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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

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