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| Shirer H.N. — Nonlinear Hydrodynamic Modeling: A Mathematical Introduction |
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| Предметный указатель |
Adjoint system 336 338
Alexander — Yorke Continuation 225 263
Alexander — Yorke Continuation, altering time-step in 243 246
Alexander — Yorke Continuation, and degenerate systems 256 257 263
Alexander — Yorke Continuation, complex variables in 237 238
Alexander — Yorke Continuation, correcting at each time-step 241 242
Alexander — Yorke Continuation, correcting failure of 243 246
Alexander — Yorke Continuation, differential equations for 236 239
Alexander — Yorke Continuation, examples of in two equations 246 255
Alexander — Yorke Continuation, flow chart for 247
Alexander — Yorke Continuation, in larger spectral models 255 263
Alexander — Yorke Continuation, multiple roots detected by 243 245 262
Alexander — Yorke Continuation, numerical integration in 240 246
Alexander — Yorke Continuation, theory and development of 226 239
Alexander — Yorke Continuation, track-jumping in 244 246
Alteration of an unfolding, minimal 187
Alteration of an unfolding, simple 186 187
Altering unfoldings 182 193
Altering unfoldings, steps in 186 187
Amplitude of waveform 49 54
Amplitude, polynomial equations for 271 277
Amplitude, vacillation of waveforms 51 53
Antipodal points 234 235
Antisymmetric property of Jacobian 27
Antisymmetry of interaction coefficients 460 465
Aperiodic flows 386
Aperiodic solutions 391 392 399 409 411
Aspect ratio, cell (see Cell aspect ratio)
Aspect ratio, domain (see Domain aspect ratio)
Asymptotic orbital stability, definition for 358
Asymptotic orbital stability, necessary condition for 365 367
Asymptotically stable solution 70 71 73 74 96
Atmosphere, finite-dimensional model of 444 445
Atmosphere, infinite-dimensional model of 445 (see also Canonical partial differential equations for the atmosphere)
Attracting direction 79 85 416 420 468 469
Attractor 80 84 412 443
Attractor and truncation dependency 408 411 441 442
Attractor and turbulence 386 392
Attractor as residing in limit set of unstable manifold 79 84 469 470 486
Attractor in infinite-dimensional case 80 85 224 425 484
Attractor, basin 81 82 417
Attractor, classifying 418 421 424 428
Attractor, definition of 412
Attractor, density of points on 423
Attractor, diagnosing structure of 412 443
Attractor, dimension of 412 426 457
Attractor, global 464
Attractor, Lorenz (see Lorenz attractor)
Attractor, minimal 464
Attractor, model form and structure of 506 507
Attractor, restricted 438 439
Attractor, shape of 84
Attractor, strange (see Strange attractor)
Available potential energy in partial differential system 32 108 109 281 299 449 450
Available potential energy in spectral model 40 284 303
Band splitting 393 395
Basic solution, definition of 87
Basic solution, exchanging stability with primary branch 94 99 110
Basic solution, nontrivial 114 115
Basic solution, stability properties of 102 111
Basic solution, uniqueness of 106 111
Basin of attraction 81 82 417
Basin of attraction for Newton's method 243 246
Basis functions and unstable manifolds 486 496
Basis functions for linear system with time-dependent coefficients 361 376 377
Basis functions for linear system with time-independent coefficients 73
Basis functions for rectangular domain 57 60
Basis functions, finding for spectral model 55 60 457 459 485 504
Basis functions, finding optimum for a flow regime 500 501
Basis functions, produced by Proper Orthogonal Decomposition Theorem 496 501
Bezout's theorem 227 228 233 234 256 257
Bifurcating two-torus (see Two-torus)
Bifurcation analysis, stationary solutions 86 105
Bifurcation analysis, temporally periodic solutions 372 383
Bifurcation and stability exchange 92 105 269 466 469
Bifurcation diagram for convection 34 200 206 211 214 219 223 410
Bifurcation diagram for temporally periodic solutions 272 274 276 328 330
Bifurcation diagram, goal form for 209 218
Bifurcation diagram, hypothetical primary 113 123
Bifurcation diagram, hypothetical secondary 205 216 219 223 409 411
Bifurcation equation 90 92
Bifurcation point as butterfly point 194
Bifurcation point as cusp point 194
Bifurcation point as singular point 128 172 194
Bifurcation point, definition of 34 87 269
Bifurcation point, double-valued and secondary branching 200 206 211 214 215 401
Bifurcation point, exchange of stability at 34 37 77 92 100 128 130 269 325 373—383
Bifurcation point, found by Hurwitz Theorem 78 145 508 509
Bifurcation point, Hopf (see Hopf bifurcation point)
Bifurcation point, identifying with Lyapunov exponents 425 426 428 432 435—436
Bifurcation point, locating on stationary solution 34 37 42 90 92 269
Bifurcation point, locating on temporally periodic solution 373 383
Bifurcation point, minimum values of 111 131 138 143 163 218 222 305—307 312 314 402 403
Bifurcation point, ordering and hierarchy of transitions 146 154 218 222 402 403
Bifurcation point, preferred values of (see Bifurcation point minimum
Bifurcation point, primary (defined) 87
Bifurcation point, secondary (defined) 87
Bifurcation point, secondary and double primary bifurcation point 204 206 211 214 215 401
Bifurcation point, single (defined) 93
Bifurcation point, well diagram for 133 138 151
Bifurcation, expected (see Bifurcation points minimum and
Bifurcation, from periodic solution 372 383
Bifurcation, harmonic 375 376 430
Bifurcation, Hopf (see Hopf bifurcation)
Bifurcation, period-doubling 375 376 390 395 402 403 429 432
Bifurcation, periodic 326 341 355 357 367—372
Bifurcation, simple examples of stationary 92 100
Bifurcation, snap-through 96 97 100 117 330 331
Bifurcation, subcritical (see Subcritically branching solutions)
Bifurcation, subharmonic (see Bifurcation period-doubling)
Bifurcation, supercritical (see Supercritically branching solutions)
Bifurcation, three types from periodic solutions 375 382
Bifurcation, three types from stationary solutions 92 100
Bifurcation, transcritical (see Transcritically branching solutions)
Boundary conditions for atmosphere 448 449
Boundary conditions for convection 30 31 281
Boundary conditions for orthogonal functions 56 57
Boundary conditions for rotating convection 142
Boundary conditions, free 31
Boundary dissipation terms in partial differential equations 449 451
Boundary value problem 55 60 106 111
Boundary value problem for the atmosphere 447 450
Boundary values, incorporating spatially or temporally varying 449 459
Boundary, conducting 30 281
Boundary, cyclically continuous 31 108 281 299
Boundary, energy loss at 449
Boundary, nonconducting 30
Boundary, rigid 30 281
Boundary, stress-free 30 281
Boussinesq system, dimensionless 28 119 185 190 192
Boussinesq system, global 448
Boussinesq system, linearized form 35
Boussinesq system, Lorenz form 29
Boussinesq system, Shirer and Dutton form 28
Boussinesq system, two-dimensional 25 26
Boussinesq system, with background wind 278 282 298 300 308 309
Branching (see Bifurcation)
Branching behavior, generic 112 130 204 215 270 277
Branching behavior, in partial differential equations as implied by spectral'model 222 223
Branching behavior, qualitative changes in 119 130
Branching behavior, under model enlargement 409 411
Branching stationary solutions, allowable forms for 112 118
Branching stationary solutions, expected stabilities of 92 100 103 105 202 206 214—218
Branching stationary solutions, expected wavelengths for 131 163
Branching stationary solutions, power series expansions for 326 334 336
Branching stationary solutions, simple examples of 92 100
Branching temporally periodic solutions, allowable forms for 270 277 328 330
Branching temporally periodic solutions, asymptotic orbital stability of 358
Branching temporally periodic solutions, computing branching direction of 325 341
Branching temporally periodic solutions, creation from stationary point 266 268 325 326
| Branching temporally periodic solutions, deduced from stability analysis of stationary solutions 144 154 264 277 288 291 325—354
Branching temporally periodic solutions, expected stability of 325 355 357
Branching temporally periodic solutions, expected wavelengths and orientations for 292 324
Branching temporally periodic solutions, limiting frequency of 146 148 269 270 287 305 307 308 312 322 327 333 373 383
Branching temporally periodic solutions, limiting period of 373 374
Branching temporally periodic solutions, polynomial equation for amplitude 271 277 290 327 329
Branching temporally periodic solutions, represented as a power series 325 341
Branching temporally periodic solutions, simple examples of 326 341 355 357 367 372
Branching temporally periodic solutions, stability analysis for (see Stability analysis for branching temporally periodic solutions)
Branching temporally periodic solutions, structural stability of 269
Branching temporally periodic solutions, uniqueness of 269
Broad-band spectrum 387 388
Butterfly point 178 194 196
Canonical partial differential equations for the atmopshere, condition for stability in 455
Canonical partial differential equations for the atmopshere, dimensionless equations 448
Canonical partial differential equations for the atmopshere, energetics of 449 450
Canonical partial differential equations for the atmopshere, Jeffreys' Theorem for 451
Canonical partial differential equations for the atmopshere, limits on predictability for 454
Canonical partial differential equations for the atmopshere, origin of instability in 453 454
Canonical partial differential equations for the atmopshere, Trapping Theorem for 451 452
Canonical partial differential equations for the atmopshere, uniqueness of solutions to 454 455
Canonical ratio 359 396
Canonical spectral model for the global atmosphere, energy equations for 462 463
Canonical spectral model for the global atmosphere, equations for 460
Canonical spectral model for the global atmosphere, existence of solutions to truncated versions of 462 463
Canonical spectral model for the global atmosphere, Jeffreys' Theorem for 464
Canonical spectral model for the global atmosphere, limit and recurrent points in 464
Canonical spectral model for the global atmosphere, Trapping Theorem for 463 464
Canonical spectral model for the global atmosphere, truncated versions of 460 464
Cantor set 398 405 413
Cantor set, fractal dimension of 413
Cantor set, structure and stange attractor 414 421
Catalytic terms and secondary branching 207 208 211
Catastrophe theory (see Contact catastrophe theory)
Catastrophic behavior in stationary, solutions 129 130
Cell aspect ratio, definition of 133
Cell aspect ratio, effects on critical Rayleigh number 37 133 138 143 157 292 295 305 308 314 324
Cell aspect ratio, effects on critical Reynolds number 312 324
Cell aspect ratio, preferred values of 136 143 144 148 157 292 295 305 307 314 324
Cell aspect ratio, solution dependence on 290
Cell broadening 197 198 211 223
Cell frequency 146 290
Cell period 146
Cell wavelength, preferred 136 138 144 153 314
Cell wavelength, variations in 197 198 222 223
Center manifold and Hopf bifurcation 267
Center subspace 468
Chang and Shirer model 206 224 435 439
Chang and Shirer model and modeling principles 208 210
Chang and Shirer model as a family of models 219 224
Chang and Shirer model as solution to 11
Chang and Shirer model as suitable extension of Lorenz model 197 198 210
Chang and Shirer model, bifurcation points in 208 214
Chang and Shirer model, branching in 209 218 435 439
Chang and Shirer model, chaotic solutions in 435 439
Chang and Shirer model, coefficient model 435 439 440 442
Chang and Shirer model, conductive solution stability 208
Chang and Shirer model, dimensions of attractors in 435 439
Chang and Shirer model, expansions for 206
Chang and Shirer model, goal form in 211 218 408 410
Chang and Shirer model, Hopf bifurcation points in 435 436
Chang and Shirer model, Lyapunov exponent spectrum in 438
Chang and Shirer model, multiple applications of 219 223
Chang and Shirer model, primary branches in 208 435
Chang and Shirer model, secondary branching in 211 218
Chang and Shirer model, stream function diagram for 201
Chang and Shirer model, sudden transition to chaos in 400 435 439
Chang and Shirer model, summary of solution regimes in 429
Chaos, abrupt transition to 400 408 411 427
Chaos, and M-frequency quasi-periodic motion on an M-torus 424 425
Chaos, deterministic 423 438
Chaos, intermittent 395 429 432
Chaos, necessary conditions for 418 425
Chaos, routes to (see Routes to chaos)
Chaos, smooth transition to 400 404 427
Chaos, well versus weakly developed 423 424 436 442
Chaotic attractor as modeling turbulence 392 401 404
Chaotic attractor in specific low-order models 428 432 435 439 441—442
Chaotic attractor, dimensions of 421 424
Chaotic attractor, spurious 410 411 442
Chaotic solutions 386 390 411
Chaotic solutions and insufficient degrees of freedom 409 411
Chaotic solutions as representing turbulence 386 391 400 408 411
Chaotic solutions, physical interpretation of 411
Characteristic equation for Chang and Shirer model 208 210
Characteristic equation for Lorenz model 41 77
Characteristic equation for model 285 287
Characteristic equation for Shirer ( 1980
Characteristic equation for stationary solutions 74 92
Characteristic equation for temporally periodic solutions 371
Characteristic equation for Veronis model 143 145
Characteristic exponents and fastest growing wave 138 140
Characteristic exponents and Lyapunov exponents 433
Characteristic exponents and stability of conductive solution 41 113 199 206 285 288 322 325
Characteristic exponents and stability of convective solution 77 79 202 206 211—224
Characteristic exponents and stability of stationary solution 74 75 264 269 270 332
Characteristic exponents as temporal eigenvalues 74 466 468
Characteristic exponents behavior when both Hopf and stationary bifurcation points occur on trivial solution 160 163
Characteristic exponents for temporally periodic solution 367
Characteristic exponents, definition of 41 74 466
Characteristic exponents, double zero-valued 200 206 211 214 215 287 288
Characteristic exponents, imaginary part of and expected bifurcating solution 373 382
Characteristic exponents, pair crossing the imaginary axis with nonzero speed 269 270 325
Characteristic exponents, tracking signs of 101 105 156 157 212 222
Characteristic multipliers 359 372
Characteristic multipliers and necessary condition for orbital stability 365 367
Characteristic multipliers and perturbation behavior 364 367
Characteristic multipliers as eigenvalues of a matrix 363
Characteristic multipliers, checking accuracy of 369 372 417
Characteristic multipliers, numerically determining 366 367
Characteristic multipliers, product of 369 372
Circulation patterns for convection 24 43 45 46 201 293
Circulation sense, expected 122
Classical Rayleigh-Benard convection as Boussinesq system 23 37 119
Classical Rayleigh-Benard convection, bifurcation diagram for 34 200 206 211 214 219 223 409—411
Classical Rayleigh-Benard convection, bifurcation in 34 37 198 206
Classical Rayleigh-Benard convection, changes of wavenumber in 198 218 223
Classical Rayleigh-Benard convection, energetics for 32 33 108 111
Classical Rayleigh-Benard convection, laboratory experiments for 7 8 384 390
Classical Rayleigh-Benard convection, observed characteristics of 7 8 23 197 292
Classical Rayleigh-Benard convection, partial differential equations for 28 31 61 107
Classical Rayleigh-Benard convection, physical interpretation of onset of convection in 110 111
Classical Rayleigh-Benard convection, preferred wavelengths in 131 140 222 223 292 293
Classical Rayleigh-Benard convection, secondary branching in 198 206 218 224
Classical Rayleigh-Benard convection, spectral models for (see Chang and Shirer model Curry Elevencoefficient Lorenz Marcus Saltzman and
Classical Rayleigh-Benard convection, transition to turbulence in 384 390 400
Classical Rayleigh-Benard convection, trivial solution in 70
Classical Rayleigh-Benard convection, unfoldings in 194 195
Classifying attractors 418 421 424 428
Closed orbits 266 273 356 357 373 382 425 427 464
Cloud streets 278 290 292 324
Cloud streets, comparison of pure modes for 314 321
Cloud streets, coordinate system for 279 280 293 294
Cloud streets, energetics of 297 300 303 304 310 311
Cloud streets, inflection point instability for (see Inflection point instability)
Cloud streets, linking of parallel and thermal, mechanisms for 308 322 324
Cloud streets, parallel instability for (see Parallel instability)
Cloud streets, preferred alignment for 293 294 297 318 324
Cloud streets, Rayleigh-Benard instability for (see Thermal instability)
Cloud streets, three mechanisms for formation of 296 297 314 321
Codimension of a singularity 178 196
Codimension of a subset of a space 237
Computational methods for finding optimum spectral models 485 504
Conductive basic state 24 25 108 278
Conductive solution, stability properties of (see Conductive solution Stability
Contact catastrophe theory 164 196
Contact equivalence 169
Contact transformation, definition of 169
Contact transformation, need for 168 169 177
Contact transformation, steps in 169 171
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