PM. See Poincaré map
Poincaré map, 329, 451
experimental, 332
for autonomous systems, 331
for periodically driven systems, 330
invariant manifolds, 351
orbit, 351
phase of, 331
saddle-point, 351
stable/unstable manifolds, 351
Poincaré stability, 297
Pointwise dimension, 339
Poisson’s ratio for engineering
materials, 496
Polar form
of amplitude function, 133
of Hopf bifurcation, 278, 283
of Laplacian operator, 485
of membrane equation, 492
Polygenic forces, 22
Positional bias, 416–418
Positive definite
EVP, 66, 68
matrix, 51, 480
Post-buckling, 55
Potential
magnetic, 99, 358
Potential barrier, 359, 397
Potential energy
for pendulum type systems, 396
of beam in bending, 15, 23
of pendulum type systems, 211, 240
of stretched string, 243
Predictability, 336
horizon, 337, 338
Predictive tools, 348
Pre-tensioned beam, 150
Primary resonance, 132, 134, 140, 153, 161,
213, 214, 222, 252
Principal minor, 480
Pseudo-arclength continuation, 141, 299, 304
Pseudo-phase-space, 328
Pythagoras, vii
Q
Quadratic
damping, 99
map, 342, 343
tangency, 281, 283
Quasi-equilibrium, 416, 443
Quasiperiodic
oscillations, 293, 328, 344
route to chaos, 343
R
Radius of gyration, 152
Rayleigh, 67
method, 18
oscillator, 171, 173
quotient, accuracy of, 86
quotient, 18, 67, 70, 71, 74, 75
Rayleigh, Lord, vii
Rayleigh-Ritz method, 74, 76
Reciprocity theorem, Maxwell’s, 42
Regular
orbit, 107
point, 106
Resonance
anti-, 10
autoparametric, 210
buildup, 34
combination, 132
external, 210, 372
frequency, 4
internal, 210, 214, 220, 372
primary (external), 153
primary parametric, 132
secondary, 161
subharmonic, 161
superharmonic, 161
Resonance capture, 210
Resonant
terms, 119, 122
vibrations, 4
Response
backbone, 141
curve, 4
peak, 140
Rings, 491
Ritz method, 19
Robots, 377
Rocking objects, 368
Rods
axial vibrations of, 59, 486
torsional vibrations of, 487
Rotary inertia, nontrivial effects of, 257
Routes to chaos, 341
intermittency, 345
period-doubling, 341
quasiperiodic, 343
Ruelle-Takens-Newhouse, 343
transient, 345
Routh-Hurwitz criterion, 232, 480
Rubber
band, twisted, 381
elasticity, 99
Ruelle-Takens-Newhouse route to chaos, 343
526
Index
Poincaré map, 329, 451
experimental, 332
for autonomous systems, 331
for periodically driven systems, 330
invariant manifolds, 351
orbit, 351
phase of, 331
saddle-point, 351
stable/unstable manifolds, 351
Poincaré stability, 297
Pointwise dimension, 339
Poisson’s ratio for engineering
materials, 496
Polar form
of amplitude function, 133
of Hopf bifurcation, 278, 283
of Laplacian operator, 485
of membrane equation, 492
Polygenic forces, 22
Positional bias, 416–418
Positive definite
EVP, 66, 68
matrix, 51, 480
Post-buckling, 55
Potential
magnetic, 99, 358
Potential barrier, 359, 397
Potential energy
for pendulum type systems, 396
of beam in bending, 15, 23
of pendulum type systems, 211, 240
of stretched string, 243
Predictability, 336
horizon, 337, 338
Predictive tools, 348
Pre-tensioned beam, 150
Primary resonance, 132, 134, 140, 153, 161,
213, 214, 222, 252
Principal minor, 480
Pseudo-arclength continuation, 141, 299, 304
Pseudo-phase-space, 328
Pythagoras, vii
Q
Quadratic
damping, 99
map, 342, 343
tangency, 281, 283
Quasi-equilibrium, 416, 443
Quasiperiodic
oscillations, 293, 328, 344
route to chaos, 343
R
Radius of gyration, 152
Rayleigh, 67
method, 18
oscillator, 171, 173
quotient, accuracy of, 86
quotient, 18, 67, 70, 71, 74, 75
Rayleigh, Lord, vii
Rayleigh-Ritz method, 74, 76
Reciprocity theorem, Maxwell’s, 42
Regular
orbit, 107
point, 106
Resonance
anti-, 10
autoparametric, 210
buildup, 34
combination, 132
external, 210, 372
frequency, 4
internal, 210, 214, 220, 372
primary (external), 153
primary parametric, 132
secondary, 161
subharmonic, 161
superharmonic, 161
Resonance capture, 210
Resonant
terms, 119, 122
vibrations, 4
Response
backbone, 141
curve, 4
peak, 140
Rings, 491
Ritz method, 19
Robots, 377
Rocking objects, 368
Rods
axial vibrations of, 59, 486
torsional vibrations of, 487
Rotary inertia, nontrivial effects of, 257
Routes to chaos, 341
intermittency, 345
period-doubling, 341
quasiperiodic, 343
Ruelle-Takens-Newhouse, 343
transient, 345
Routh-Hurwitz criterion, 232, 480
Rubber
band, twisted, 381
elasticity, 99
Ruelle-Takens-Newhouse route to chaos, 343
526
Index