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| Hale J.K., Kocak H. — Dynamics and Bifurcations |
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| Предметный указатель |
Piecewise linear vector field 519
Pitchfork bifurcation 34 35 38 205 207
Pitchfork bifurcation and symmetry 319 405
Pitchfork bifurcation in Lorenz 521
Pitchfork bifurcation in maps 471
Pitchfork bifurcation, numerical computation of 59 60
Pitchfork bifurcation, unfolding of 53
Poincare map 376 378
Poincare map in four-space 533
Poincare map in three-space 512
Poincare map of Duffing 502 503 507
Poincare map of one-periodic equation 116 119 130 121 134 136
Poincare map of one-periodic linear system 258
Poincare map of periodically forced oscillator 500
Poincare map of toral flow 149 151
Poincare map, derivative of 130 136 377
Poincare map, monotonicity of 376 577
Poincare — Andronov — Hopf bifurcation for maps 473—483 474 451 493r
Poincare — Andronov — Hopf bifurcation for maps in delayed logistic 476 478
Poincare — Andronov — Hopf bifurcation for maps, subcritical 475
Poincare — Andronov — Hopf bifurcation for maps, supercritical 475
Poincare — Andronov — Hopf bifurcation for vector fields 211 344—356 344 351 362r 410 515
Poincare — Andronov — Hopf bifurcation for vector fields in FitzHugh neuron model 360
Poincare — Andronov — Hopf bifurcation for vector fields in Lorenz 521
Poincare — Andronov — Hopf bifurcation for vector fields in partial differential equations 363r
Poincare — Andronov — Hopf bifurcation for vector fields in predator-prey model 361
Poincare — Andronov — Hopf bifurcation for vector fields in Rayleigh 355
Poincare — Andronov — Hopf bifurcation for vector fields in Van der Pol 346 348
Poincare — Andronov — Hopf bifurcation for vector fields, generic 359 361 401
Poincare — Andronov — Hopf bifurcation for vector fields, nongeneric 361
Poincare — Andronov — Hopf bifurcation for vector fields, subcritical 346 354
Poincare — Andronov — Hopf bifurcation for vector fields, supercritical 346 354
Poincare — Bendixson theorem 366 367 368 369 370 387r 415
Poincare — Bendixson theorem on surfaces 387r
Poisson bracket 524
Pole placement 277
Position variable 524
Positive definite function 278 279
Positive definite Hamiltonian 531 532
Positive definite quadratic form 239 244 283 279
Positive orbit 9 10 11
Positive orbit of maps 444
Positively invariant set 287 367 369 371
potential energy 414
Potential function 12 13 14 15 190 191 414
Potential function with degenerate critical point 425
Potential function with equal maxima 429
Potential function, bounded 420 421 432
Potential function, generic 422
Potential function, maximum of 423
Potential function, minimum of 423
Potential function, two-parameter 431
Predator prey 171 180 182 215r 361
Predator prey, discrete 483
Predator prey, first integral for 195 196 202
Principal matrix solution 219—222
Product system 185 187 192 214 309
Product system as gradient system 438
Product system in polar coordinates 193
Product system, linear 187 189 191
Product system, simplest 185
Purely imaginary eigenvalues 251 334—364 335 339 362
Quadratic form 125 239 279 283
Quadratic form, positive definite 239 244 283 279
Quadratic system, planar see “Hilbert”
Qualitative structure of the flow 27
Qualitative theory 65r
Quasi-hyperbolic equilibrium 398
Quasi-hyperbolic periodic orbit 398
Quasiperiodic 132r
Rayleigh equation 355
Rayleigh — Bernard convection 104r
Reaction-diffusion equation 434 435 439 440r
Regular value 525
Resonance 255
Resonance, strong 481 482 488 493r
Restricted three-body problem 531 535r
Riccati equation 126 127 132r
Root computing 71 75 76
Root of unity 456 474 480 481
Rotation interval 166r
Rotation number 155 165r
Rotation number in standard circle map 159
Rotation number, computation of 165r
Rotation number, irrational 157 164
Rotation number, rational 154—156 160 162
Saddle 188 191 239 241 272 275 284 292
Saddle connection 302 304 392 “Homoclinic”)
Saddle connection and symmetry 403
Saddle connection breaking 303 304 401 403
Saddle node, elementary 397
Saddle of map 415 447 457
Saddle-center bifurcation 425 426
Saddle-focus see “Sil’nikov”
Saddle-node bifurcation 26 28 36 44 204 206 316 401 437
Saddle-node bifurcation and even symmetry 320
| Saddle-node bifurcation in Henon 470
Saddle-node bifurcation in standard circle map 161 162
Saddle-node bifurcation of fixed point 86 468
Saddle-node bifurcation of periodic orbit 383 384 401
Saddle-node bifurcation of periodic solution 141 143 145
Saddle-node bifurcation on a loop 401 402 403
Sard’s theorem 423 542
scalar 4
Scalar autonomous equation 3—66
Scalar maps 67—104 155—166
Scalar nonautonomous equation 107—155
Second-order differential equation 170 (see also “Conservative system”)
Sensitive dependence on initial data 520 521
Separation of variables 4
Separatrix 398
Shadowing 104r
Sharkovskii’s theorem 99 102 103r
Sharkovskii’s theorem on the circle 155
Shift map 504 509r 517
Sil’nikov orbit 515 518 519 522r
Similar matrices 237
Similar matrices, eigenvalues of 238 246
Singularity theory 64r 439 440r
Sink 239 240 445 447
Soft spring 204
Source 239 447 457
Spherical pendulum 534r
Stability theory 65r
Stability theory of elliptic fixed point 486
Stability theory versus continuous dependence on initial data 18
Stability theory with purely imaginary eigenvalues 334—343
Stability theory with zero eigenvalue 308—313
Stable equilibrium point 17 18 266 313 340
Stable fixed point 73 444 486
Stable manifold of fixed point 457 459
Stable manifold, global 297 298
Stable manifold, local 293 294 295 296 299 300 305r
Stair-step diagram 72 74
Standard circle map 157—164 165r
Standard circle map and universality 164
Standard circle map, bifurcation diagram of 160
Standard circle map, rotation number of 159
Standard circle map, saddle-node bifurcation in 161
Steady state solution 11 (see also “Equilibrium”)
Steepest descent 440r
Step size 68
Stereographic projection 529
Strange Attractor 459 460 514 520
Strong resonance 481 482
Structural stability 63—64 391 392 393 410r
Structural stability for diffeomorphisms 493r
Structural stability in conservative systems 423
Structural stability on torus 156 165r
Structural stability, genericity of 393
Structural stability, Peixoto’s theorem 392 393 411r
Subcritical bifurcation 34 346 475
Submanifold 397 541
Supercritical bifurcation 34 346 475
Superposition principle 218
Symbol set 504
Symbolic dynamics 509r
Symmetry and bifurcations 319 320 403 408 411r
Symplectic integration 535r
Taylor’s Theorem 540
Tent map 102
Three-sphere 529 530
Three-sphere, foliation of 529 534r
Time 4
Topological equivalence 62 64 65r 301 391
Topological equivalence of linear maps 454r
Topological equivalence of linear systems 237 240 241 242 245 264r
Topological equivalence of maps 102
topology see “C” and “Whitney”
Torus 148 149
Torus in forced Duffing 502 503
Torus in forced Van der Pol 499 500
Torus, flow on 153 166r
Torus, Hamiltonian 525 528 529 530
Torus, map on 492
Trajectory 8 9 108 109 176 177
Transcritical bifurcation 30
Transcritical bifurcation and symmetry 404 405
Transcritical bifurcation of fixed point 86
Transversal 249
Transversal homoclinic point 458 459 503 505 506 509r
Transversal homoclinic point and shift map 504—505
Transversal homoclinic point in Duffing 502 503 506 507 509r
Transversal homoclinic point in Henon 459
Transversal section see “Local”
Travelling wave 299
Triangle inequality 175
Tunnel diode 173
Turbulence 519
Twist map 487 502 532
Twist theorem 487 491 493
Two-parameter local bifurcations see “Codimension-two bifurcations”
Two-step methods 464
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