Hale J.K., Kocak H. — Dynamics and Bifurcations :: Электронная библиотека попечительского совета мехмата МГУ

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Hale J.K., Kocak H. — Dynamics and Bifurcations

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Название: Dynamics and Bifurcations

Авторы: Hale J.K., Kocak H.

Аннотация:

This comprehensive textbook is designed to take undergraduate and beginning graduate students of mathematics, science, and engineering from the rudimentary beginnings to the exciting frontiers of dynamical systems and their applications. It is a masterful exposition of the foundations of ordinary differential and difference equations from the contemporary viewpoint of dynamical systems and bifurcations. In both conception and execution, the authors implemented a fresh approach to mathematical narration. Fundamental ideas are explained in simple settings, the ramifications of theorems are explored for specific equations, and above all, the subject is related in the guise of a mathematical epic. With its insightful and engaging style, as well as its numerous computer-drawn illustrations of notable equations of theoretical and practical importance, this unique book will simply captivate the attention of students and instructors alike. 345 illustrations.

Язык:

Рубрика: Физика/Динамические системы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 582

Добавлена в каталог: 10.09.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

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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