An Introduction to Dynamical Systems - 2nd Edition

Index 729

diagonalizable, 714 forced Duf ng equation, 317
diffeomorphism, 167, 564 forced oscillator, 317
differentiably conjugate, 407 forward orbit, 355
dimension fractal, 669
actal dimension, 670
box, 671 frequency measure, 524
correlation, 681 frequency plot, 528
fractal. 670 friction, 183
Hausdorff, 670 full shift on N-symbols, 442
full shift on two symbols, 603
Lyapunov, 682 full shift space on two symbols, 637
similarity, 678 full two-sided shift on nite number of
direction elds, 129
dissipative, 183 symbols, 622
distance, 712 fundamental domain, 421
distance from a point to a set, 710 fundamental matrix solution, 15
divergence, 243 fundamental set of solutions, 15
double-well potential, 174
doubling map, 337, 356, 439 game theory, evolutionary, 202
doubling the periods, 433, 436 generalized eigenvector, 714
Du ing equation, 200 generalized eigenvectors, 68
forced, 317 generic, 717
Dulac criterion, 248 geometric horseshoe, 598
dynamical systems, xviii geometric series, 437
Gleick, J ., xx
eigenvalue, 543, 713 globally asymptotically stable, 115, 128
eigenvalues of a xed point, 137 gradient, 196
eigenvector, 543 gradient system of differential
electric circuit, 264
elliptic center, 30 equations, 196
energy, 173 graph of a function, 358
epidemic model, 156, 158, 589 graphical method of iteration, 365
c-chain, 493, 631, 638, 665 Grassberger, P., 680
equilibrium point, 4, 81, 120 Grobman—Hartman Theorem, 596
equivalent ows, 167 Gronwa].l's inequality, 101
Euler method, 85
Hamiltonian differential equations, xix
higher dimensions, 91 hard spring, 200
eventually positive stochastic matrix, harmonic oscillator

581 coupled, 45
evolutionary game theory, 202 damped, 49
expanding factor, 467 uncoupled, 44
expanding, piecewise, 467 Hartman Theorem, 596
harvesting, 125
Feigenbaum constant, 402 Hausdorff dimension, 670
Feigenbaum, M., xix, 400 Hayashi, S., 718
Fibonacci recurrence relation, 472, 474 Hénon attractor, 350
rst return map, 216 Hénon map, 350, 561, 566, 571, 624,
rst variation equation, 82, 102
xed point, 4, 81, 119, 120, 354 636, 646, 649, 653, 675, 681, 682
xed point for iteration, 356 Hénon, M., 350
Herman, M., xix
our beetle, 349, 586 heteroclinic orbit, 178, 179
ow, 77, 80 Heun method, 86
folding, 454
food chain, 190 higher dimensions, 92

730 Index

Hirsch, M., 264 Kupka—Smale property, 720
histogram, 528
homeomorphism, 167, 406, 564 L-stable, 115, 368, 555
homoclinic bifurcation, 213, 245 orbitally, 214
homoclinic orbit, 177
homoclinic point, 636 Lanford, O.E., 402
Hopf bifurcation, 235 Lasota, A., 530
hyperbolic, 559 least period, 356
hyperbolic xed point, 137
hyperbolic periodic orbit, 256 Lebesgue measure, 516
hyperbolic toral automorphism, 575 length of an interval, 443
hyperbolicity, 612, 632 Leontief, 55

IFS, 685 level set, 174
image, 564
improved Euler method, 86 Li, T.Y., xviii, 293, 424, 493, 498, 530
Lienard equation, 229, 265
higher dimensions, 92 limit cycle, 215
in ation, 54
initial condition, 13 limit set
input-output, 55 or-limit set, 110, 565
integral of motion, 4
interior, 286, 424, 492, 709 w-limit set, 110, 488
invariant, 112
linear combination, 14, 714
negatively, 112 linear transformation, 541
positively, 112, 379, 439 linearized stability, 557
invariant measure, 517 linearized system, 137
invariant set, 439 linearly conjugate, 407
basic, 647, 665
inverse, 564 linearly independent, 544, 714
irrational rotation, 660
i.rreducible, 581 linearly independent set of solutions, 15
irreducible transition graph, 468 Liouville formula, 15, 63, 248, 282, 325
irreducible transition matrix, 472 Lipschitz, 97
irreducible word, 429 local di eomorphism, 595
isocline, 126 local stable manifold, 138, 570, 608
isolated invariant set, 640 local unstable manifold, 139
isolating neighborhood, 640
iterate, 256 logistic differential equation, 120
iterated-function system, 685 logistic equation, 79
with probabilities, 692 logistic function, 345, 363
iterates of a function, 354 Lorenz differential equations, 297
itinerary, 426, 430
map, 444, 603, 629 Lorenz, E., xix, 291, 331

Jordan canonical form, 67, 715 Lotka, A.J., xix
Lotka—Volterra equations, 145, 169,
Kaplan, J., 680, 682
Keynwian IS-LM model, 58 201, 202, 269
kinetic energy, 173
King Oscar's prize, 662 Lyapunov dimension, 682
Koch curve, 687 Lyapunov exponent, 321
Kolmogorov, A.N., xix, 671 Lyapunov function, 186, 210

weak, 186 -

Lyapunov stable, 115, 368, 555

M-recta.ngle, 613
Mandelbrot, B., 669
manifold, 139, 567, 712

local stable, 138
local unstable, 139
stable, 138, 570
unstable, 138
Maple, 25, 89, 129, 177
Markov chain, 351, 580

Index 731

Markov partition, 466, 621 orbitally asymptotically stable, 214, 262
Markov rectangles, 621 orbitally L-stable, 214, 262
Mathematica, 25, 89, 129, 177 Oregonator system, 263
Mather, J ., 663 oscillator, 173
Matlab, 25, 89, 129, 177
matrix of partial derivatives, 82, 137, coupled harmonic, 45
forced, 317
557, 706 uncoupled harmonic, 44
May, H., xix, 345 undamped nonlinear, 200
McGehee, B., 663 Oseledec Multiplicative Ergodic
measure, 515
Theorem, 656
Borel, 515 Ottino, J., 537
frequency, 524
Lebesgue, 516 partition, 428
natural, 527, 656 pendulum, 177
measure preserving, 517
metastasis of tumor cells, 52 with damping, 184
metric, 442 perfect, 482, 711
metric space, 689, 712 period, 5, 81, 214, 356
middle-third Cantor set, 457 period doubling bifurcation, 397
Milnor attractor, 289, 494 period doubling cascade, 399
Milnor, J., 289, 494 period-n poi:nt for iteration, 356
mod, 70, 254, 356 periodic, 5, 214
modulo, 70, 254, 356 periodic orbit, 5, 81, 214
Moser, J ., xix, 662
stable, 256
natural measure, 527, 656 periodic point, 356
negatively invariant, 112 periodic sink, 214, 556
neighborhood, 524, 709 periodic source, 215
neural network, 208 permutation matrix, 472
Newton map, 348, 373, 585 Perron—Frobenius operator, 529
Newton method for roots, 347 Perron—Frobenius theorem, 520
Newton, 1., xvii phase plane, 23
node, 26 phase portrait, 23, 126, 129
nonhomogeneous linear system, 49 phase space, 23
nonlinear center, 170, 180 Picard iteration scheme, 77, 99
nonrecti able, 688 piecewise expanding, 467
nonresonance, 596 Poincaré map, 216, 217, 231, 251, 255,
norm of a matrix, 61, 554, 716
nowhere dense, 482, 710 273
nullclines, 126 Lorenz equations, 307
numerical methods, 84 Poincaré, H., xvii, 636, 662
PoincarcLBendixson theorem, 219
w-limit set, 110, 488 populations, 150, 412, 477
one degree of freedom, 173 competitive, 145
one to one, 406, 564 epidemic, 156, 158, 589
onto, 406 food chain, 190
open, 286, 492, 709 Hassel model, 413
open ball, 286, 555, 708 predator—prey, 169, 241, 265
orbit, 80, 355, 565 Ricker model, 413, 416, 477
SIR model, 156, 158
forward, 355 SIS model, 589
periodic, 356 Verhulst model, 412
orbitally w-attracting, 214 positively invariant, 112, 379, 439
potential energy, 173
predator—prey system, 169, 241, 265

732 Index

principal Lyapunov exponents, 329 shift space, 468, 604
probability transition matrix, 581 full two-sided, 622
Procaccia, I., 680
pseudo-orbit, 631, 665 Siegel, C. L., xix
Pugh, C., 718 Sierpinski gasket, 686
similarity, 686
quasiperiodic, 45, 52, 295, 330, 660 similarity dimension, 678
quasiperiodic function, 48 Sinai, Ya., 622
Sinai—Ruelle—Bowen measure, 656
rationally independent, 47
reducible word, 429 Singer, D., 386
repeated eigenvalues, 35 sink, 115, 368
repelling
linear map, 553
xed point, 119, 123 SIR model, 156, 158
periodic point, 368, 556, 565 SIS model, 589
repelling xed point, 116
repelling periodic point, 556, 565 Sitnikov, K., xix, 662
replicator system of differential six-twelve potential, 200
Smale horseshoe, 598
equations, 203 Smale, S., xviii, 575, 598, 662, 717
Ricker model, 416 soft spring, 200
rooftop map, 490 solution of a linear equation, 13
Rossler attractor, 313 source, 116, 368, 556
Rossler, O., xx
rotary solutions, 179 linear map, 553
Ruelle, D., xviii, 316 sphere, 709
Runge—Kutta method, 87 stable

higher dimensions, 93 periodic orbit, 256
stable eigenspace, 573
saddle, 24 stable xed point, 115
linear map, 547 stable focus, 32, 552

saddle xed point, 137 stable manifold, 115, 135, 138, 570
saddle periodic point, 559 stable node, 26
Sauer, T., 513, 660
scaling dimension, 678 linear map, 545
Schwarzian derivative, 388 stable subspace, 139
second-order scalar equations, 41 stair step method of iteration, 365
self-excited oscillator, 229 Sternberg Theorem, 596
self-similar, 669, 684 Stewart, 1., xx
semiconjugacy, 407 stochastic matrix, 581
semistable, 123, 215, 368 strange attractor, xviii, 316
sensitive dependence on initial stretching, 454
stretching factor, 467
conditions, JO(, 8, 291, 452, 641 string, 430
at points in a set, 291, 452 strongly attracting, 187
when restricted to a set, 291, 452 subcritical bifurcation, 233
separation of variables, 77 subshift of nite type, 465, 468, 471
separatrix, 567 '
set difference, 708 622, 637
shadowed, 631 superattracting, 369, 586
Sharkovskii ordering, 431 supercritical bifurcation, 233
Shatkovskii, A. N., 424 support of a measure, 515
shed map, 465 symbol space, 442, 603, 604
shift map, 442, 604 symbolic dynamics, 423
symbols, 428

Takens, F., xviii, 316
tangent space to a manifold, 719

Index 733

tangent vectors to a manifold, 719 waterwheel model of Lorenz equations,
tent map, 316, 359, 443 332
tent map of slope 1', 384, 455
ternary expansion, 438 weakly ttf bti g. 123, 137
test function, 191, 219 W93-kl)’ Fepelling xed Point. 124
time plot of the solution, 22 William. R-. 311. 495
time-dependent Word. 430
wf°I15klIm. 15
(llf; )l;11lSl8.l equation, 85, 91, 257, 317, Xia, Z“ 663

linear differential equation, 49, 59, 83, yakubu, A_A_| 539
103. 165. 248. 282. 324 Yorke, J., xviii, 293, 424, 493, 493, 513

wpolosiwl wmiuswy. 407 530, 660, eso, es2

topological Markov chain, 471
topologically conjugate, 167, 407, 595
topologically equivalent, 167
topologically transitive, 287, 439, 492
totally disconnected, 711
trace, 63, 713
trajectory, 80
transition graph, 428

irreducible, 468
transition matrix, 471
transitive, 287, 439, 492
transpose, 12, 713
transversal, 217, 231, 254
transverse, 636
transverse manifolds, 719
trapping region, 286, 304, 492, 640
Tresser, C., xix, 400

Tucker, W., xx, 297

Ueda, Y., xx
unemployment, 54
uniformly hyperbolic, 682
unstable, 115

periodic orbit, 256
unstable eigenspace, 573
unstable focus, 552
unstable manifold, 115, 135, 138, 570
imstable node, 27

linear map, 547
unstable periodic point, 368, 555
unstable subspace, 139

Van der Pol equation, 229, 264
Variation of parameters, 50
vector eld for the system of equations,

95, 126
Volterra, V., x.ix
volume change, 247, 281



Published Titles in This Series

19 R.. Clark Robinson, An lntroduction to Dynamical Systems: Continuous and Discrete,
Second Edition, 2012

18 Joseph L. Taylor, Foundations of Analysis, 2012
17 Peter Duren, Invitation to Classical Analysis, 2012
16 Joseph L. Taylor, Complex Variables, 2011
15 Mark A. Pinsky, Partial Differential Equations and Boundary-Value Problems with

Applications, Third Edition, 1998
14 Michael E. Taylor, lntroduction to Differential Equations, 2011
13 Randall Pruim, Foundations and Applications of Statistics, 2011
12 John P. D’Angelo, An lntroduction to Complex Analysis and Geometry, 2010
11 Mark R.. Sepanski, Algebra, 2010
10 Sue E. Goodman, Beginning Topology, 2005
9 Ronald Solomon, Abstract Algebra, 2003
8 I. Martin Isaacs, Geometry for College Students, 2001
7 Victor Goodman and Joseph Stamp i, The Mathematics of Finance, 2001
6 Michael A. Bean, Probability: The Science of Uncertainty, 2001
5 Patrick M. Fitzpatrick, Advanced Calculus, Second Edition, 2006
4 Gerald B. Folland, Fourier Analysis and its Applications, 1992
3 Bettina Richmond and Thomas Richmond, A Discrete Transition to Advanced

Mathematics, 2004
2 David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of Scienti c

Computing, Third Edition, 2002
1 Edward D. Gaughan, lntroduction to Analysis, Fifth Edition, 1998