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









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: Dynamics and Bifurcations

: Hale J.K., Kocak H.

:

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.


: en

: / /

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ed2k: ed2k stats

: 1991

: 582

: 10.09.2005

: | | ID
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$C^0$, $C^1$ distance      390
$C^1$, $C^r$ topology      390 410r
$C^1$, $C^r$ topology versus $C^0$ topology      391
$C^2$ topology      422
$C^n$ function      6
$C^n$ function, canonical form      223 (see also Jordan normal form)
$S^3$      see Three-sphere
Abelian integral      408
Adjoint equation      254255
Alpha-limit point      183
Alpha-limit point of maps      445
Alpha-limit set      14 183 366 368
Alpha-limit set of maps      445
Analytic vector field      331 339
Animation      530
Annulus      369 487
Area-preserving flow      423
Area-preserving map      260 484493 484 487 493r 502
Area-preserving map, Henon      491
Area-preserving map, linear      484 485
Area-preserving map, piecewise linear      492
Arnold tongues      160 165
Asymptotic phase      342
Asymptotic phase and hyperbolicity      344
Asymptotically stable equilibrium point      17 18 266 267
Asymptotically stable fixed point      73 444 455
Attractor      394 395 439
Attractor dimension      537
Attractor, strange      520 (see also Chaos)
Autonomous      4
Averaging      146r
Basin of attraction      277 281 286
Bendixsons criterion      373 374
Bifurcation and symmetry      see Symmetry
Bifurcation at infinity      411r
Bifurcation diagram      28
Bifurcation diagram of harmonic oscillators      210 527
Bifurcation diagram of hysteresis      31 38
Bifurcation diagram of integrable Hamiltonians      527
Bifurcation diagram of logistic map      100
Bifurcation diagram of Poincare Andronov Hopf      212 359
Bifurcation diagram of saddle node      28 36 38
Bifurcation diagram of subcritical pitchfork      35
Bifurcation diagram of supercritical pitchfork      34 38
Bifurcation diagram of transcritical      30
Bifurcation diagram of Van der Pol      351
Bifurcation diagram of vertical      210
Bifurcation diagram, computation of      5661 57 59 60 65r 357360
Bifurcation equation      315
Bifurcation function      315 317 330
Bifurcation function and symmetry      317318
Bifurcation function versus center manifold      330
Bifurcation of conservative systems      425431
Bifurcation of equilibrium      see Pitchfork Poincare Saddle
Bifurcation of fixed point      see Period doubling Poincare Saddle
Bifurcation of gradient systems      436440
Bifurcation of linear systems      247253
Bifurcation of one-periodic equation      133146
Bifurcation of periodic orbit      382386
Bifurcation point      27
Bifurcation set      525
Bifurcation value      27
Birkhoff normal form      486
Boundary-value problem      112
Breaking a saddle connection      210 213 303 304 401 402
Cantor function      159
Catastrophe theory      64r 439
Center      179 180 181 343 416
Center and numerical solutions      463 464 467
Center in Hamiltonian systems      356
Center manifold      321332 321 326 327 332r
Center manifold approximating      325 328
Center manifold for maps      493r
Center manifold versus bifurcation function      330
Center manifold, attraction of      322
Center manifold, depending on parameter      325
Center manifold, nonanalytic      331
Center manifold, nonuniqueness of      330 322
Center theorem of Liapunov      339r 340
Central difference algorithm      464 466
Cetaevs instability theorem      284 305r
Chain rule      540
Chaos      97 99 103r 506 517 520 522r 535r
Chaos in Cremona map      489
Chaos in delayed logistic map      479
Chaos in Duffing      503 507
Chaos in Henon      460
Chaos in Henon Heiles      533
Chaos in logistic map      97 100
Chaos in Lorenz      520
Chaos, onset of      506
Characteristic multipliers      259 260 261 263
Characteristic multipliers, invariance of      261
Characteristic polynomial      230
Characteristic polynomial and trace and determinant      236
Chemical instability      360
Circle map      149 150166 165r 487
Circle map of forced oscillations      501
Circle map, periodic orbits of      155
Circle map, standard      157165 158
Circle, phase portrait on      21 22 54 55
Closed orbit      see Periodic orbit
Codimension-one bifurcation      45 249250 250 251 252
Codimension-one bifurcation with two parameters      46
Codimension-one bifurcation, all planar generic      396403 401 402 411r
Codimension-one singularity      249
Codimension-one submanifold      249 397 541
Codimension-two bifurcation      251 252 405412 407 409 431
Codimension-two singularity      251
Commuting matrices      223
Competing species      171 181
Complete integrability      525 526 531
Complete integrability, analytic versus smooth      535r
Complete integrability, nongenericity of      534r
Composed focus, elementary      398
Computer graphics      529 534r
Conjugate gradient method      440r
Conservation of energy      198
Conservative system structural stability      423
Conservative system, bifurcations in      425432
Conservative system, generic      422
Conservative system, period of      203
Conservative system, second-order      194 198 214 413432 414
Conserved quantity      524 (see also First integral)
Constant energy surface      524
Constant momentum surface      525
Continuation of solution      542
Continued fraction      164
Controllable      277
Cremona map      486 488 489 490 491 493
Cremona map, chaos in      489 490
Cremona map, inverse of      491
Critical point      11 415 433 440r
Critical point, degenerate      425 426 438
Critical point, nondegenerate      415 433 438
Critical value      415 525
Cross section      see Local transversal
Cusp      35 36 37 49 84 86 428
Cylinder      118 119 121
Cylinder, invariant      498 499 525
Degenerate critical point      425 426 438
Degenerate equilibrium, cubic      46 53
Degenerate equilibrium, quadratic      44 53
Degenerate equilibrium, quartic      53
Degenerate fixed point, cubic      84
Degenerate fixed point, quadratic      84
Delayed logistic map      455 456 460 493r
Delayed logistic map, chaos in      479
Delayed logistic map, Poincare Andronov Hopf bifurcation in      476 478
Denjoys theorem      155 165
Dense      152 248 251
Dense orbit on torus      154
Devils staircase      159 165r
Diffeomorphism      61 186
Differentiable equivalence      61 238
Dimension, four      IX 523536
Dimension, one      VIII 3104
Dimension, one and one half      VIII 105166
Dimension, three      IX 511522
Dimension, two      VIII 169494
Dimension, two and one half      IX 495509 497
Direction field      8 9
Dissipative system      394 394396 411r 508 521
Distance      175
Double zero eigenvalue      239 242 246 406 408 411r
Double zero eigenvalue, unfolding      406 407 411r
Double zero eigenvalue, unfolding linear      250 252
Double zero eigenvalue, unfolding with odd symmetry      408 409 411r
Double zero eigenvalue, unfolding with origin fixed      408
Duffings equation      416 418 501509
Duffings equation, forced      501 503 509r
Duffings equation, forced and damped      506 507
Duffings equation, homoclinic tangency in      506
Duffings equation, period-doubling in      514
Dulac criterion      373
Dulac function      373 374 387r
Dynamical system      VII 7 23r 65r
Ecological models      215r (see also Competing species Delayed Logistic and
Eigenvalue      229 231
Eigenvalue with negative real part      263 266 267
Eigenvalue, complex      234
Eigenvalue, equal      232
Eigenvalue, purely imaginary      333364
Eigenvalue, real distinct      232
Eigenvalue, zero      307332
Eigenvector      229 231 232237
electrical circuits      172 172174 215r
Elementary composed focus      398 399 401
Elementary homoclinic loop      398 401
Elementary saddle node      397 398 401
Elliptic fixed point      485 502 503
Elliptic fixed point at resonance      488 494r
Elliptic fixed point, stability of      486 493r
Elliptic integral      204
Elliptic integral, numerical computation of      458
Elliptic Umbilic      439
Energy      414
Energy-momentum mapping      525 526
Energy-momentum mapping, bifurcation diagram of      527
Energy-momentum mapping, level set of      525 527
Energy-momentum surface      525 528
entropy      537
Equilibrium point      11 178
Equilibrium point, asymptotically stable      17 266
Equilibrium point, hyperbolic      19 301 391
Equilibrium point, nonhyperbolic      308 334
Equilibrium point, quasi-hyperbolic      398
Equilibrium point, stable      17 266
Equilibrium point, unstable      17 266
Ergodic orbit on torus      528
Ergodic theory      537
Eulers algorithm      68 104r 463
Eulers algorithm and hyperbolic limit cycle      467
Eulers algorithm and linear systems      184 463 464
Eulers algorithm and logistic map      68
Eulers algorithm on circle      157
Existence and uniqueness of solutions      6 22r 218 542 543r
Faradays law      172
Feedback control      277
Fibonacci numbers      164
Figure eight      416
Figure eight in Duffing      502
First integral      194 196 202 524
First integral for maps      453
First integral, analytic versus smooth      534
First integral, detecting      198
First integral, local      194 203
First integral, nonexistence of      197 535r
First return map      see Poincare map
First-order structurally unstable vector field      397 398 399 400
FISH      416 417
Fishers Equation      299
FitzHugh neuron model      360
Fixed point      72 444
Fixed point, asymptotically stable      73 444 455
Fixed point, hyperbolic      76 82 83 454
Fixed point, nonbyperbolic      76 77 78 79 84
Fixed point, stable      73 444
Fixed point, unstable      73 444 455
Flip bifurcation      see Period-doubling
Floating point arithmetic and dynamics      104r 493r
Floquet representation      263
Flow      7
Flow box      186 187
Flow box theorem      185 186 203 375
Flow on torus      147154 153 166r
Fluctuating environment      126
Fold bifurcation      34 (see also Cusp)
Foliation      526
Forced Duffing      501 (see also Duffing)
Forced linear oscillator      503
Forced Van der Pol      498 (see also Van der Pol)
Forcing function, periodic      498
Fourier series      123
Fractals      494r
Fredholms alternative      117 255
Frequency locking      165r
Functionally independent      525
Fundamental matrix solution      219 235
Fundamental matrix solution of one-periodic linear system      257
Galaxy model      531
Galileo      174
General position      406
GENERIC      248 252 264r
Generic codimension-one bifurcations      396 401 402 411r
Generic gradient vector field      438
Generic potential function      422 423
Generic properties of Hamiltonians      534r
Generic structural stability      393 411r
Global analysis      23r
Global attractor      394 395
Golden mean      164
Gradient      188 280
Gradient vector field      1214 15 23 188 432442 433 435 436 440r
Gradient vector field, bifurcations of      436
Gradient vector field, generic      438
Gradient vector field, linear      438
Gradient vector field, omega-limit set of      434
Gradient vector field, unfolding of      439
Grobman Hartman theorem      301 305r 493r
Gronwalls inequality      269 270
Hamiltonian system      198 423 494r 523536 524 534r
Hamiltonian system, completely integrable      525 526 534
Hamiltonian system, nonintegrable      531 533
Hamiltons equations      524
Hard spring      204
harmonic oscillator      see Linear harmonic oscillator
Hartman Grobman theorem      see Grobman Hartman
Harvesting, constant      40
Harvesting, periodic      128
Harvesting, proportional      40
Henon map      458 459 460 493r
Henon map and logistic map      461 465
Henon map and Poincare Andronov Hopf bifurcation      483
Henon map, area-preserving      491
Henon map, attractor of      459 460 461
Henon map, inverse of      461
Henon map, period 7 orbit of      471
Henon map, period doubling in      471 472
Henon map, saddle-node bifurcation in      470
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