Alligood K., Sauer T., Yorke J.A. Chaos: An Introduction to Dynamical Systems
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594
Index
␣
-limit set, 331
acceleration, 45
accessible point, 433
affine contraction map, 157
amplification factor, 572
Andronov-Hopf bifurcation, see Hopf
bifurcation
angular velocity, 53
antimonotonicity, 536
area-contracting, 243, 470
area-preserving, 471
Arnold, V., 476
aspect ratio, 190
asteroid, 47, 50
astronomical unit, 100, 101
asymptotically stable, 290
attractor, 75, 235, 281
nonhyperbolic, 452
two-piece, 75
autonomous, 275
base 3 arithmetic, 152
basin, 11, 52, 56, 129
infinite, 133
of infinity, 12, 59, 61, 131, 161, 167, 184
of pendulum, 54, 55
of stable equilibrium, 307
riddled, 170
basin boundary, 83, 84, 527
fractal, 164, 166, 183
basin cell, 435
beetles, 39
Belousov-Zhabotinskii reaction, 143, 549
595
I NDEX
bi-infinite, 213
bifurcation, 18, 19, 447
flip, 451
generic, 513
Hopf, 483
nongeneric, 460, 468
period-doubling, 21, 74, 451, 494, 499
pitchfork, 459, 467
saddle-node, 21, 450, 464
subcritical, 486
supercritical, 486
tangency, 21
transcritical, 458, 467
value, 448
bifurcation diagram, 18, 21, 74, 450
billions and billions, 101
binary numbers, 111
birth rate, 39
bit, 111
blow-up in finite time, 280, 283
Bolzano-Weierstrass Theorem, 338
boom and bust, 40
Borel measure, 251
Borel set, 251
boundary, 433
branch
of bifurcation diagram, 506
bronze ribbon, 229
Brouwer Fixed-Point Theorem, 210
Brouwer, L., 432
cannibalism, 39
Cantor set, 150, 186, 244, 411
box dimension of, 177
middle-3/5, 554
middle-half, 186
middle-third, 158, 177, 554
cascade, 18, 499, 520, 526
Chua’s circuit, 534
experimental, 532
reverse, 533
Cascade Theorem, 520
cat map, 95, 97, 197
Cayley-Hamilton Theorem, 559
celestial mechanics, 46, 101
cell, 435
center
equilibrium, 491
chain rule, 15, 16, 71, 118
change of coordinates, 62, 67, 80, 561
chaos, 27, 82, 99
chaotic attractor, 232, 235, 247
chaotic orbit, 106, 110, 196
of flow, 385
chaotic set, 235
chemical reaction, 143, 321
Chua’s circuit, 375
crisis, 421
Lyapunov exponents of, 390
period-doubling, 377
synchronization of, 391
circle map, 92
closed curve
simple, 333
closed set, 186, 337
cobweb plot, 5–7, 12
coexistence
of species, 312, 326
coexisting attractors, 355
Colorado Corollary, 211
comet
Giacobini-Zinner, 103
commensurate, 352
competing species, 310
complement, 250
complex multiplication, 167
composition property
of autonomous equations, 339
conjugate, 115, 527
connected, 520
connected set, 339
connecting arc, 334, 335, 407
conservation of energy, 300
continuable, 460
continuous dependence, 280, 295, 296
convection rolls, 362, 363
cooling
Newton’s law, 44
correlation dimension, 180
correlation function, 180
Couette-Taylor experiment, 538
countable, 121, 123, 153
596
I NDEX
countably infinite, 153
covering rule, 126, 245
crisis, 18, 414
boundary, 420, 444
interior, 416
critical point, 32, 114, 167, 306, 527
is period three, 248, 260
cycle, 333
damping, 54
death rate, 39
decreasing, 37
delay coordinates, 145, 541
delay plot, 538
delta diamond, 342
dengue fever, 325
dense, 122, 204
dense orbit, 247
density
of measure, 257
derivative
partial, 60, 69
determinant, 67, 76, 94
deterministic, 2, 294
diagonalizable, 559
diamond construction, 343
dichotomy
spaghetti-lasagna, 205
diffeomorphism, 403
Digital Orrery, 99
dilation, 66
dimension
box-counting, 174, 545, 555
correlation, 180
discontinuity, 25
discrete-time, 50
disk, 58
dissipative, 366, 519
flow, 385
distance
Euclidean, 58
domain, 77
double-loop attractor, 368
double scroll attractor, 390
double scroll chaotic attractor,
377
double-well Duffing equation, 303,
378–380, 406, 412
doughnut, 79, 92
dynamical system, 1
continuous-time, 2
discrete-time, 2
eigenvalue, 62, 87, 558
simple, 259
eigenvalues
complex, 66, 90, 91, 289, 491
distinct, 63, 88
repeated, 65, 89, 90
eigenvector, 62, 87
electrodissolution, 496
ellipse, 47, 63, 69, 87, 89
ellipsoid, 563
elliptical orbit, 46
embedding, 543
embedding dimension, 543
energy function, 491
ENIAC, 361
epsilon neighborhood, 9, 11, 58
error
one-step, 571
total, 570
ESP, see exchange of stability principle
Euler method, 568, 573
modified, 573
eventually periodic, 25,94
exchange of stability principle, 515
existence of solutions, 296
expanding interval map, 245
exponential convergence, 11
exponential decay, 274
exponential growth, 3, 100
exponential separation, 106
exponential separation time, 101
extinction, 310
Fatou’s Theorem, 167
faucet
leaky, 442
Feigenbaum’s constant, 502, 525
Feigenbaum, M., 500
Fibonacci number, 95, 97
597
I NDEX
figure-eight, 338
final state prediction, 184
first-return map, 318
fixed point, 5,62
attracting, 10
elliptic, 473
repelling, 10
Fixed-Point Theorem, 137, 208
flip repeller, 506
Floquet multipliers, 479
flow, 277, 331
forced damped pendulum, 187, 228, 239,
400
area contraction rate, 385
basin cell, 438
bifurcation, 504
horseshoe in, 220
forward invariant set, 308
forward limit set, 234
for flow, 331
Fourier series, 362
fractal, 51, 52, 56, 149
fat, 179
fractal dimension, 172, 188
fractal structure
in mixing, 269
fraction
of iterates, 253
friction, 54
generalized eigenspace, 293
generic, 510, 546
global synchronization, 389
globally asymptotically stable, 290
gravity, 45, 55
Gronwall inequality, 296
halo orbit, 103
Hamiltonian, 50, 491
heart
of cockroach, 541
H
´
enon attractor, 61, 163, 174, 232
box dimension of, 178
Lyapunov dimension of, 205
H
´
enon map, 51, 70, 72–74
area-contracting, 470
area-preserving, 473
basin cell, 437
bifurcation, 456
cascade, 503
crisis, 418
inverse, 77
stable manifold, 407
two-piece attractor, 418
H
´
enon, M., 50
heteroclinic, 410
Hodgkin-Huxley equations, 357
homeomorphism, 403
homoclinic, 48, 50, 85, 86, 87, 409, 422,
440
homoclinic point
transverse, 413
Hooke’s law, 321
Hopf bifurcation, 375, 483
subcritical, 358, 484, 488, 496
supercritical, 486
Hopf, E., 485
horizontally aligned, 425
horse, 64
horseshoe, 521
contains chaotic set, 236
near homoclinic point, 411
horseshoe map, 216, 440
chaotic orbits of, 221
hyperbolic, 68,70
equilibrium point, 298
hyperbolic horseshoe, 411
hysteresis, 486, 496
Ikeda attractor, 232
Ikeda map, 202, 414
crisis, 416
Ikeda measure, 250
Implicit Function Theorem, 97, 463
Inclination Lemma, see Lambda
Lemma
increasing, 37
infinite
uncountably, 32, 35
infinite series, 48
inflection point, 452
initial value, 5
598
I NDEX
initial value problem, 276, 568
interdrip time, 443
interior, 433
intersection theory, 553
initial condition, 28
invariant, 339
invariant circle, 478
invariant measure, 251
nonunique, 261
of logistic map, 264
invariant set, 86, 196
inverse map, 76, 77, 81
invertible, 77
irrational number, 93
irrational rotation, 112
irreducible, 240
isolated, 336
iterated function system, 157
itinerary, 28, 36, 113, 123–124, 214, 218
Jacobian matrix, 60, 69, 70, 72, 74, 76, 80,
97, 298, 401
Jordan Curve Theorem, 344
Julia set, 167
as fractal basin boundary, 169
KAM curve, 476
King Oscar II, 46, 85, 99
Lagrange point, 103
lakes of Wada, 432
Lambda Lemma, 415, 438
LaSalle’s Corollary, 309, 341
laser, 191, 394, 532
Lebesgue measure, 250
letters
as omega limit sets, 353
level curve, 301
level set, 314
lie across, 211, 224
limit point, 186, 337
linear
differential equation, 275
linear approximation, 60
linear stability, 68, 69
linearization, 68
Liouville’s Formula, 384
Lipschitz constant, 296, 354
local stable set, 423
local synchronization, 389
locally continuable, 460
logistic differential equation, 279
slope field, 280
logistic growth, 4
logistic map, 13, 17, 18, 51, 106, 112
bifurcation, 448, 456
chaos, 121
conjugate to tent map, 116, 264
crisis, 422
dense orbit, 123
Lyapunov exponent, 121
Lorenz, E., 359
Lorenz attractor, 363
Lyapunov dimension of, 366
Lyapunov exponents of, 363, 366
reconstructed from time series, 546
symmetry of, 363
Lorenz equations, 362
are dissipative, 385
subcritical Hopf for, 489
time-T map, 383
Lorenz tent map, 367
Lorenz windows, 367
Lotka-Volterra equations, 309
Lyapunov Center Theorem, 492
Lyapunov dimension, 203
of corn syrup, 271
Lyapunov exponent, 100, 106, 107, 107,
194, 270
calculation of, 200
for flow, 381
for logistic family, 237
positive, 110
zero along direction of flow, 384
Lyapunov function, 305, 340, 392
strict, 305
Lyapunov number, 87, 106, 107, 107,
194
Lyapunov stable, 290
Mandelbrot, B., 149
Mandelbrot set, 167
599
I NDEX
manifold, 430
stable, 48
topological, 79
unstable, 48
with boundary, 79
map, 5
linear, 62, 62
nonlinear, 69
Markov matrix, 260
Markov partition, 212, 245
mathematical modeling, 44
Mean Value Theorem, 510
measure
atomic, 255
Borel, 251
invariant, 251
Lebesgue, 250
natural, 253
measure zero, 151, 180, 511, 546
method of analogues, 539
method of nullclines, 310
M
¨
obius band, 79
Monotonicity Lemma, 344
mortality rate, 40
Moser, J., 476
mosquito, 325
n-body problem, 46
NASA, 103
Navier-Stokes equations, 539
near-collisions, 48
neuronal circuit, 355
Newton
law of cooling, 44, 274
laws of motion, 45, 46, 48, 53
node, 450
non-periodic, 32
nonautonomous, 275
nonlinear, 51
nonlinear optics, 394
nullclines, 310–314
obliquity, 101
of Mars, 103
ODE method
adaptive step, 574
ODE solver, 568
Euler, 568
multi-step, 575
order of, 572
Runge-Kutta, 568
Runge-Kutta-Fehlberg, 574
omega-limit set
for flow, 331
one-parameter family, 448
generic, 514
one-to-one, 76, 543
one-to-one correspondence, 153
one-way coupling, 391
open set, 433
orbit, 5
periodic, 13
shadowing, 225
orbiting masses, 46
order, 528
of differential equation, 275
of ODE solver, 572
order-n subinterval, 245
orientation-preserving, 503
orthogonal, 87
orthogonal matrix, 565
parabola, 22
partition, 125
party trick, 142
path connected, 434
pendulum, 52–54, 56, 228, 274,
334
damped, 308
forced damped, 55, 84, 85
perfect set, 186
perfect shuffle, 142
period
of periodic orbit of flow, 333
period-doubling, 236, 373, 442
period-doubling cascade, 368, 375
period-halving, 375
period-three, 23, 135
implies all periods, 127
implies sensitive dependence, 32
saddle node bifurcation, 453
window, 19
600
I NDEX
periodic
asymptotically, 108
periodic forcing, 55
periodic orbit, 49, 58
of flow, 317, 333
periodic point, 13
periodic points
infinitely many, 92
periodic table, 24, 37, 215
periodic window, 236, 526
phase plane, 285
phase portrait, 277
phase space, 285
piecewise expanding, 256
piecewise linear, 245
piecewise smooth, 256
piecewise-linear nonlinearity, 375
plane map, 49
planetary collision, 99
Pluto, 100
Poincar
´
e-Bendixson Theorem, 337, 341,
386
Poincar
´
e, H., 46, 85, 99
Poincar
´
emap,49, 50, 479
Poincar
´
e section, 145, 190
polar coordinates, 404
population model, 1, 39
potential energy, 300
potential well, 302
Prandtl number, 362
predator-prey
Hopf bifurcation for, 495
predator-prey model, 313
predictability, 99
probability measure, 251
projectile, 45
quadratic map, 170, 500
bifurcation, 449
quasiperiodic, 199, 352
rabbit, 168
rain gauge technique, 250
random number generator, 252
random process, 2
range, 77
rational number, 26, 93
rational numbers, 154
Rayleigh number, 362
Rayleigh-Benard convection, 360
regular repeller, 506
renormalization, 526
repeller, 451, 506
repelling, 281
resonance, 101
Riemann sum, 179
RLC circuit, 375
R
¨
ossler attractor
Lyapunov exponents of, 371
R
¨
ossler equations, 371
bifurcation diagram, 374
rotation, 66
rotation number, 476
roundoff error, 222
Runge-Kutta method, 568
Runge-Kutta-Fehlberg method, 574
saddle, 68, 76, 84, 89, 400
equilibrium point, 286, 291
flip, 80,85
regular, 80,85
Saltzman, B., 360
Sard’s Theorem, 511
satellite, 46, 103
Schwarzian derivative, 132, 502
self-intersection, 544
self-similar, 149, 162, 525
semi-group property, 339
semi-major axis, 65
semilog, 100
sensitive dependence, 25, 26, 27, 30–32,
36, 58, 99, 107, 362
sensitive point, 27
separation of variables, 282
shadowing, 225
Sharkovskii’s Theorem, 135
shift map, 124, 221
Sierpinski carpet, 160
Sierpinski gasket, 159, 179
similar matrix, 67,67
simple closed curve, 344
single-well potential, 303
601