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| Shirer H.N. — Nonlinear Hydrodynamic Modeling: A Mathematical Introduction |
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| Предметный указатель |
Stability, orbital 357 359
Stability, periodic solutions 325 355 357
Stability, solutions 355 359
Stable manifold of a fixed point and constructing spectral models 84 85 472 486
Stable manifold of a fixed point and hierarchies of transitions 82 84 464 465
Stable manifold of a fixed point and locations of attractors 465
Stable manifold of a fixed point and solution components 469 470
Stable manifold of a fixed point, as a divider of phase space 80 82
Stable manifold of a fixed point, as a stable subspace 486
Stable manifold of a fixed point, definition of global 79 472
Stable manifold of a fixed point, example of 471
Stable manifold of a fixed point, global 469 472
Stable manifold of a fixed point, local 469
Stable manifold of a fixed point, space 79 85 405 416
Stable Manifold Theorem 469 474
Stable manifolds relative to evolving of 473
Stable manifolds relative to evolving, as integral manifolds 496
Stable manifolds relative to evolving, coordinates for 482 484 487 488
Stable manifolds relative to evolving, definitions of 472 474
Stable manifolds relative to evolving, finding coordinates for 479 484 494
Stable manifolds relative to evolving, finding eigenvectors spanning 473 477 478 488 494
Stable manifolds relative to evolving, simple example of finding 479 484
Stable manifolds relative to evolving, tangent spaces for 473 477 478 488 489 493 494
Stable solutions and observability 70 71
Stable solutions, temporal behavior near 34 70 71
Stable subspace of a fixed point 486
Stable subspace, definition of 468 469
Stable subspace, example of 470 471
Statically stable fluid 107 111
Stationary point of the flow 49 50
Stationary solutions and transition to turbulence 390 402
Stationary solutions of 33 106
Stationary solutions, allowable forms for 112 118
Stationary solutions, bifurcation analysis of 42 86 105
Stationary solutions, branching (see Branching stationary solutions)
Stationary solutions, catastrophic jumps in 129 130
Stationary solutions, continuation of 466
Stationary solutions, degree of 100
Stationary solutions, existence of unique 106 111
Stationary solutions, expected number of 227 233 234 256 257
Stationary solutions, expected stabilities of 92 100 103 105 202 206 214 218
Stationary solutions, expected wavelengths for 131 163
Stationary solutions, finding all numerically 225 263
Stationary solutions, general stability properties of 100 105
Stationary solutions, hyperbolic 469 474
Stationary solutions, phas e-lo eked 50
Stationary solutions, physical applicability of 129 130
Stationary solutions, polynomial equations for 106 167 172 177 178 225
Stationary solutions, preserving configuration of 177 178
Stationary solutions, qualitative behavior of 124 130 177 178
Stationary solutions, stability analysis of 70 79 92 100 465 469
Stationary solutions, surface 126 128
Stationary solutions, typical branching forms for 106 130
Steady circulation in convection 23
Steady flow, as a regime 8 384 390
Stensrud and Shirer model, bifurcation points in 312 314
Stensrud and Shirer model, energetics of 310 311
Stensrud and Shirer model, equations for 309
Stensrud and Shirer model, expansions for 309
Stensrud and Shirer model, inflection point mode in 314 321
Stensrud and Shirer model, linear analysis of 312 314
Stokes problem, spectral model 55 458 459 502
Strange attractor and predictability 11 404 407
Strange attractor and routes to chaos 390 392 399 400 425 427
Strange attractor, as Lorenz attractor 84 404 411 428 432
Strange attractor, Cantor set-like structure of 414 421
Strange attractor, dimensions of 413 414 418 425 428
Strange attractor, possible modal dependence of 408 411
Strange attractor, restricted 438 439
Strange attractor, sheeted structure of 418 419
Strategy of science 1
Stratocumulus decks 292
Stream function 26
Structural stability 21 164 165 269 421
Structure of flows, complicated spatial and temporal 384 385
Structure of solutions, describing 444
Structure of solutions, preservation of 197
Subcritical Hopf bifurcation and transition to turbulence 390 399 400 408 411 427 428
Subcritically branching solutions, adequate modeling of 117
Subcritically branching solutions, stationary 95 97 103 104 116 118 153 163
Subcritically branching solutions, temporally periodic 153 269 270 325 355 357
Subharmonic bifurcation 375 376 390 395 402 403 429 432
Successive Hopf bifurcations and transition to turbulence 390 392 401 403 424 427
Supercritically branching solutions, stationary 93 94 103 104 153 163 203
Supercritically branching solutions, temporally periodic 153 269 273 325 355 357
Supercritically branching torus 382
Swallowtail point 178
Symbolic manipulator output 67 69
System, adjoint 336 338
System, complete set of roots of 234 236
System, degenerate 234
System, dynamical (see Dynamical system)
System, homogeneous 336 338
System, inhomogeneous 2
System, ir-periodic 336 338
System, physically equivalent 182 191 193
System, self-adjoint 339
Tangent space of stable manifolds and Ruelle's Theorem 473 477 478 481 483 487
Tangent space of stable manifolds, definition of 489
Tangent space of stable manifolds, simple example of finding 479 484
Tangent space of stable manifolds, vectors spanning 488 489 493 494
Tangent space of unstable manifolds and Ruelle's Theorem 476 477 479 481 483 487
Tangent space of unstable manifolds, simple example of finding 479 484
Tangent space of unstable manifolds, vectors spanning 488 496
Taylor number 141
Taylor's theorem 170 180 192 241
Temporally periodic flow, as a regime 8 384 390
Temporally periodic solutions for 292 324
Temporally periodic solutions, as n-cylces 393 395
Temporally periodic solutions, bifurcation diagram for 272 274 276 328 330
Temporally periodic solutions, bifurcation from 372 383
Temporally periodic solutions, branching (see Branching temporally periodic solutions)
Temporally periodic solutions, concept of stability for 355 359
Temporally periodic solutions, determining stability of 357 372
Temporally periodic solutions, existence of 2
Temporally periodic solutions, ir-periodic 336 338
Temporally periodic solutions, on a torus 379 380 396 398 402 403
Temporally periodic solutions, origin of in convection 281 284
Temporally periodic solutions, phase-locked 388 390 403
Temporally periodic solutions, representing in spectral model 278 282 284 326
Temporally periodic solutions, sinusoidal forms for 288 291 Limit and
Temporally periodic solutions, stability 359 366
Tensor analysis 488
Terrain height, varying 195 196
Theory, definition of 2
Theory, development of 444
Thermal advection term 39
Thermal forcing 449
Thermal instability 296 297 300 308 314 321
Thermal instability, developing from 282 285 301 302
Thermal instability, energy source for 300 304
Thermal instability, filtering 309
Thermal instability, necessary condition for 306
Thermometric conductivity 26
Three-cycles and intermittency 395
Three-torus, as signal of chaotic attractor 390 392 401 403 427
time function 492 494
Time-independent system 87
Topological hydrodynamics, introduction to 444 504
Topological hydrodynamics, invariant two- (see Invariant two-torus)
Torus, n 391 424 426
Torus, pathological wrinkling of 390 395 399 402 403 426 427
Torus, three- (see Three-torus)
Torus, two- (see Two-torus)
Torus, wrinkled 399
Total energy, in partial differential system 32 108 281 299 450
Total energy, in spectral model 40 284 302 303 310
Trajectories and unstable manifolds 469 470 482 488
Trajectories, convergence or divergence of nearby 417 418
Trajectories, in phase space 49 54 79 85
| Trajectories, manifolds 79 84 469 470 483 484 486 487
Trajectories, recurrence properties of 420 423
Trajectories, recurrent 405
Trajectories, when attractor is chaotic 405 407 417 424 436 438
Transcritically branching solutions, stationary 97 100 103 105 121 123
Transcritically branching solutions, temporally periodic 269 274 275
Transition to turbulence 7 8 355 373 384 411 506
Transition to turbulence and principal frequencies in flow 384 390
Transition to turbulence, experimental results for 386 390 411
Transition to turbulence, hypothesized mathematical routes for 389 403
Transition to turbulence, modeling of 384 386 389 408 411
Transition to turbulence, principal ways of 389
Transition types and parameters 119 130
Transitions, bifurcation 86
Transitions, discontinuous 96 97 100 125 130
Transitions, hierarchies of 82 84 197 198 218 223 408 411 464 465
Transitions, smooth 94 100 125 130
Transitions, sudden 125 130
Translation of waveforms 50 52
Transversality condition, statement of 179
Transversality condition, unfoldings 185 193
Trapping Theorem for partial differential equations 451 452
Trapping Theorem for truncated spectral models 463 465 503 504
Trident branching form for stationary solutions 44 119 130 163 164 183 184
Trident branching form for temporally periodic solutions 272 273 376 377
Trident branching form, splitting 122 130 164 165 183
Triple Hopf bifurcation point 392 401 403 427
Triple root 125
Trivial solution, bifurcation points on 87 92
Trivial solution, branching from 114 118
Trivial solution, definition of 70
Trivial solution, stability of 70 71
Trivial solution, uniqueness of 91 106 119 271 Conductive and
Turbulence as fundamentally finite-dimensional 391 411
Turbulence as represented by chaotic solutions 386 391 400 408 411
Turbulence, convective (see Convective turbulence)
Turbulence, indicators of 386
Turbulence, Leray 385
Turbulence, modes 391
Turbulence, origin of 384
Turbulence, power spectra for 386 389
Turbulence, Ruelle — Takens 385
Turbulence, suitable representations of 386 408 411
Turbulence, two types of 385
Turbulent flow, as a regime 384 390
Turbulent flow, distinguishing characteristics of 384 391
Turbulent motion, clearly defined onset of 391 392
Two-torus, asymptotic stability of 381 382
Two-torus, bifurcation from 390 392 402 403
Two-torus, invariant 374 382
Two-torus, periodic solution 374 382
Two-torus, principal directions of 380
Two-torus, quasi-periodic 424 427
Two-torus, quasi-periodic solution on 376 381 389 392
Two-torus, stable solutions on 382 425 427
Two-torus, supercritically branching 382
Two-torus, temporally periodic solution on 376 383 388 392
Two-torus, unstable solutions on 382 425 427
Unfolded physical systems 195
Unfolding, alteration of 186 187
Unfolding, altering an (see Altering unfoldings)
Unfolding, function 174 175 180
Unfolding, matrix 180
Unfolding, motivations for 177 178
Unfolding, parameters 174 175 182 196
Unfolding, versal (see Versal unfolding)
Unfoldings and classifying the routes to chaos 400 404
Unfoldings, corank one 178
Unforced flows 106
Uniqueness of solutions and Implicit Function Theorem 86 90
Uniqueness of solutions, in partial differential equations 445 447 453 455
Uniqueness of solutions, loss of and bifurcation 90 92
Uniqueness of solutions, occurrence of 106 111
Uniqueness of solutions, Unit matrix 371
Unobservable solution, perturbations 34
Unstable manifold of a fixed point and constructing spectral models 84 85 472 486
Unstable manifold of a fixed point and hierarchies of transitions 82 84 464 465
Unstable manifold of a fixed point and location of attractors 80 84 464 465
Unstable manifold of a fixed point and solution components 469 470
Unstable manifold of a fixed point and structure of attractors 469 470
Unstable manifold of a fixed point forcing and dimension of 486
Unstable manifold of a fixed point, as finite-dimensional 80 84
Unstable manifold of a fixed point, as unstable subspace 486
Unstable manifold of a fixed point, definition of global 79 472
Unstable manifold of a fixed point, example of 470 471
Unstable manifold of a fixed point, global 469 472
Unstable manifold of a fixed point, local 469
Unstable manifold of a fixed point, space 79 85 405 416
Unstable manifold of a fixed point, strength of and Lyapunov exponents 433
Unstable manifold, invariant 486 487
Unstable manifolds of a point in phase, space 495 496
Unstable manifolds relative to evolving and reduction of spectral models 485 496
Unstable manifolds relative to evolving, coordinates for 473 474 482 484 487 488
Unstable manifolds relative to evolving, definition of 472
Unstable manifolds relative to evolving, finding coordinates for 479 484 488 496
Unstable manifolds relative to evolving, finding eigenvectors spanning 474 477 479 488 496
Unstable manifolds relative to evolving, simple example of finding 479 484
Unstable manifolds relative to evolving, tangent space for 476 477 479 481 483 488 496
Unstable solutions and predictability problems 82 84
Unstable solutions, as decision points 82 84 382 407 433
Unstable solutions, as observable 82 84
Unstable solutions, as unobservable 70 71
Unstable solutions, temporal behavior near 34 70 71 75
Unstable subspace of a fixed point 486
Unstable subspace, definition of 468 469
Unstable subspace, example of 470 471
Variables, number of needed in a model 412
Variance, explanation of 496 501 504
Variational equations 72 92
Veronis model and modeling principles 143 145
Veronis model, as primary branch in a larger model 155 157
Veronis model, bifurcation diagram for 159
Veronis model, bifurcation points in 143 154
Veronis model, characteristic equation for 143
Veronis model, conductive solution in 142 144 154 160 163
Veronis model, convective solutions in 153 163
Veronis model, equations for 142
Veronis model, expansions for 141
Veronis model, expected solution types in 150 163
Veronis model, generalized form for 140 163
Veronis model, Hopf bifurcation in 144 154 159 163
Veronis model, model 301
Veronis model, periodic solution in 146 148
Veronis model, power series analysis 326
Veronis model, solution in 153 163
Veronis model, temporally periodic solution in 159 160
Veronis model, unfolding of 194 195
Versal unfolding 178 181
Versal unfolding, contact equivalent forms for 182 193
Vertical heat flux 109 111 144 150 318 450
Vertical heating rate 23 122 124 184 185 195
Vertical momentum flux 282
Vertically varying wind profile for rolls 315
Vertically varying wind profile, effects of on convection 281 291 296 305 308
Vertically varying wind profile, effects on rolls 293 297 315 318 319
Vertically varying wind profile, Ekman 319 324
Vertically varying wind profile, Fourier coefficients of 284 302 310
Vertically varying wind profile, incorporating in model 278 282 293 294
Vertically varying wind profile, linear 314 315
Vertically varying wind profile, quadratic 315
Vertically varying wind profile, rotating into roll coordinate system 278 282 293 294
Vertically varying wind profile, sinusoidal 315 319
Vickroy and Dutton model 194 196
viscosity 26
Viscosity, effects of 106 107
Viscosity, molecular versus eddy 449
Viscous flows 106
Vorticity, equation 26 66
Vorticity, mean square 385
Wave, fastest growing 131 138 140 212
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