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| Hale J.K., Kocak H. — Dynamics and Bifurcations |
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| Предметный указатель |
Henon — Heiles Hamiltonian 531 533 534 534r
Henon — Heiles Hamiltonian, generalized 534
hessian 278 433 542
Heteroclinic orbit 184 200 392 435 436
Heteroclinic orbit breaking 292 303 304 428 430 431 437
Hilbert’s sixteenth problem 365 385 388r
Hill’s equation 264r
Homeomorphism 62 63 64 237
Homeomorphism of linear systems 245
Homoclinic bifurcation 210 385 386 388r 401 402 407 408
Homoclinic orbit 184 202 210 212 213 392 416 417 425—430 439 522r
Homoclinic orbit as limit set 366 368
Homoclinic orbit breaking 210 213 292
Homoclinic orbit in three-space 517 518
Homoclinic orbit, elementary 398 399 401
Homoclinic orbit, period of 214
Homoclinic point 458 493r
Homoclinic point in forced damped Duffing 506
Homoclinic point tangency 506 509r
Homoclinic point, transversal 458 459 502 504 509r
Hopf bifurcation theorem see “Poincare — Andronov — Hopf”
Hopf map 534r
Horseshoe 509r
Hydroplane 40
Hyperbolic equilibrium point 19 20 26 43 301 391
Hyperbolic equilibrium point, saddle 239
Hyperbolic equilibrium point, sink 239
Hyperbolic equilibrium point, source 239
Hyperbolic fixed point 76 82 83 454
Hyperbolic linear map 445—449 451
Hyperbolic linear systems 237 238
Hyperbolic linear systems, genericity of 248
Hyperbolic periodic orbit 377 392
Hyperbolic periodic solution 131
Hyperbolic Umbilic 439
Hyperbolicity and finiteness 382
Hysteresis 30 31 32 38 58 59
Hysteresis loop 32
Implicit function theorem 41 42 43 47 80 344 541
Independent variable 4
Index theory, Conley, homotopy 305r
Indirect control 287
Inertial manifold 332r
Infinite dimensional dynamical system 538
Initial-value problem 4 175
Instability from Cetaev’s theorem 284
Instability from linearization 272 274
Integrable Hamiltonian 523—530 525 527 534r
Intermediate-value theorem 80 539
Invariance principle 287—291 288 305r 395 434
Invariant closed curve 473 475 478 481 482 493r
Invariant closed curve, disintegration of 479 489 490
Invariant cylinder 498 499
Invariant manifold see “Center manifold” “Stable
Invariant set 287 305r
Invariant set for maps 445
Invariant set, negatively 287
Invariant set, positively 287
Invariant torus bifurcation 514 516 522r
Invariant torus Hamiltonian 528
Invariant torus, disintegration of 532
Inverse function theorem 541
Irrational flow on torus 154
Islands 489
Isolating block 305r
Jacobian matrix 267 454
Jordan curve theorem 366 376 387r
Jordan normal form 223 228 231 263r
Jordan normal form and linear equivalence 237
Jordan normal form and maps 445
Kam theorem 494r (see also “Twist theorem”)
Kepler motion 424
kinetic energy 414
Kirchkoff’s law 173
Lagrange libration point 531
Lagrange multiplier 526
Level set 195 278 414 524—531
Level set crossing 280
Liapunov exponents 537
Liapunov function 277—280 281 305r
Liapunov function for maps 462
Liapunov function, converse theorems for 283 305r
Liapunov function, strict 281
Liapunov — Schmidt method 332r 362r
Liapunov’s thesis 313
Lienard’s equation 173
Limit cycle 181 193 366 384 385 386 387r 408 498
Limit cycle and Euler 467
Limit cycle in three-space 513 516
linear combination 218
Linear gradient system 438
Linear harmonic oscillator 171 177 179 182 184 195 196
Linear harmonic oscillator and numerics 183 184 463 464 467
Linear harmonic oscillator, forced 503
Linear harmonic oscillator, pair of 526 527 530 534r
Linear independence 219
Linear map 73 74 445 445—453
Linear map, hyperbolic 445—449 451
Linear map, nonhyperbolic 448 453
Linear map, topological classification of 454r
Linear periodic equation 123 256—263
Linear system 218 218—255
Linear system and Euler’s algorithm 183 463
Linear system with one-periodic coefficients 256—263
Linear system, bifurcations of 247—253
Linear system, canonical 223 228—236
Linear system, hyperbolic 237—238 240
Linear system, nonhomogeneous 253—254
Linear system, nonhyperbolic 239
Linear system, stiff 236
Linear system, topological classification of 237—247
Linear variational equation 19 130 267
Linearization 19 265—277 267 301
Linearization at a fixed point 73 454 460
Linearization of a periodic orbit 381
Linearization of pendulum 271 275
Linearization, asymptotic stability from 266—272 269 271
Linearization, instability from 272—275 273 275
Linearization, when insufficient 275
Linearly equivalent 237 238
Liouville’s formula 256
Lissajous figure 529
Local transversal section 375 376 382 500 512 513
Logistic differential equation 68 69
Logistic differential equation and Euler 68
Logistic differential equation, generalized 128
Logistic differential equation, pair of 192
Logistic differential equation, periodic 128
Logistic map 92 92—104 100
Logistic map and Henon 461 465
Logistic map, bifurcation diagram of 100
Logistic map, chaos in 99
Logistic map, period-doubling in 95 97 98 99 101
Logistic map, strong resonances of 483
Lorenz equations 519 520 521 522r
Lozi map 461
Lyapunov see “Liapunov”
MAP 67—103 72 443—495
Map, monotone 81 103r
Mathieu’s equation 259 263 264r 492
Mathieu’s equation, characteristic multipliers of 260
Mathieu’s equation, zones of stability of 261 262
Matrix exponential 220 (see also “Principal matrix solution”)
Matrix exponential of commuting matrices 223 226
Matrix solution 219
Maximal interval of existence 6 7 109 110 542
Mean-value theorem 20 539
Measure zero 525
Mechanics 215r 534r
Membrane 435 436 439
| Metric for bi-infinite sequences 504
Mixed difference algorithm 468 471
MOD 149
Modulus 451
Momentum variable 524
Monotone map 81
Morse function 278 440r
Morse lemma 542
Negative definite function 278
Negative orbit 9 10 11
Negative orbit of maps 444
Nerve axon equation 522r
Neutral damping 424
Newton’s law 170
Newton’s method 71 75 76 104r
Node 188
Nonautonomous 107—144 108 111 256—263 497—510
Nondegenerate critical point 415 433 438
Nonhomogeneous linear system 253
Nonhyperbolic equilibrium 308
Nonhyperbolic equilibrium and computing 311
Nonhyperbolic fixed point 76 77 78 79 84
Nonhyperbolic linear map 448 453
Nonhyperbolic periodic orbit 381
Nonhyperbolic periodic solution 137
Nonintegrable Hamiltonian 531 533
Nonperiodic orbit 99 460 479 503 520
Nonresonance condition 474 480 486
Norm 175 215r 390 410r
Norm equivalence 175 176
Norm, Euclidean 175 176
Norm, sup 175 176
Normal form theory 146r 411r
Numerical analysis and dynamics 493r
O, big and small 539
Odd symmetry 408
Ohm’s Law 172
Omega-limit point 183
Omega-limit point of map 445
Omega-limit set 14 183 291 366 193
Omega-limit set and center manifold 327
Omega-limit set of a set 394
Omega-limit set of gradient system 434
Omega-limit set of map 445
Omega-limit set, connectedness of 288
Omega-limit setm homoclinic loop as 368
One-parameter bifurcations see “Codimension-one bifurcations”
One-periodic equation 113 113—146 132r
One-periodic equation on cylinder 118—121
One-periodic equation, bifurcations of 133—146
One-periodic equation, hyperbolic solution of 127 131
One-periodic equation, limit sets of 116
One-periodic equation, linear 123 256—264
One-periodic equation, nonhyperbolic solution of 135—140 137
One-periodic equation, Poincare map of 116 119 121 134
One-periodic equation, saddle-node bifurcation in 145
One-periodic equation, translation invariance of 114
Orbit 9 10 11 177 178
Orbit of maps 444
Orbit of nonautonomous equation 108
Orbit structure 27 62
Orbit, negative 9 10 11 178
Orbit, positive 9 10 11 178
Orbitally asymptotically stable 341
Orbitally stable 341
Orbitally unstable 341
Ordinary point 185 187
Organizing center 537
Parabolic 485
Parallel flow on torus 150 151 500 528
Pendulum, planar 170 174 180 195 199 201 281
Pendulum, planar, damped 269 271 273 275
Pendulum, planar, damped with torque 317
Pendulum, planar, first integral for 195 196 200 281
Pendulum, planar, on cylinder 201 202
Pendulum, planar, period of 203
Pendulum, planar, rotated 331 432
Pendulum, planar, spherical 534r
Period 54 56 87 114 179 182
Period in conservative system 203
Period near homoclinic loop 214
Period of pendulum 203
Period-doubling bifurcation 87—92 88 468
Period-doubling bifurcation in Duffing 514
Period-doubling bifurcation in Henon 471 472
Period-doubling bifurcation in logistic map 94—97 95 98 100 101 103r
Period-doubling bifurcation in Lorenz 522
Period-doubling bifurcation in three-space 512 513 514 522r
Period-doubling bifurcation, bifurcation diagram of 87
Periodic equation see “One-periodic equation”
Periodic orbit 179 365—388
Periodic orbit bifurcation 382
Periodic orbit from homoclinic loop 385 386
Periodic orbit of maps 87 445
Periodic orbit, hyperbolic 377 392
Periodic orbit, linearization of 381
Periodic orbit, nonexistence of 373
Periodic orbit, nonhyperbolic 381
Periodic orbit, quasi-hyperbolic 398 399 401
Periodic orbit, saddle-node bifurcation of 384 401
Periodic point of maps 87 445
Periodic solution 54 56 114 116 179
Periodic solution, asymptotically stable 114—115
Periodic solution, derivative of Poincare map 130 136
Periodic solution, hyperbolic 127 131
Periodic solution, nonhyperbolic 131 137 141 143
Periodic solution, stable 114—115
Periodic solution, unstable 114—115
Periodic vector field 21
Perturbation 26
Phase portrait 10 12 13 14 178
Phase portrait of linear systems 242
Phase portrait on circle 22 55
Phaser IX 8
PHASER, equations in the library of: anosov 492
PHASER, equations in the library of: arnold 165
PHASER, equations in the library of: cremona 491
PHASER, equations in the library of: discubic 102
PHASER, equations in the library of: dislin2d 453
PHASER, equations in the library of: dispprey 484
PHASER, equations in the library of: dzero1 409
PHASER, equations in the library of: dzero2 410
PHASER, equations in the library of: dzero3 410
PHASER, equations in the library of: forceduf 504
PHASER, equations in the library of: forcevdp 501
PHASER, equations in the library of: gauss 468
PHASER, equations in the library of: gingerman 492
PHASER, equations in the library of: gradient 440
PHASER, equations in the library of: harmoscil 530
PHASER, equations in the library of: henheile 534
PHASER, equations in the library of: henon 461
PHASER, equations in the library of: hilbert2 385
PHASER, equations in the library of: hilbert4 388
PHASER, equations in the library of: hopf 214
PHASER, equations in the library of: icremona 491
PHASER, equations in the library of: linear2d 184 214
PHASER, equations in the library of: lorenz 522
PHASER, equations in the library of: lozi 461
PHASER, equations in the library of: mathieu 492
PHASER, equations in the library of: mod 155
PHASER, equations in the library of: pendulum 184
PHASER, equations in the library of: pitchfork 214
PHASER, equations in the library of: predprey 184
PHASER, equations in the library of: rossler 514
PHASER, equations in the library of: saddlenod 214
PHASER, equations in the library of: silnikov 518
PHASER, equations in the library of: silnikov2 519
PHASER, equations in the library of: singer 103
PHASER, equations in the library of: tent 103
PHASER, equations in the library of: zeroim 515
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