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Index
A-stable, 301, 302absolutely continuous
conditional measure, 459measure, 459
absolutely stable, 301absorbing state, 510Ackerberg–O’Malley resonance, 69action, 646
functional, 514action-angle coordinates, 273, 316,
646active node, 698Adams–Bashforth method, 299adaptive
dynamics, evolution, 680mesh, 308network, 697self-organized criticality, 700
adiabaticinvariant, 648theorem, 695
adjoint operator, 481adjoint representation, 333admissible control, 606admissible time change
hysteresis operator, 635advection–reaction–diffusion, 572affine coordinates, 174Airy equation, 107Allee effect, 559alpha limit set, 527amplitude, 278
equation, 279, 283, 286equation, PDEs, 614
analytic continuation, 363
angle, 646annealed network, 697appreciable hyperreal, 640approximate identity, 580approximation
geometrical optics, 262physical optics, 262zeroth-order, 102
ARD, 572Arrhenius–Eyring–Kramers law,
517Arzela–Ascoli, 610asymptotic
expansion, 15expansion, Gevrey, 269matching, 240, 243, 248matching condition, 245moment of falling, 364moment of jumping, 364phase, 340rate foliation, 46sequence, 15sequence, logarithmic, 107series, 16, 92
asymptoticallyflat (Gevrey), 269flat (Poincare), 269
asymptoticsPoincare, 270via blowup, fold, 262
atlas, 28attracting normally hyperbolic, 55attractor, 436
Axiom A, 460Lorenz, 468
© Springer International Publishing Switzerland 2015C. Kuehn, Multiple Time Scale Dynamics, Applied MathematicalSciences 191, DOI 10.1007/978-3-319-12316-5
799
800 INDEX
maximal, 436strange, 436
attractor–repeller pair, 527autocatalator, 674auxiliary canard, 420average, 542average connectivity, 699averaged equation, 266averaging, 265
fast subsystem, 267theorem, 266
Axiom A attractor, 460
B-tipping, 655Bezout’s lemma, 379backward
differentiation formula, 302, 626Kolmogorov equation, 481, 517
Banachcontraction mapping theorem, 35manifold, 588space, 37
Barkley model, 684basin of attraction, 514BDF, 302, 626Belousov–Zhabotinskii reaction, 398,
673Bernoulli shift, 438Berstein-type inequality, 488beyond all orders, 15, 248bi-infinite sequences, 438bifurcation, 53
critical transition, 656cusp, 63diagram, singular, 409dynamic, 63fold, 62Hopf, 63, 361, 404pitchfork, 265, 386saddle node of limit cycles, 213saddle-node, 62transcritical, 63
bilinear oscillator, 607block diagram, 665blowdown, 187blowup
asymptotics, fold, 262directional, 168directional, repeated, 170directional, rescaled, 168example, polar, R2, 161method, 160planar fold, 180polar, R2, 160polar, Rn, 162polar, rescaled, 164polar, weighted, 173quasihomogeneous, 173repeated, construction of, 165transcritical point, 231weighted, directional, 176, 177
Boltzmann constant, 497Bornholdt–Rohlf model, 698boundary
condition, separated, 303function, 257function method, 255layer, 693manifold, 149map, 536
boundary value problem, 303two-point, 303, 393two-point, fast–slow, 149two-point, linear, 76, 94
box, 546Brownian motion, 478, 497, 506
scaling law, 482bundle
normal bundle, 23plane bundle, 561tangent bundle, 23transversal bundle, 30
Burgers’ equation, 563burst
initiation, 416termination, 416
bursting, 398elliptic, 417fold-homoclinic, 416square-wave, 416subHopf–foldcycle, 417
BZ reaction, 673
INDEX 801
C-slow exit/entrance point, 543canard, 200
auxiliary, 420, 423cycle, 212delayed Hopf, 361explosion, 212, 315, 418, 643faux, 215, 218in R
3, node, saddle, 219jump-back, 211jump-forward, 212maximal, 200, 355, 419point, 200, 418, 632secondary, 226, 228, 420secondary, turning point, 228singular, 215, 401singular cycle, 211turning point, Gevrey, 273vrai, 218with head, 212without head, 211
candidate, 64, 462, 655canonical
adaptive dynamics, 680transformation, 646
cap, 547carrier density, 590carrier wave, 613celestial mechanics, 685cell problem, 598center
folded, 218manifold, 49
chain recurrent, 542chaotic, 435, 456, 458characteristic polynomial, 300
first, 299second, 299
characteristics, 563chart, 28
classical, 220fold point, 181rescaling, 220
Chern number, 561classical chart, 220close vector fields, 28cocycle property, 506
coevolutionary network, 697coherence resonance, 479cohomological Conley index, 531collapse time, 365collocation, 307
conditions, 307points, 307
compact perturbation, 591complete orbit, 437complex
fast–slow system, 271phase, 365
composite expansion, 246, 520computational singular perturbation
condition, 334fast fiber, 340method, 331one-step method, 332, 334two-step method, 332, 334
conditionasymptotic matching, 245
conditional measure, 458conditioning, 308
constant, 310ill, 308well, 308
cone, 440condition, 440, 454
conjugatetopologically, 438
Conley index, 531connecting homomorphism, 536connecting orbit, 528connection
matrix, 538problem, 149
conservation law, 562constrained ODE, 620continuation, 312
natural, 314problem, 313property, Conley index, 532related by, 526
contraction mapping theorem, 35control
admissible, 606
802 INDEX
curve, 272theory, 88
convex hull, 373coordinates
affine, 174chart, 28homogeneous, 174
corner layer, 255corrector problem, 598correlation, 504
function, 503cost, 516covariance ellipsoid, 485critical
set, 12, 229set, for 1D map, 457transition, 654, 655
critical manifold, 12, 54normally hyperbolic, 48random, 509S-shaped, 188tame branch, 388
CSP, 331condition, 334fast fiber, 340one-step method, 334two-step method, 334
CU-system, 653curvature, 233cusp point, 63, 167, 469
DAE, 11, 620linear, 623
Dafermos regularization, 568Dahlquist
barrier theorem, 301test equation, 300
damped harmonic oscillator, 274delay
equation, 602fold, 360
delay equation, 586delayed Hopf bifurcation, 361, 407delta-distribution, 580Descartes, folium, 376desingularization, 167
diffeomorphism, 29differential
-algebraic equation, 11, 620form, 129, 646form, basic, 130form, projective, 130inclusion, 605
differentiation index, 620diffusion
matrix, 472, 481, 514process, 480term, 480
Diophantine equation, 379direct sum, 23directional blowup, 168
transition maps, 183discontinuous vector field, 628disordered series, 104distinguished limit, 251domain
of validity, 244sectorial domain, 269
dominant balance, 251, 393Dorodnitsyn’s formula, 111double limit, 494double-well potential, 512Drazin inverse, 623drift, 480drop curve, 463drop point, 109duck, 200, 212Duffing equation, 275Duhamel’s principle, 484, 487, 594dynamic bifurcation, 63Dynkin’s equation, 518
early-warning sign, 654ecology, 679edge of a network, 696eigenspace
center, 48stable, 48unstable, 48
eigenvaluenontrivial, 47operator, 554
INDEX 803
problem, operator, 554singular, 207trivial, 47, 555
eikonal equation, 261ellipsoid, 485elliptic
bursting, 417operator, 595
entrance point, slow, 541envelope, 613
equation, 614epsilon-embedding method, 624equation-free, 344equicontinuity, 610error
equidistribution, 308global, 296HMM, 320local truncation, 296, 320
essential spectrum, 555Estrin–Kubin model, 667Euclidean norm, 4Euler method
explicit, 296implicit, 299modified, 298
Euler–Lagrange equations, 515Euler–Maruyama, 318, 478Evans function, 555, 557evolution operator, 309evolutionary dynamics, 680exchange lemma, 129
C0-version, 129C1-version, 129differential form version, 130MFDE, 577strong version, 148with exponentially small error,
148exchange of stability, 231excitable system, 684, 700excitation, 691existential transfer, 639exit
guide, slow, 543point, immediate, 544
point, slow, 541set, 530time, 481, 514
expansioncomposite, 246in 1D maps, 457inner, 241outer, 241post-Newtonian, 602small noise, 493
explicit method, 298, 299exponential dichotomy, 556extended domain of validity, 244extended system, 117exterior product, 130Eyring–Kramers law, 517
family of cones, 440fast
flow, 12, 47rotating phase, 273subspace, 345subsystem, 12, 54time, 8variable, 8vector field, 12wave, 142, 351
fast–slowdifferential inclusion, 607Markov chain, 511random differential equation, 507system, 8system, complex, 271vector field, 8
faux canard, 215, 218feasible control, 606feedback system, 665Fenichel
normal form, 73, 74theorem, classical fast–slow, 48theorem, fast–slow systems, 55,
328theorem, foliations, 45theorem, general manifolds, 28,
42FHN, 11
804 INDEX
fiberbundle, 45stable, 45, 46unstable, 45
Field–Koros–Noyes mechanism, 673Filippov convex method, 628Fillipov system, 606finite differences, 306finite smoothness, 65first
-order corrector, 598exit-time, 481, 514integral, 202, 646Lyapunov coefficient, 209, 372,
408Lyapunov exponent, 435
FitzHugh–Nagumo equation, 112D ODE, 10, 233, 314, 3722D PDE, 10, 120, 573, 6493D ODE, 141, 350, 406, 534, 650lattice model, 573symmetry, 650
FitzHugh-Nagumo equation3D ODE, 11, 67
FKN mechanism, 673flow, 11, 584
box, 75skew-product, 507
fluid dynamics, 691Fokker–Planck equation, 473, 481fold
bifurcation, 62blowup, 180curve, 80, 353curve, nondegenerate, 80curve, normal form, 80in the slow flow, 433nonautonomous, 103point, 62, 632point, generic, 77, 103, 178, 359,
386, 489point, nondegenerate, 62point, normal form, 77, 80, 359saddle, 450
fold-homoclinic bursting, 416
foldedcenter, 218focus, 218, 410node, 218, 353, 400, 408, 410saddle, 218, 410saddle, example, 214saddle-node, 218, 404, 410singularity, 409singularity in R
2, 198singularity in R
3, 216singularity, generic, R2, 198
foliation, 45asymptotic rate, 46
folium of Descartes, 376forward Kolmogorov equation, 481frozen
component size, 700node, 698
full system, 12function
input-output, 365functional differential equation
mixed, 574retarded, 586
functional equation, 39functional response, 559fundamental solution, 309, 483, 556,
696
gain, 578block, 665medium, 676
galaxy, 640Gaspard–Nicolis–Rossler model, 399gating variable, 669gauge function, 15Gauss points, 307Gaussian random variable, 484generating function, 647generator, 481
infinitesimal, 482Markov chain, 510
generic, 621-parameter family, 198delayed Hopf, 363fold point, 77, 103
INDEX 805
geometricdesingularization, 160singular perturbation theory, 53,
56geometrical optics, 262Gevrey
asymptotic expansion, 269order, 269
Gierer–Meinhardt equation, 682Ginzburg–Landau equation, 615global error, 296good rate function, 514gradient, 5, 132
system, 512, 516graph, 33, 696
transform, 34, 35Green’s function, 309Gronwall’s lemma, 32
generalized, 138GSPT, 53
Henon map, 455half-return map, 447, 462halo, 641Hamiltonian
Schrodinger equation, 695system, 273, 316, 646, 652system, time-dependent, 647
harmonic average, 599harmonic oscillator, 274, 276, 646
damped, 274Hausdorff distance, 55heat equation, 585Heaviside function, 578, 690height
of an edge, 384of the Newton diagram, 384
Hermite polynomial, 222Hermitian conjugate, 590Hessenberg index-1, 621Hessian, 163heteroclinic
cycle, 628orbit, 114, 527orbit, connection problem, 150
heterogeneous, 596
heterogeneous multiscale methods,315
hidden constraint, 622high-gain amplifier, 666HMM, 315Hodgkin–Huxley equations, 669Holling functional response, 559holomorphic, 270homoclinic orbit, 114, 405
computation, FHN, 350double loop, 471existence, FHN, 142Lorenz-type model, 470MFDE, 576n-homoclinic, 407
homogeneous coordinates, 174homogenization, 595, 596homogenized tensor, 596homological Conley index, 531homology, 531homotopy, 470
method, computation, 354Hopf bifurcation, 63, 361
delayed, 361, 407in van der Pol-type system, 208singular, 315, 643singular, R2, 206, 207singular, R3, 404subcritical, 414, 416theorem, 209
horizontalpart, 439, 441surface, 440
horizontally cylindric, 441horseshoe map, 437hydra, 681hyperbolic
conservation law, 571manifold, 41matrix, 48normally, 41splitting, 41
hyperreal number, 638, 640hysteresis, 633
operator, 635
806 INDEX
ILDM, 345
ill conditioned, 308
implicit
Euler method, 299
function theorem, 59
method, 299
index
bundle, 550
DAE, 620
Morse, 556
of nilpotency, 623
pair, 530
pair, singular, 544
triple, 536
induced derivation, 146
inequality
Lojasiewicz, 166
inertial manifold, 595, 601
infinitely
close, 641
large, 638
small, 638
infinitesimal generator, 481, 482
infinitesimal hyperreal, 640
inflection point, 233
inflowing
invariant manifold, 21
inhibition, 691
initial conditions
sensitive dependence, 435
inner
expansion, 241
layer, 242
product, 5
input-output function, 365
integer programming, 388
integrating factor, 137
integrator block, 665
integrodifferential equation, 578,590
interaction kernel, 578
interior layer, 150, 242, 253
intersection
transverse, 142
interval, 529interval arithmetic, 449intrinsic low-dimensional manifold,
345invariance equation, 328, 336, 604,
632invariant
globally, 20locally, 49, 55manifold, 22manifold, random, 508measure, 458negatively, 45positively, 46probability measure, 317set, 587subspaces, blowup, 184
inverse function theorem, 30isola, 485isolating
block, 531neighborhood, 526, 540neighborhood, singular, 541
Ito integral, 480Ito’s formula, 480, 483iterative method
slow manifold, 347
Jeffrey–Hamel flow, 691jet, 163jump, 97
-back canards, 211-forward canards, 212-on set, 151matrix, 510point, 109, 188, 191
Kepler’s second law, 686kinematic viscosity, 692Kirchhoff’s law, 668Kolmogorov’s
backward equation, 481, 517forward equation, 481
Koper model, 354, 408symmetric, 409
Kramers law, 517
INDEX 807
Lagrangian viewpoint, 693Landau notation
big-O, 14little-o, 14
Langevin equation, 497LAO, 399Laplace method, 490Laplace transform, 501Laplacian, 5, 611large deviation
principle, 514theory, 514
large-amplitude oscillation, 399laser, 590, 676lattice differential equation, 573layer
equations, 12inner, 242interior, 150, 242, 253outer, 242problem, 12
LDP, 514leaf, 45Lebesgue space, 587Lienard transformation, 9Lienard transformation, 573Lie bracket, 333light amplitude, 590limit
distinguished, 251set, alpha, 527set, omega, 527
limited distance apart, 640limited hyperreal, 640Lin’s method, 653Lindstedt’s method, 276linear
k-step method, 299DAE, 623multistep method, 298recurrence relation, 299stability, traveling wave, 555system, 86
link, 696Liouville equation, 472Lipschitz
set-valued map, 606local
truncation error, 296, 320vector field, 177
locally invariant manifold, 22, 55logarithmic equivalence, 515logistic map, 433Lojasiewicz inequality, 166Lorenz attractor, 468Lyapunov
-type numbers, 23, 40coefficient, Hopf bifurcation, 209,
408equation, 484exponent, 435exponent, first, 435number, 435
macro solver, 316macroscopic variable, 316magnifying glass, 85, 644manifold, 6, 54
boundary, 149center-stable, semiflow, 589center-unstable, semiflow, 589critical, 12, 54hyperbolic, 41inertial, 595, 601inflowing invariant, 21intrinsic low-dimensional, 345invariant, 22locally invariant, 22normally hyperbolic, 41, 48, 54overflowing invariant, 22slow, 56stable, 42unstable, 42
maprank-one, 456singularly perturbed, 455
Markov chain, 510martingale, 488mass action kinetics, 674matched asymptotic expansion, 240,
243, 248maximal canard, 200
808 INDEX
Maxwell–Bloch equation, 593measure
F -invariant, 458-preserving, 506acc, 459conditional, 458physical, 459SRB, 459
Melnikovmethod, 228, 684theory, 156
membrane potential, 668mesh, 306
adaptation, 308metastable, 481method
of characteristics, 563of multiple scales, 277
metric dynamical system, 506MFDE, 574Michaelis–Menten kinetics, 688Michaelis–Menten–Henri, 338micro solver, 316microscopic variable, 316mild solution, 594Minkowski sum, 373mitotic oscillator, 688mixed functional differential equation,
574mixed-mode oscillations, 398mixing, 436, 458MMH, 338MMO, 398modified Euler method, 298modulation equation, 614moment
of falling, 364of jumping, 364
Morris–Lecar model, 413, 416, 689Morse
decomposition, 529index, 556
movingframe, 339window, 658
multi-index, 280, 377
multiplescales method, 277, 596shooting, 306
multiplier, 437multistep method, 298, 625
natural continuation, 314near-identity transformation, 273negatively invariant, 45neighborhood
isolating, 526Nernst potential, 668Netushil’s principle, 635networks, 696neural competition, 485neural field, 578neuroscience, 668Newell–Whitehead–Segel equation,
614Newton
diagram, 374diagram, height, 384law of gravitation, 686method, 307open polygon, 375polygon, 373
Newtonian expansion, 602nilpotency index, 623node
folded, 218, 353of a graph, 696
noise level, 493noise-induced
effect, 479transition, 479
nondegeneratefold curve, 80fold point, 62
nonsmooth system, 628nonstandard analysis, 637nonwandering set, 443norm, 4
sup-norm, 37Euclidean, 4
normalbundle, 23
INDEX 809
form, 71hyperbolicity, semiflow, 588space, 23switching condition, 80, 191, 461
normal formFenichel, 74fold point, 77, 359Hopf bifurcation, 361linear system, 87theory, 284
normally hyperbolicattracting, 55manifold, 41, 48, 54planar system, 98repelling, 55saddle type, 55splitting, 41
numerical continuation, 312
ODE, 3omega limit set, 527one-step method, 298operator norm, 587orbit
complete, 437heteroclinic, 114homoclinic, 114
order, 298higher than, 14not lower than, 14partial, 529total, 529
Oregonator, 674Ornstein–Uhlenbeck process, 484, 503oscillation
bursting, 398large-amplitude, 399mixed-mode, 398relaxation, 97relaxation, simple, 97small-amplitude, 399
outerexpansion, 241layer, 242
overflowing invariant manifold, 22overlap domain, 244
parthorizontal, 439, 441vertical, 439, 441
partial order, 529PDE, 4period
of oscillation, vdP, 108of relaxation oscillation, R3, 194
perturbationregular, 95singular, 95
phasecomplex, 365
physical measure, 459physical optics, 262, 695pitchfork bifurcation, 265, 386Planck’s constant, 694plane bundle, 561plastic deformation, 667play operator, 633Poincare asymptotics, 270Poincare–Lindstedt method, 276point spectrum, 555polar blowup
R2, 160
Rn, 162
rescaled, 164weighted, 173weighted, rescaled, 173
population inversion, 676positive averages, strictly, 542positively invariant, 46, 587
random set, 507post-Newtonian expansion, 602potential
double-well, 512quasi-, 516
powerpositive, 458ramp, 372transformation, 377transformation, unimodular, 377
predator–prey, 558, 679predictor–corrector, 313principal solution, 309, 483, 556,
696
810 INDEX
probabilitydensity, 480distribution, 480
projective space, 174propagation failure, 575propagator, 309pullback, 130pushforward, 130, 162
set, 544
Q-matrix, 510quantum adiabatic theorem, 695quantum Hamiltonian, 695quartic oscillator, 261quasihomogeneous
blowup, 173fold, 180function, 172vector field, 172, 393
quasipotential, 516quasistationary distribution, 512
Radon–Nikodym derivative, 459ramp, 370random
critical manifold, 509differential equation, 507dynamical system, 506equilibrium point, 509invariant manifold, 508set, 507
rank-one map, 456Rankine–Hugoniot condition, 565rarefaction, 566rate
function, 514independence, 635independent memory, 635
Rayleigh’s equation, 280RDE, 507RDS, 506reaction–diffusion equation, 554, 586recall, 29rectangle, 441rectification lemma, 75recurrence relation, 299
reduced flow, 11reduced problem, 11reflection coefficient, 695region of absolute stability, 301regular
pair, 623perturbation, 95point, fast–slow, 58
regularizationDafermos, 568nonsmooth, 630viscous, 567
related by continuation, 526relative homology, 531relaxation oscillation, 97, 188
asymptotics in R3, 194
existence in R2, 188
existence in R3, 190
period, vdP, 108simple, 97, 109
renormalization, 285renormalization group, 279
condition, 280, 283equation, 286equation, kth-order, 289transformation, 286transformation, kth-order, 289
repeated blowup, 165repelling normally hyperbolic, 55replica, 318rescaling, 83
chart, 220time, 195
resonance, 219Ackerberg–O’Malley, 69coherence, 479stochastic, 479
resonant term, 284retarded functional differential
equation, 586, 600Reynolds number, 692Riccati equation, 107Riemann
–Dafermos solutions, 568problem, 567
INDEX 811
Rosenzweig–MacArthur model, 558,679
rotation
sector of, 403, 424Runge–Kutta method, 298
saddle-focus, 407-node bifurcation, 62
-node, folded, 218-type slow manifold, 310, 348
node of limit cycles, 213type, normally hyperbolic, 55folded, 218
sample path, 318, 478SAO, 399scale-invariant conservation laws, 568
scaling law, 658Brownian motion, 482
fold, 360Schilder’s theorem, 515Schrodinger equation, 260, 694
Schurdecomposition, 345vector, 345
SDE, 478secondary canard, 226, 228, 420section, 30, 33
sector of rotation, 403, 424sectorial domain, 269
secular terms, 275, 687self-organized criticality, 700semiexplicit system, 620
semiflow, 584semigroup, 584sensitive dependence, 435
separation time, 365series
asymptotic, 92disordered, 104uniformly convergent, 92
set-valued map, 605Lipschitz, 606
shadow, 641
Sharkovskyordering, 434theorem, 434
shiftBernoulli, 438
shock, 564shooting
multiple, 306simple, 305
simple shooting, 305Sinai–Ruelle–Bowen measure, 459single-neuron models, 668singular
bifurcation diagram, 409, 649canard, 215canard cycle, 211eigenvalue, 207faux canard, 215Hopf bifurcation, 315, 643Hopf bifurcation, R2, 206, 207Hopf bifurcation, R3, 404, 408index pair, 544isolating neighborhood, 541limit, 12perturbation, 95point, fast–slow, 58point, singularity theory, 58trajectory, 65transition matrix, 539
singular perturbationcomputational, 331geometric theory, 56problem, 95
singularityfolded, in R
3, 216transcritical, 229
singularly perturbed map, 455skew-product flow, 507sleep–wake cycle, 689sliding
flow, 629window, 658
slowentrance point, 541entrance point, strict, 545exit guide, 543
812 INDEX
exit point, 541flow, 11, 47, 54, 58manifold, 56manifold, finite smoothness, 65manifold, iterative, 347manifold, saddle-type, 310, 348subspace, 345subsystem, 11, 54time, 8variable, 8vector field, 11wave, 142
slowing down, 657Smale horseshoe, 437, 443
existence theorem, 441small-amplitude oscillation, 399smooth manifold, 54SMST algorithm, 348Sobolev space, 515special
flow, delay equation, 601linear group, 377solution, delay equation, 601
spectralgap, 696gap, RDEs, 508problem, 554
spectrum, 554essential, 555point, 555
spike, 399, 669spiral wave, 684splitting, 439
hyperbolic, 41normally hyperbolic, 41
square-wave bursting, 416SRB measure, 459stability polynomial, 300stable
absolutely, 301fiber, 45, 46manifold theorem, 20subspace, 20
standardpart, hyperreal, 641rectangle, 439
standing wave, 694state space form method, 625static network, 697stiff differential equation, 297stiffness ratio, 297stochastic resonance, 479Stokes number, 693Stommel flow, 625, 693Stommel–Cessi model, 659stop operator, 637strained coordinate, 276strange attractor, 436strict slow entrance point, 545strictly positive averages, 542structural stability, 443subHopf–foldcycle bursting, 417subordinate, 458subspace
fast, 345slow, 345stable, 20unstable, 20
sufficiently small, 8sup-norm, 37superslow, 418support function, 609surface
horizontal, 440vertical, 440
Swift–Hohenberg equation, 611symbol sequence for MMOs, 399symbolic dynamics, 438symmetry
FHN, 650periodically forced vdP, 447
symplectic 2-form, 646system
extended, 117of first approximation, 83, 85, 222
systems biology, 688
tame branch, 388tangent
bundle, 23invariant manifolds, 413prediction, 313
INDEX 813
test function, 565thermal conductivity tensor, 595thermohaline circulation, 658three-time-scale, 417Tikhonov’s theorem, 56time
rescaling, 195scale, 8step, 296
topologicallyconjugate, 438transitive, 436
total order, 529tourbillon, 408trait, 680transcendentally small, 248transcritical point, 63, 229transfer principle, 639transformation
canonical, 646power, 377
transitionfunction, nonsmooth, 630map, 177maps, blowup, 183matrix, 539matrix, singular, 539point, 655probability, 481, 511
transitivetopologically, 436
translation invariance, 555transmission coefficient, 695transport equation, 261, 594transversal
bundle, 30intersection, 142
transversality, 29condition, fold, 77condition, fold curve, 80parametric, 1D maps, 457
trapezoid rule method, 299traveling
back, 120front, 120pulse, 120
pulse, neural field, 579wave, 10, 120wave train, 120wave, nonlinear stability, 555wave, spectral stability, 555
trivial eigenvalue, 555truncation, 380tube, 546tubular neighborhood, 188tunneling, 694turning point, 62, 69, 272, 304, 695
of canards, 228twist, 401two-body problem, 685two-scale convergence, 599two-timing, 277, 687
ultrafilter, 639uniformity lemma, 26, 32unimodal map, 433unimodular power transformation,
377universal transfer, 639unlimited hyperreal, 640unstable
fiber, 45subspace, 20
upper semicontinuous, 608
van der Pol equation, 9, 16constant forcing, 9, 201, 233, 234,
305, 478, 642equation, 194periodically forced, 10, 13, 82, 97,
190, 222, 442, 460stochastic, 478, 494unforced, 9, 13, 59, 96, 102, 108,
252, 330, 541, 545variable
macroscopic, 316microscopic, 316
variation of constants, 484, 594variational equation, 117, 451vdP, 9vector bundle, 23vector exponent, 377
814 INDEX
vector fielddiscontinuous, 628local, 177perturbed, 28quasihomogeneous, 172,
393unperturbed, 28
vertex, 696vertical
part, 439, 441surface, 440
vertically cylindric, 441viscosity, 692viscous profile criterion, 567viscous regularization, 567vrai canard, 218
wandering set, 443Wang–Young theory, 456wave
fast, 142function, 694number, 612slow, 142speed, 10, 120vector, 612
way-in/way-out map, 365Wazewski property, 532weak solution
conservation law, 565Weber equation, 221wedge product, 130Weierstrass canonical form, 623weighted
directional blowup, 176, 177polar blowup, 173
well conditioned, 308Wentzel–Kramers–Brillouin, 259white noise, 472Whitney sum, 562Wiener
measure, 506space, 506
WKB, 259, 518delayed Hopf, 371
Yamada model, 678
ZDP, 342zero section, 30zero-derivative principle, 342zoom, 85