heiles and Toda-Hamiltonian systems: Poincare maps for a nonintegrable and integrable system\0......Page 92
6.12 Abstract area-preserving maps on R^2 and T^2\0......Page 98
6.13 Maps of sets: The folds and kinks of a periodically forced conservative oscillator\0......Page 108
6.14 Maps on a lattice: Rannou\'s study of the standard map and generalization on a N x N lattice\0......Page 111
6.15 Dynamic entropies and information production: The shuffling of partitions in phase space\0......Page 119
6.16 Epilogue: order-order, order-chaos, chaos-chaos in the house!\0......Page 136
7 Models based on third order differential systems\0......Page 143
7.1 Linear third order equations: Characterization of fixed points\0......Page 144
7.2 Nonlinear flows\0......Page 150
7.3 The Lorenz model\0......Page 156
7.4 Lorenz chaotic dynamics\0......Page 163
7.5 A \'Lorenz-dynamic\'.fuid system\0......Page 169
7.6 Dynamo dynamics\0......Page 173
7.7 The Lorenz homoclinic and heteroclinic bifurcations\0......Page 180
7.8 The Lorenz-Hopf bifurcation: The subcritical bifurcation\0......Page 195
7.9 Lorenz dynamics for various parameter values\0......Page 196
7.10 The Lyapunov exponents\0......Page 208
7.11 Rossler\'s models\0......Page 216
7.12 Lyapunov exponents and the dimension of a strange attractor\0......Page 235
7.13 Open systems - chemical oscillations\0......Page 240
8 \'Moderate-order\' systems\0......Page 249
8.1 Linear systems\0......Page 250
8.2 Turing\'s linear chemical morphogenesis system\0......Page 258
8.3 \'Integrable\' Hamiltonian systems\0......Page 264
8.4 The Kolmogrov-Arnold-Moser theorem: `Near-integrable\' systems\0......Page 271
8.5 Poincare\'s and Fermi\'s theorems; Arnold diffusion.\0......Page 274
8.6 The Fermi-Pasta-Ulam phenomenon and equipartitioning\0......Page 277
8.7 Molecular models: Polynomial potentials; Toda\'s exponential potential\0......Page 286
8.8 Toda\'s solitary waves in a lattice\0......Page 292
8.9 The dynamics of various Toda lattices\0......Page 299
8.10 The Painleve property and integrability conjecture\0......Page 312
8.11 Chemical oscillations and dissipative open-system structures\0......Page 322
8.12 Smale\'s analysis of Turing\'s morphogenic system\0......Page 329
8.13 Embedding the dynamics of high-order dissipative systems in lower dimensional R\0......Page 335
8.14 Some dynamics of living systems\0......Page 348
8.15 Epilogue: open systems; open sesame!\0......Page 361
9 Solitaires: solitons and nonsolitons\0......Page 366
9.1 The continuum limit of lattices and \'solitaire\' solutions\0......Page 368
9.2 Riemann invariants and the Korteweg-de Vries (KdV) equation\0......Page 376
9.3 A comparison of the Burgers and KdV equations: Dissipation vs dispersion\0......Page 379
9.4 The exact solution of Burgers equation - The Hopf-Cole transformation\0......Page 383
9.5 A brief history leading to the inverse scattering transform (1ST)\0......Page 385
9.6 The general solution of the KdV equation\0......Page 391
9.7 Pure soliton solutions: KdV equation; the Landau-Lifshitz subset\0......Page 399
9.8 The Lax formulation: The KdV example\0......Page 405
9.9 The sine-Gordon equation: Kinks; topological solitons; breather modes\0......Page 408
9.10 Hirota\'s \'direct method\' in soliton theory\0......Page 416
9.11 The AKNS formulation of the 1ST\0......Page 424
9.12 Some Backlund transformations between difference equations\0......Page 428
9.13 Invariant Backlund transformations\0......Page 431
9.14 Infinite number of conservation laws\0......Page 436
9.15 Onward: Higher dimensions; resonant interactions\0......Page 438
10.1 An overview: Lagrangian and Eulerian models, with continuous and discrete variables and functions\0......Page 445
10.2 Some coupled maps (CM)\0......Page 449
10.3 Coupled lattice maps (CLM = CA)\0......Page 463
10.4 General cellular automata (CA): Von Neumann\'s question; Ulam\'s suggestion; dynamics defined\0......Page 472
10.5 \'Legal\' cellular automata: Quiescent and symmetry conditions\0......Page 475
10.6 A general association for legal CA: Some possible physical relations to CA rules, states and configurations\0......Page 477
10.7 Simple examples: \'Self-reproduction\'\0......Page 478
10.8 Neighborhood configurations and dynamic rules: Rule number; `totalistic\' CA\0......Page 480
10.9 Several classifications of CA dynamic properties: Four qualitative categories of dynamics\0......Page 482
10.10 Entropies of one-dimensional CA: Some possibilities and shortcomings of entropy measurements of `chaos\' or `turbulence\'\0......Page 491
10.11 Particle-like dynamics from partial CA rules: Colliding `particles\' with and without delays; oscillating `molecule\'\0......Page 496
10.12 Two-dimensional CA: Von Neumann neighborhoods; Fredkin\'s \'self-reproducing\' rule\0......Page 500
10.13 Garden-of-Eden configurations: Configurations which cannot dynamical arise; Moore\'s theorem\0......Page 502
10.14 J.H. Conway\'s \'Life\': Moore neighborhood; snakes, ponds, blinkers, beehives, barges, barberpoles, eaters, gliders and glider guns\0......Page 505
10.15 Excitable medium: Quiescent, excited, and refractory states\0......Page 509
10.16 Invertible CA and physical dynamics: Invertible vis-d-vis reversible dynamics; the description of nature, how can it best be achieved?\0......Page 511
Appendix: von Neumann\'s questions\0......Page 517
Epilogue: `Understanding\' complex systems: Order; organization; Endnote-models, causality, irreversibility\0......Page 523
J On the Cartwright-Littlewood and Levinson studies of the forced relaxation oscillator\0......Page 547
K Smale\'s horseshoe map\0......Page 557
L Notes on the Kolmogorov-Arnold-Moser theorem\0......Page 561
Bibliography\0......Page 571
References added at 1991 reprinting\0......Page 639
Cumulative index (Volumes 1 and 2)\0......Page 641