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Lichtenberg A.J., Liebermen M.A. Regular and Chaotic Dynamics
Lichtenberg A.J., Liebermen M.A.  Regular and Chaotic Dynamics









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: Regular and Chaotic Dynamics

: Lichtenberg A.J., Liebermen M.A.

:

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.


: en

: /

:

ed2k: ed2k stats

: second edition

: 1992

: 692

: 18.08.2008

: | | ID
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$f(\alpha)$      see Scaling index spectrum
Aamodt, R.E.      364 650
Abarbanel, H.D.I.      328 607
Ablowitz, M.J.      42
Accelerator modes, effect on diffusion      344 345349
Accelerator modes, for standard mapping      253
action      171172
Action integral, relation to integral invariant      1617
Action space      375380
Action space, for billiards problem      381383
Action space, for resonance streaming      410 417
Action, change in resonance crossing      369372
Action, change, for driven pendulum      233234
Action-angle variables      21 ff
Action-angle variables, for central force      35
Action-angle variables, for coupling resonance      395396
Action-angle variables, for harmonic oscillator      23
Action-angle variables, for pendulum      27 82
Action-angle variables, for two resonances      282283
Adiabatic barrier      see Barrier transition
Adiabatic invariant      16 24 71 93ff
Adiabatic invariant, by canonical methods      93ff
Adiabatic invariant, by noncanonical methods      102ff
Adiabatic invariant, from Lie transformations      141ff
Adiabatic invariant, of harmonic oscillator      see Harmonic oscillator
Ahlers, G.      635
Alexeev, M.V.      645
Alfven, H.      4 648
Angelopoulos, V.      240
Anosov, D.V.      305 610
Anosov-systems      see C-systems
Antonsen, T.M., Jr.      (654 Ott
Area-preserving transformation, condition for perturbed twist map      203
Area-preserving transformation, definition      19
Arnold diffusion      6062 238 373 401 646 647 649
Arnold diffusion, calculation of      386ff
Arnold diffusion, examples of      380ff
Arnold diffusion, for coupling resonance      393ff
Arnold diffusion, for many resonances      397ff
Arnold tongue      528
Arnold web      61 379383
Arnold, V.I.      3 5 14 59 60 61 66 163 175 185 188 197 198 302 306 307 373 380 528
Asteroid belt      646
Asymptotic expansions      see Expansions asymptotic
Attractors, basin boundaries      51ff 558ff
Attractors, basins of      461
Attractors, crises      see Crises
Attractors, definition of      461 ff
Attractors, semiattractor      576578
Attractors, simple      see Fixed point Sink Limit
Attractors, strange      see Strange attractors
Autocorrelation      342
Autocorrelation, for reconstruction      599
Autonomous Hamiltonian      see Hamiltonian autonomous
Avalanches      see Self-organized critically
Averaging, applied to harmonic oscillator      106109
Averaging, Kruskal's method      103106
Averaging, method of      71
Averaging, multiple time scale      103
Averaging, over slower of fast phases      111
Avez, A.      59 163 185 188 302 306 307
Bachelert, S.      (652 Vidal
Bak, P.      527 528 530 532 534 620 621 623 624 625 628
Baker's transformation      303 607
Bardet, R.      650
Barrier transition      246
Barrier transition, for electron cyclotron heating      650
Barrier transition, for Fermi acceleration      225
Barrier transition, for separatrix mapping      237
Barrier transition, for standard mapping      256
Barrier transition, for two resonances      287288
Barrier transition, summary of criteria for      291 ff
Basin of attraction      see Attractors
Baxter, D.C.      650
Beam-beam interaction      647
Behringer, R.P.      635
Belousov Zabotinsky reaction      652653
Belykh, V.N.      534
Ben-Misrachi, A.      538 545
Benettin, G.      245 298 299 301 305 311 315 316 317 319 438 445
Bensimon, D.      172 594
Benson, S.V.      (635 636 Gollub
Berge, P.      547
Berk, H.L.      651
Berman, C.P.      449
Berman, G.P.      654
Berman, R.H.      255
Bernoulli shifts      307309
Bernstein, G.M.      534 535 536 537 572 573
Bernstein, I.B.      650
Berry, M.V.      150 185 193 293 303 653 654
Bers, A.      87 126 (364 Ram
Berz, M.      173
Bhattacharee, A.      (342 Cary
Bialek, J.      256 279 280
Bifurcation theory for Hamiltonian maps      508ff
Bifurcations      461 ff
Bifurcations, exchange of stability      463464 484
Bifurcations, for one-dimensional quadratic map      482485
Bifurcations, for two-dimensional dissipative quadratic map      510512
Bifurcations, for two-dimensional Hamiltonian map      512ff
Bifurcations, Hopf      463464 465 541 634635
Bifurcations, of Fermi mapping      226229
Bifurcations, of standard mapping      254256
Bifurcations, period-doubling      209 463 see
Bifurcations, pitchfork      463464 484ff 511 540
Bifurcations, reverse      see Reverse bifurcations
Bifurcations, tangent (saddle-node)      209 462 464 496 538
Bifurcations, transcritical      463
Billiards problem      381ff
Billiards problem, coupling resonance      385386 393397
Billiards problem, thick layers      382 387389
Billiards problem, thin layers      382 389393
Birkhoff, G.D.      2 93 185 290 645
Bivins, R.L.      439 443 449
Bochelart, S.      (638 Roux
Boghosian, B.M.      45
Bogoliubov, N.N.      1 93 103 648
Boozer, A.      422
Born, M.      2 72 81 134 653
Bounds, T.C.      42 154 158 159 160 201 273 513
Bowen, R.      610
Brahic, A.      216
Brambilla, M.      427
Bridges, R.      581
Brillouin, L.      224
Brock, W.A.      598
Broomhead, P.S.      599 601 602
Broucke, R.      209 210
Bruhwiler, D.L.      364 370 371 372
Bruns, H.      645
Bunimovich, L.A.      67 342
Byers, J.A.      650 (650 Smith
C-systems      305307
Canonical perturbation      see Perturbation theory canonical
Canonical transformation, general theory      7ff
Canonical transformation, to rotating coordinates      111
Canonical variables      8
Cantor set      63 374375 588
Cantor set, two-scale      588589 593594
Cantorus      195 328
Capacity      see Fractal dimension
Capel, H.W.      (202 Roberts (219 van
Carlson, J.M.      620 625 626 627 629
Carreras, B.      436 437 438
Cary, J.R.      43 134 (134 Kaufman 136 151 152 342 364 369 370 371 372 425 651
Casartelli, M.      (311 Benettin
Casati, G.      312 506 644 653 654
Casimir      45ff 213215
Caustics      653
Central force      3336
Chaiken, J.      640
Chaitin, GJ.      309
Chandrasekhar, S.      293 408 648
Chao, A.W.      210
Characteristic exponents      see Liapunov exponents
Characteristic function      343
Chate, H.      619
Chen, C      523 524 525
Chernikov, A.A.      217
Chhabra, A.      597
Chillingworth, D.R.J.      561
Chirikov, B.V.      4 29 57 62 99 109 114 151 163 169 182 183 216 219 234 236 242 244 247 253 254 256 258 264 298 (312 Casati 313 315 334 364 369 (373 Gadiyak 374 385 392 393 395 397 398 399 403 404 405 406 412 413 418 437 438 443 444 570 644 646 647 648 649 (653 Casati
Chua, L.O.      458
Churchill, R.C.      645
Circle map      525ff
Coarse-graining      89 647 653
Cohen, R.H.      94 (216 226 254 328 Lichtenberg 406 412 413 416 648 649
Collet, P.      458 485 504 505 509 514
complexity      309
Conditionally periodic motion      22
Conditionally periodic motion, for central force      33
Conjugate variables      8
Constant of the motion      11
Continued fractions      273274 286
Contopoulos, G      179 326 645 646
Contraction of phase space volume      459466
Converse KAM theorem      289291
Coordinates, delay      598ff
Coordinates, generalized      8
Coordinates, noncanonical      44ff
Coordinates, rotating      109ff
Correlation dimension      see Fractal dimension
Correlation function, for quadratic map      506507
Correlations      342ff 506
Courant, E.D.      32 224 646 647
Crawford, J, D.      328 462 466 468
Cremers, J.      608 610
Crises      523 548ff
Crises, boundary crisis      551ff
Crises, exterior crisis      550
Crises, hysteresis      555
Crises, in one-dimensional maps      549551
Crises, interior crisis      550 553
Crises, transient chaos near      579ff
Crutchfield, J.      (68 Farmer 469 470 (478 Packard 557 605 608 614 619 638
Curry, J.H.      473 633 635
Dangelmayr, G.      640
Darboux's theorem      45
de Vogeleare, R.      172 201
Decomposable system      295 305
Degeneracy, accidental      113114 119 122123 178
Degeneracy, intrinsic      113117 178179
Degeneracy, intrinsic, for Kepler problem      36
Degeneracy, transition from accidental to intrinsic      124125
Degn, H.      652
Delay coordinates      578ff
Deprit, A.      2 134 136 645
Devil's staircase      526
Dewar, R.L.      134 136
Differential delay equation, Mackey Glass      598 603606
Differential equations, Floquet form for      31
Differential equations, for dissipative flows      458
Differential equations, linear      29ff
Differential equations, with periodic coefficients      3032
Diffusion      see also Fokker Planck equation
Diffusion coefficient      322
Diffusion coefficient, relation to friction coefficient      322
Diffusion for many degrees of freedom      437ff
Diffusion for many degrees of freedom, Fermi Pasta Ulam system      440
Diffusion for many degrees of freedom, for attracting sheets      440441
Diffusion for many degrees of freedom, for Lennard Jones potential      441442
Diffusion in toroidal magnetic fields      412 418ff
Diffusion in toroidal magnetic fields, magnetic islands      422425
Diffusion in toroidal magnetic fields, magnetic surfaces      420422
Diffusion in toroidal magnetic fields, mapping for      428ff
Diffusion in toroidal magnetic fields, numerical results      433435
Diffusion in toroidal magnetic fields, static fields      425ff
Diffusion in toroidal magnetic fields, tearing and disruptions in tokamaks      436438
Diffusion in toroidal magnetic fields, time varying fields      428ff
Diffusion of a parameter      413ff
Diffusion of a parameter, for oscillation center      414416
Diffusion of a parameter, in toroidal magnetic field      428ff
Diffusion of a parameter, mapping for      413415
Diffusion of a parameter, relation to resonance streaming      416418
Diffusion tensor      408411
Diffusion, across resonance layers      386387
Diffusion, along resonance layers      373 386 399ff see
Diffusion, Arnold      see Arnold diffusion
Diffusion, banana      412 427428 434
Diffusion, coupling resonance      393ff
Diffusion, extrinsic      see Extrinsic diffusion
Diffusion, Fourier path technique      329ff
Diffusion, higher order      329ff
Diffusion, in action space      320ff
Diffusion, in billiards problem      see Billiards problem
Diffusion, many resonance      397ff
Diffusion, modulational      see Modulational diffusion
Diffusion, neoclassical      428
Diffusion, oscillation center      see Resonance streaming
Diffusion, Pfirsch Schliiter      428
Diffusion, plateau      427428 433435
Diffusion, pseudo-classical      433
Diffusion, quasilinear      see Quasilinear diffusion
Diffusion, self-consistent problem      436ff
Digital phase-locked loop, circle map representation      534537
Digital phase-locked loop, transient chaos in      572
DIMENSION      see Fractal dimension
Dipole, axisymmetric magnetic      648
Dissipative systems      62ff 570ff
Dissipative systems, resonance streaming in      412
Dorizzi, B.      43
Doveil, F.      249 274 281 284 286 287 289
Dragt, A.J.      134 168 186 187 648 649
Driscoll, C.F.      439 448 449
Dubois, M.      548
Duffing's equation      565567
Duffing's equation, chaotic motion in      567
Dunnett, D.A.      72 110 128 147
Earthquake model      625628
Ecke, R.E.      528 529
Eckmann, J.-P.      458 485 504 509 513 514 598 635 (635 Collet
Edmonds, P.H.      650
Eigenvalues      196ff
Eigenvalues, for Liapunov exponents      300
Eigenvalues, for two-dimensional mappings      200ff
Eigenvalues, of cat mapping      306307
Eigenvalues, symmetry of      197199
Eigenvectors      196ff
Eigenvectors, for Liapunov exponents      300
Eigenvectors, for two-dimensional mappings      200ff
Eigenvectors, of cat mapping      306307
Einstein, A.      101
Eldridge, O.      650
Electron cyclotron resonance heating      650651
Elliptic orbits, for two-dimensional mapping      205206
Elliptic points      see Fixed point elliptic
Eminhizer, C.R.      152 154 195 201
Energy surface      61 376380 406407
EnergyCasimir method      213215
entropy      308309 446 609 see
Equipartition of energy      445447 452453
Equipartition of energy, relaxation time      447
Ergodic systems      5960 295ff 308
Ergodic systems, baker's transformation      5960
Ergodic systems, hard sphere gas      5960
Ergodic systems, Lorenz attractor      65ff
Errors, numerical      see Numerical errors
Errors, roundoff      see Roundoff errors
Escande, D.F.      250 274 279 281 284 286 287 289 (371 Cary
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