Beams (cont.)
orthogonally clamped, 227, 375
spring-tensioned, 464
Beating response, 248, 254
Beck’s column, 228
Bernoulli, Daniel, vii
Bernoulli-Euler
beam, 487
theory, 487
Biasing. See High-Frequency (HF) effects
Bifurcation, 115, 145, 273
catastrophic, 294
codimension, 274
codimension one, 271, 274, 280, 292, 293,
316
codimension two, 274, 292, 294
continuous, 294
dangerous, 294
diagram, 274
discontinuous, 294
divergence, 237
dynamic, 294
explosive, 294
flip, 293
fold, 277
for a Duffing equation, 310
for the autoparametric vibration absorber,
315
for the follower-loaded double pendulum,
317
for the pendulum with a moving support,
313
generic, 275
global, 341, 369
grouping according to effect, 293
Hopf, 145, 278
into chaos, 341
inverse tangent, 346
necessary condition for, 274
Neimark, 293
of periodic solutions, 292
parameter, 272
period-doubling, 293
pitchfork, 275
point, 145, 273, 275, 280, 281, 286, 303
saddle-node, 277
secondary Hopf, 293
set, 273
stability to perturbations, 294, 297
stable/unstable, 294
static, 294
subcritical (hard), 145
supercritical (soft), 145
tangent, 277, 293
transcritical, 277, 293
turning point, 277, 293
unfolding, 274
value, 145, 273
Bifurcational stability, 297
Bifurcation diagrams
Hopf, subcritical, 279
Hopf, supercritical, 279
pitchfork, perturbed, 295
pitchfork, subcritical, 276
pitchfork, supercritical, 276
saddle-node, 277
saddle-node, perturbed, 295
transcritical, 278
Bi-linear spring, 101
Blue sky catastrophe, 294
Bouncing ball, 368
Boundary conditions
essential, 19, 63
natural, 63
suppressible, 63
Bounded stability, 296
Brumberg’s pipe, 401
Buckling, 49, 53
due to gravity, 57
Euler elastica, 380
mode, 55, 276
post-, 55
Bursting oscillations, 147
C
Capacity dimension, 339
Catastrophic bifurcation, 294
Center, 107, 115
Center manifold, 274, 284, 285
approximation to, 288, 290
local graph for, 287
reduction, 284, 287
theorem, 284, 285
Chaos
asymptotic, 378
controlling, 381
destroys information, 337
detecting, 327
deterministic, 323
elastostatical, 378
Hamiltonian, 330
hyper-, 335
life on the edge of, 383
precluding, 358
predicting, 348
spatial, 381
spatiotemporal, 381
transient, 349, 371
518
Index
orthogonally clamped, 227, 375
spring-tensioned, 464
Beating response, 248, 254
Beck’s column, 228
Bernoulli, Daniel, vii
Bernoulli-Euler
beam, 487
theory, 487
Biasing. See High-Frequency (HF) effects
Bifurcation, 115, 145, 273
catastrophic, 294
codimension, 274
codimension one, 271, 274, 280, 292, 293,
316
codimension two, 274, 292, 294
continuous, 294
dangerous, 294
diagram, 274
discontinuous, 294
divergence, 237
dynamic, 294
explosive, 294
flip, 293
fold, 277
for a Duffing equation, 310
for the autoparametric vibration absorber,
315
for the follower-loaded double pendulum,
317
for the pendulum with a moving support,
313
generic, 275
global, 341, 369
grouping according to effect, 293
Hopf, 145, 278
into chaos, 341
inverse tangent, 346
necessary condition for, 274
Neimark, 293
of periodic solutions, 292
parameter, 272
period-doubling, 293
pitchfork, 275
point, 145, 273, 275, 280, 281, 286, 303
saddle-node, 277
secondary Hopf, 293
set, 273
stability to perturbations, 294, 297
stable/unstable, 294
static, 294
subcritical (hard), 145
supercritical (soft), 145
tangent, 277, 293
transcritical, 277, 293
turning point, 277, 293
unfolding, 274
value, 145, 273
Bifurcational stability, 297
Bifurcation diagrams
Hopf, subcritical, 279
Hopf, supercritical, 279
pitchfork, perturbed, 295
pitchfork, subcritical, 276
pitchfork, supercritical, 276
saddle-node, 277
saddle-node, perturbed, 295
transcritical, 278
Bi-linear spring, 101
Blue sky catastrophe, 294
Bouncing ball, 368
Boundary conditions
essential, 19, 63
natural, 63
suppressible, 63
Bounded stability, 296
Brumberg’s pipe, 401
Buckling, 49, 53
due to gravity, 57
Euler elastica, 380
mode, 55, 276
post-, 55
Bursting oscillations, 147
C
Capacity dimension, 339
Catastrophic bifurcation, 294
Center, 107, 115
Center manifold, 274, 284, 285
approximation to, 288, 290
local graph for, 287
reduction, 284, 287
theorem, 284, 285
Chaos
asymptotic, 378
controlling, 381
destroys information, 337
detecting, 327
deterministic, 323
elastostatical, 378
Hamiltonian, 330
hyper-, 335
life on the edge of, 383
precluding, 358
predicting, 348
spatial, 381
spatiotemporal, 381
transient, 349, 371
518
Index