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Murdock J. — Perturbations: Theory and Methods
Murdock J. — Perturbations: Theory and Methods

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Название: Perturbations: Theory and Methods

Автор: Murdock J.

Аннотация:

This user-friendly work presents perturbation theory for the solution of algebraic and differential equations, especially ordinary differential equations. It covers all of the methods commonly used in both regular and singular perturbations: Taylor series, Lindstedt series, strained parameters, multiple scales, averaging, matching and WKB methods. Its unique feature is the attention given to the interpretation of results. Considerable attention is given to the mathematical validity of the methods presented and to the estimation of error in a perturbation approximation. Makes accessible to all users of perturbations theory many of the subtler points often known only by specialists.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 509

Добавлена в каталог: 14.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Adiabatic invariance of action      439 446 455 460
Airy function      411 457
Almost periodic function      258
Amol'd tongue      see Resonance horn
Analytic function      466
Anatomy, human      60 97 322 383
Approximation: asymptotic      68 72
Approximation: one-term, two-term, etc.      34 359
Approximation: order of      see Order
Approximation: zeroth, first, second, etc.      see Order
Asymptotic      see Approximation; Series; Uniformity
Attractor      320
Autonomous      488
Averaging      228 275
Averaging, first order      264 275
Averaging, higher order      243 315
Averaging, justification of      276 281 318
Averaging, relation to multiple scales      228 273
Averaging, stroboscopic      287 317
Bessel funclion      411 457
Bifurcation      15 18 94 129
Bifurcation diagram      18
Bifurcation, Hopf      176 224 415
Bifurcation, limit point      see Bifurcation pair
Bifurcation, pair      19 25 129 306
Bifurcation, pair, of limit cycles      185
Bifurcation, pitchfork      39 80 129 132
Bifurcation, saddle-node      18 306
Bifurcation, tangent      see Bifurcation pair
Bifurcation, transcritical      44 129 132
Bootstrapping      145
Boundary layer problem      343 417
Boundary value problem      86 115 135 417 479
Brainwashing      84
Canard      407 413 416
Catastrophe      79 213
Characteristic (length, time, etc.)      see Dimensional analysts
Common part      365 376
Compact set      468
Comparison equation      381 387
Composite solution      358 362 376 378
Connection problem      459
Conservative      164
Continuation      25 27
Convex set      496
Correction layer condition      377
Correction method      344 376 400 422
Critical problem      185 187 197
Damping      345
Damping, critical, overcritical, undercritical      86
Damping, strong, weak      86 92 144
Degenerate      19 49 55
Degenerate boundary value problem      88
Degenerate nearly      21 24 50 194
Determining equation      180 197 199 297
Detuning      196 262
Differential equation: second order      477
Differential equation: system of      148 480 485
Differential inequality      142. See also Gronwall's inequality
Diffusion-reaction problem      92 104
Dilation      see Rescaled coordinate
Dimensional analysis      89 96 346
Discontinuous solution      398 409 413
Dissipation      see Damping
Dream      45 136
Duffing equation: derivation of      90 99
Duffing equation: forced      186 202 262 302 308
Duffing equation: free (unforced)      114 168 174 261 292
Dynamical systems      224 323
Eigenvalue      122 441
Eigenvalue, large      448
Eigenvalue, nonlinear      121
Elastica      94
Elbow neighborhood      322
Energy      164
Entrainment      187 189 196 198 225
Entrainment, domain      see Resonance horn
Entrainment, index      189 196
Equilibrium point      see Rest point
Error      see also Remainder
Error estimation      30 135 163 266 281 381 449
Error factor      444 450
Existence: importance of      12 17
Existence: of periodic solutions      170 180 199 293
Existence: of solutions to boundary value problems      87 115 137 418 479
Existence: of solutions to initial value problems      84 479 485
Expanding interval      163
expansion      see also Limit process expansion: Series operator
Expansion, operator: inner, outer      359 364
Expansion, operator: intermediate      372
Expansion, operator: middle      393
Exponentially small      see Transcendentally small
Extension theorem      375 416
Fast angle      332
First return map      see Poincare map
Flow property      488
Fluid flow      425
Forcing frequency      187
Formal: order      13 66
Formal: power series      12
Formal: reasoning      8
Formal: solution      13
Fredholm: alternative      223
Fredholm: integral equation      484
Free frequency      187
Friction      see Damping
Gauge function      33 65. gauges
Gauge function, distinguished      36
Gauge function, significant      37
Green's function: for ordinary differential equations      137 484
Green's function: for partial differential equations      133
Gronwall's inequality      143 152 267 291 384
Guiding: center      288
Guiding: system      286 316 319
Heuristic      8
Hill's equation      277
honesty      438
Hooke's law      90 345
Hyperbolic      320
Implicit: differentiation      470 471
Implicit: function theorem      26 469
Impulse (initial momentum)      344
In-track error      174 242
Initial layer problem      343
Initial value problem      84 106 140 227 343 479 485
Inner-outer solution      362
Inner: equation      349 420
Inner: expansion operator      359 364
Inner: layer      351 356
Inner: solution      349
Inner: variable      359 420
Integral: equation      137 141 384 453 483
Integral: inequality      138 141 267 384
Intermediate variable      372
Internal layer problem      344
Irrational number      190
Jungle      7
Kernel      141 484.
Kolmogorov — Amol'd — Moser (KAM) theory      332
Langer transformation      458
Least period      188
Level curve      166
Lie transform      339
Limit cycle      176 406. solution
Limit process expansion      75 352 374
Lindstedt method      157
Linearization      489
Lion      7 150
Liouville — Green transformation      450
Lipschitz constant      495
Logic      497
Matching      344
Matching by intermediate variable      371
Matching by Van Dyke's roles      358 419
Matching derivatives      392
matching rules (IVMR)      373
Mathieu equation      186 202 277
Matrix: exponential of      150
Matrix: norm of      see Norm
Method of      see individual heading (e.g.for method of averaging see
Molasses, influenced by position of earth in orbit about sun      426
Multiple scales      216 227
Multiple scales for initial and boundary layers      377. See also Correction method
Multiple scales, first order method      233
Multiple scales, higher order methods      230 242
Multiple scales, justification of      238
Multiple scales, relation to averaging      228 273
Near-identity transformation      283
Nearly linear      see Nonlinear weakly
Negation      501
Noncritical problem      185
Nondegenerate      12 49
Nondegenerate, boundary value problem      88 115 120
Nondimensional      see Dimensional analysis
Nonlinear: strongly      95
Nonlinear: weakly      84 86 101
Nonoscillatory      85 120 418 435
Nonsense      61 68
Nonstandard analysis      407
Nonuniformity      see Uniformity
Norm: Euclidean      150 495
Norm: matrix      150 152 154
Norm: operator      152
Norm: vector      154 495
Orbit      166 178 187 397 480 488
Orbital error      173
Order: asymptotic      29 61 68
Order: formal      13
Order: leading      34
Order: of magnitude      60 61 68
Order: zeroth, first, second, etc.      34 228 229 288 289
Oscillation      85 406. solution
Oscillation, relaxation      406
Oscillation, self-sustained      176
Oscillator      85
Oscillator, coupled      257 342
Oscillator, nearly linear      86 144
Oscillatory      see Nonoscillatory
Outer-inner solution      362
Outer: equation      350 420
Outer: expansion operator      359 364
Outer: layer      351
Outer: solution      350
Overlap: condition      353 357 372 390
Overlap: domain      374
Overlap: failure of      391
Overlap: rigorous      375
PARAMETER      64
Parameter, bifurcation      18 129
Parameter, control      49 64
Parameter, large      429 434
Parameter, natural      6 62 91 121 346
Parameter, perturbation      4 49 64 91 121 347
Parametric excitation      277
Patching method      344 351 415
Period map      188 197 287 490.
Periodic: regime      168 437
Periodic: solution      159 166 176 180 199 297
Periodic: standard form      see Standard form periodic
Perturbation: family      4 62 91 121
Perturbation: family, viewed as a path      6 62
Perturbation: parameter      see Parameter perturbation
Perturbation: regular      73
Perturbation: series      see Series perturbation
Perturbation: singular      73
Phase plane      166 396
Poincare map      179 188 197.
Pointwise      67
Potential      see Energy
Potential well      166 437
Quantifier      497
quantum mechanics      436 459
Quasiperiodic function      258
Rational number      190
recycling      146
Reduced problem      4 6 55 344 398 435
Remainder      16 23 146 464.
Repellor      320
Rescaled: coordinate      46 55 95 346 348 391 394 435
Rescaled: parameter      55 412
Residual      23 146 382
Resonance      187 191 195
Resonance, complete      332
Resonance, harmonic      197 254 261
Resonance, horn (or tongue)      193 212
Resonance, internal      329
Resonance, near      193
Resonance, subharmonic      197 215 259
Resonance, superharmonic      197
Resonance, surface      335
Resonant      187 191 195
Resonant, nearly      193
Rest point      166 170 488
Reynolds number      7 426
Riccati equation      442
Richness      56 428
Root      4
Root, unbounded      43 48
Running away inequality      150 291
Saddle      320
Salt      210
SCALE      see also Multiple scales; Rescaled
Scale, distinguished      56
Schroedinger equation      437
Secular terms      158 235
Secular terms, $T_k$-secular terms      217 236 248
Separatrix      168
Series: asymptotic      69 72
Series: asymptotic power      72 75
Series: asymptotic, addition of      74
Series: asymptotic, differentiation of      76
Series: asymptotic, integration of      78
Series: asymptotic, multiplication of      75 78
Series: asymptotic, uniqueness of      74
Series: convergent      72 466
Series: formal power      12
Series: Fourier      124 172 182 237 286 491
Series: Fourier, multiple      326 493
Series: fractional power      46 75
Series: generalized asymptotic      73 158 161 216 233
Series: Lindstedt      161 171
Series: ordinary asymptotic      74
Series: perturbation      74
Series: Poincare      74
Series: Puiseux      see Series fractional
Series: Taylor      466
Shadowing      323
shooting      88 115 117 184 197
Singular point      see Rest point
Sink      320
Slow curve      396
Slow curve, hypothesis of stability for      397 403
Slow curve, hypothesis of stability for, failure of      408
Slow time      161 229
Small denominators (divisors)      328 330 339
Source      320
Stability (including asymptotic, orbital, etc.): of limit cycle      180
Stability (including asymptotic, orbital, etc.): of linear system      486
Stability (including asymptotic, orbital, etc.): of periodic solution      193 293 301
Stability (including asymptotic, orbital, etc.): of rest point      489
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