Hirsch M., Smale S., Devaney R., Differential Equations, Dynamical Systems, and an Introduction to Chaos
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416 Index
Poincaré map , 319, 323, 373
closed orbit, 221
computing of, 12–15
description of, 11, 221, 319
graph of, 14
semi-, 265
Poincaré-Bendixson theorem
applications of, 227–230, 246
description of, 225–227
polar coordinates, 164
population
discrete logistic population model, 336
logistic population model, 4–7, 335
positive Liapunov exponents, 323
positively invariant, 200
potential energy, 280
Prandtl number, 304
predator/prey systems, 239–246
Q
quantum mechanical systems, 298–299
quasi-periodic motions, 119
R
range, 92, 94
Rayleigh number, 304
real distinct eigenvalues, 39–44, 51–53
reduced row echelon form, 79
regular point, 205
repeated eigenvalues, 47–49, 56–57, 95–101,
119–122
repelling fixed point, 330
resistor
characteristic, 259
passive, 262
return condition, 316
RLC circuit, 23, 257–261
Rössler attractor, 324–325
Runge-Kuta method, 154–155
S
saddle
definition of, 39–42, 108
higher dimensional, 173–174
nonlinear, 168–174
spiral, 113
saddle connections, 193
saddle-node bifurcation, 177–178, 179f–180f,
181, 332
scaled variables, 294
second-order equations, 23
seed of the orbit, 38f, 328
semiconjugacy, 341
semi-Poincaré map, 265
sensitive dependence on initial conditions,
305, 321
sensitivity constant, 338
separation of variables, 241
set
compact, 228–229
dense, 101
limit, 215–218
set of physical states, 259
shift map, 347–349, 370
Shil’nikov system, 359–366
singularity, 293–297
sink
definition of, 42–44, 329
higher-dimensional, 109
nonlinear, 165–168
spiral, 47, 48f
three-dimensional, 110f
slope, 117
slope field, 6
smooth dynamical system, 141
smoothness of flows, 149–150, 402
S-nullclines, 236
solutions
collision-ejection, 296–297
continuous dependence of, 147–149
equilibrium, 2
extending, 395–398
general, 34
heteroclinic, 193
homoclinic, 209, 226, 363f
near-collision, 297
periodic, 215, 363f
straight-line, 33, 52
source
definition of, 43–44
nonlinear, 165–168
spiral, 47, 48f
source fixed point, 330
space
configuration, 278
phase, 278
state, 259, 278
tangent, 286
three-parameter, 71–72
space derivative, 402
spanned, 77, 89
spiral center, 112f
spiral saddle, 113, 113f
Index 417
spiral sink, 47, 48f
spiral source, 47, 48f
spring constant, 26, 394
square matrix, 95
stability
asymptotic, 175
equilibrium point, 194
Liapunov, 194–196
nonlinear systems, 174–176
stable curve, 161
stable curve theorem, 168–169
stable equilibrium, 175
stable line, 40
stable subspace, 108, 109f
standard basis, 28, 76
state space, 259, 278
Steve. See Bob
straight-line solution, 33, 52
subspace
definition of, 77, 89–90
stable, 108, 109f
unstable, 108, 109f
symbolic dynamics, 342–347
symmetric matrix, 207
T
tangent bifurcation, 332
tangent space, 286
Taylor expansion, 166
tent map, 339–340
tertiary expansion, 351
three-parameter space, 71–72
threshold level, 238
time derivative, 402
time t map, 64
torus, 116–117
total energy, 280
total energy function, 197
trace-determinant plane, 61–63, 64f
transitive, 323
two parameters, 15–16
two-body problem, 292–293
U
undamped harmonic oscillator, 55,
114, 208
uniform convergence, 389
unrestricted voltage state, 259
unstable curve, 169
unstable line, 40
unstable subspace, 108, 109f
V
van der Pol equation, 261–270
variation of parameters, 130–131,
134
variational equation, 149–153, 172,
401
vector
definition of, 83
momentum, 281
vector field
complex, 182–184
definition of, 24
divergence of, 309
Lorenz, 306
vector notation, 21–22
Volterra-Lotka system, 240
W
web diagram, 331
Z
zero velocity curve, 286, 287f