Subject Index

1/3.41613062 . . . – convergence radius of series(5.64), 529

– spectral radius of operator (7.12), 555– value of RK,1/r(K) for constant K , 523β-neighbourhood, 443δ�(f ), 540, 557GLn(R), 443μ-Lipschitzian map, 501n, 443∂�, 443�ed(L), 194�-divisor, 229

A(�)-module At1(G), 32

Abel identity, 293Abel map, 228absolutely irreducible, 13adjoint, 373, 416Airy differential equation, 289algebraic solvability, 624, 627algebroid solution, 350, 352–356almost automorphic– extension, 144, 162, 168– flow, 144– function, 144almost complex structure, 665almost periodic flow, 143alternating group, 98analytic Hadamard–Perron theorem, 632analytical solvability, 629arbitrary growth theorem, 344arranging equivariant spectral data, 74Arzelà–Ascoli theorem, 473asymptotic cycle, 609, 654asymptotic phase, 367, 387, 394–396, 401auxiliary function, 88auxiliary G-invariant function, 7

Baker–Akhiezer function, 241Banach G-representations, 17Banach vector bundle, 19basic degree, 45basic map, 37, 44, 45

Bebutov construction, 136–138, 143, 145Bebutov flow, 136Beltrami differential, 667Beltrami equation, 667Bessel equation, 611bi-orientable, 32bifurcation, 87Birkhoff Ergodic Theorem, 145Birkhoff recurrence, 143BL-condition, 294BL-conjecture, 294BL-function, 294Blaschke-oscillatory, 308, 310Bloch, 315Bogdanov–Takens prenormal form, 673Bogdanov–Takens singularity, 618, 671Bohr almost periodicity, 143Borel transform, 651Borsuk theorem, 473boundary conditions– linear– – periodic, 444– – three-point, 444– non-linear– – separated, 444, 551– – two-point, 444boundary value problem– regular, 443– singular, 444bounded mean motion, 178Briot–Bouquet differential equation, 271, 283, 342,

343Briuno condition, 658Brouwer degree, 471Burnside ring, 28, 29B(y,β), 443

C-complementing, 44– map, 40, 45– pair, 40, 45CnT(τ,E), 510

C([0, T ] ×D,Rn), 443Camassa–Holm equation, 181

697

698 Subject Index

Cameron’s theorem, 158, 161, 167Carathéodory conditions, 529center, 3, 94, 624, 625– isolated, 3– manifold, 643, 647center–focus problem, 623chaotic behaviour, 367, 402, 407character, 143– of representation, 15characteristic– equation, 54, 77, 93, 114– operator, 54, 93– root, 54, 93– trajectory, 623Choquet theory, 157, 204closed– gap, 253– operator, 21Clunie, 337, 339, 345, 352– lemma, 322cocycle, 137, 145coincidence problems, 84community matrix, 113complementing function, 60complete reducibility theorem, 13completely continuous map, 50complex function spaces, 307, 312, 314, 320– Bergman space, 313, 314– – weighted, 313, 314– Bloch space, 307, 314– BMOA, 315– Dirichlet space, 314– – weighted, 314– Hardy space, 307, 310, 313, 332– – weighted, 332– Korenblum space, 313, 318– Nevanlinna class, 313– normal functions, 315, 322– Qp -space, 307, 314complex isotypical decomposition, 17complexification, 14condensing– field, 50– map, 50cone construction, 257conjugacy class, 11conjugation, 14continuation, 278, 356– along the curve, 276– of local solutions, 275–277continued, 336continuous family of equivariant Fredholm opera-

tors of index zero, 85

crossing numbers, 63, 71, 77, 95, 96, 101, 109cusp, 618, 674, 682

Dβ , 443defect, 337, 338defining equation, 602Denjoy cocycle, 180density of exponential dichotomy, 251desingularization, 619dicritical– davison, 618– node, 617diffeomorphism tangent to identity, 633differential– field, 343, 348–350– independence, 349– of the first kind, 226– of the second kind, 231– of the third kind, 229, 231differentially– algebraic, 349– elementary functions, 349– transcendental, 349diffusion equation, 113dihedral group, 98, 122direct growth problem, 320disconjugate, 308–310distal– extension, 161– pair, 161distance, 310divisor, 227dominating, 67– orbit types, 9, 67Duffing equation, 477dynamical spectrum, 147, 148, 151

Ecalle–Voronin moduli, 632, 636, 650Ecalle–Voronin theorem, 636, 639eigenvalue, eigenfunction, 184elementary singular point, 617equilibrium point, 104equivalence of– diffeomorphisms, 633, 636, 641, 642, 649– of linear systems, 596, 602, 608– of vector fields, 616, 656, 657equivariant Dugundji theorem, 26ergodic– measure, 144– theory, 144exceptional divisor, 617exponent of convergence, 288, 289, 293exponential dichotomy, 141, 147, 194, 200, 250

Subject Index 699

Favard property, 139, 161Favard theory, 165field, 49finite oscillation property, 290, 306finiteness degree, 318– of growth, 297first focus quantity, 628fixed singularities, 326, 335Floquet exponent, 201Floquet matrix, 153Floquet theory, 153, 155, 303flow, 142, 143– homomorphism, 143– isomorphism, 143folding homomorphism �l , 74Fredholm operator, 20Frei theorem, 318frequency module, 144function– (τ,E)-proper, 510– τ -even, 510– τ -odd, 510– approximate determining, 470– determining, 470– even, 510– odd, 510– T -periodic, 443fundamental domain, 24

G-action, 10G-equivariant field, 48G-equivariant homotopy of compact fields, 49G-equivariantly homotopic, 49G-fundamental, 51G-homotopy, 25G-invariant norm, 17G-manifold, 19G-representation, 12G-representation conjugate, 14G-space, 10G-vector bundle, 19Gauss hypergeometric equation, 596, 602generalized �-function, 235generalized cusp, 671generalized Jacobian variety, 234generalized Riemann vector, 235generalized saddle, 671generalized saddle–node, 671Gevrey series, 651Gilbert–Pearson theory, 140, 191, 205global bifurcation problems, 81global continuation of bifurcating branches, 109global Hopf bifurcation, 111– theorem, 82

global oscillation property, 290, 306good resolution, 617, 619Green’s function, 195

H -fixed-point subset, 11(H)-normality, 27Haar integral, 15Hartman argument, 210Hausdorff metric, 21Herglotz function, 207Hermite–Weber differential equation, 339Hölder theorem, 349holomorphic differential, 226, 231holomorphic foliations, 616holonomy transformation, 647, 648homoclinic orbit, 403homotopy factorization, 40Hopf bifurcation, 54Hukuhara–Kimura–Matuda theorem, 669hull, 137, 143Hutchinson model, 112hyperbolic, 310, 367, 386, 405hyperbolic distance, 311hyperbolic periodic orbits, 381

icosahedral group, 98, 124index of L, 21indicial equation, 301induction over orbit types, 23intristic dimension, 16invariant– measure, 144– set, 143inverse growth problem, 320irreducible representation, 12irregular singular point, 596, 603isolated, 94– center, 77isometric Hilbert G-representation, 17isotropy, 11isotypical component, 18isotypical decomposition, 15, 18, 55iterated order, 297, 306, 317, 318iterated type, 319

Jacobian variety, 227

Korteweg–de Vries (K-dV) equation, 181Kotani theory, 212, 215Krein–Rutman theorem, 523Kronecker flow, 144Kronecker winding, 142, 144Krylov–Bogoliubov construction, 145, 152

700 Subject Index

l-folding, 38l-th isotypical crossing number, 63L1([0, T ],Rn), 530Lamé equation, 302, 303lemma of the logarithmic derivative, 286, 351Leray–Schauder twisted degree, 48,·-, 519Li–Yorke chaos, 139, 179lifting homeomorphism, 25limit periodic, 144limit point case, 188linearization of diffeomorphism, 633, 658Liouville-type frequencies, 250Lipschitz condition, 518, 537, 539local bifurcation– � × S1-invariant, 116– invariant, 7, 68, 78, 80, 88– result, 97local existence, 271, 273, 276local index, 35local solution, 336, 356Lyapunov exponent, 146, 151, 175, 201, 202, 253

Malgrange–Sibuya theorem, 608Malmquist, 325, 328, 332, 345Malmquist theorem, 271„ 324, 326, 327, 330, 353,

354map, 26, 27Maple© input data, 102Marcus and Moore disconjugacy, 171Martinet–Ramis moduli, 642, 646, 650, 651, 654,

656Martinet–Ramis theorem, 646Melnikov function, 417Millionscikov–Vinograd examples, 139, 175Millionscikov–Vinograd type, 170minimal flow, 143minimal support, 191modulus– of quadrangle, 668– of unnulus, 668Mohon’ko, 322, 337, 339, 352– lemma, 323monodromic singular point, 623monodromy– operator, 598– transformation, 648movable singularities, 326multiplicativity property, 47, 48

N, 443necessary condition– for Hopf bifurcation, 58– for the occurrence of Hopf bifurcation, 57

negative spectrum, 95, 101, 108, 118Nevanlinna theory, 284–287, 307, 312, 319, 324,

328, 351, 352– characteristic function, 285, 286, 323, 324, 345,

351, 353– counting function, 285, 351, 352– deficiency, 286, 288– first main theorem, 285, 286, 351– logarithmic derivative lemma, 292, 297, 352– non-integrated counting function, 288, 351– proximity function, 285, 351– ramification index, 287, 337–340– second main theorem, 286, 287, 310, 340, 341Newlander–Nirenberg theorem, 666Newton–Puiseux diagram, 290node, 617non-oscillatory, 308–310non-trivial solutions, 87nonautonomous dynamical systems, 135nonlinear Schrödinger (NLS) equation, 181normal, 27normal form– for diffeomorphisms, 634, 641, 649– for linear system, 603– for vector fields, 680normal homotopy, 27n species ecosystem, 112numbers n(L,H), 28numerical shadowing, 368, 425

�-admissible, 26, 49– homotopy, 26�P,σ , 520�(β), 557orbit, 11, 143– space, 11– type, 11orbital equivalence, 616, 642, 646, 649, 674orbital linearization, 665orbital normal form, 649, 656order, 286, 288, 290–292, 294–296Ortega and Tarallo examples, 165oscillatory, 307, 308Oseledets– spectrum, 147, 151– theory, 146

p-summability, 652Painlevé differential equation, 271, 274, 275, 334– Airy solutions, 337– fifth Painlevé equation, 336, 337– first Painlevé equation, 336– first Painlevé hierarchy, 341

Subject Index 701

– fourth Painlevé equation, 336–338– higher order Painlevé equations, 340– rational solutions, 337, 339– second Painlevé equation, 336, 338– second Painlevé hierarchy, 341– sixth Painlevé equation, 336, 337– third Painlevé equation, 336–338Painlevé equation, 335Painlevé property, 334, 335, 340parametrisation– method of, 567parametrized equivariant coincidence problem, 86period matrix, 226periodic, 294, 295, 302, 303, 331– orbit, 367, 386periodic-to-periodic homoclinic orbit, 367, 402,

412, 415perturbed system, 367, 413, 416Picard–Lindelöf theorem, 475Poincaré map, 367, 382, 385, 386, 403, 405Poincaré–Dulac normal form, 629Poincaré–Dulac theorem, 601, 629Poisson recurrence, 193, 194principal solution, 171projective flow, 139, 146, 154, 155, 197proximal extension, 161, 168property (τ,E), 510proximal flow, 161proximal pair, 161pseudo orbit, 425– homoclinic, 426pseudo periodic orbit, 425

quasi-conformal map, 667quasi-periodic flow, 142, 144quasi-section, 258

R, 443R+ , 443Red(L), 194r(A), 443random dynamics, 135ray of division, 606RE±(Lω), 208, 209reaction equation, 113Recurrence Formula, 40, 43, 44regular fundamental domain, 25regular normal– homotopy, 27– map, 27regular singular point, 596–598, 600rescaling, 336resolvent, 191resonance, 601, 629

resonant diffeomorphism, 641, 649resonant one-dimensional map, 649resonant saddle, 630, 647, 656resonant singular point, 617result due to Frei, 296Riccati differential equation, 271, 274, 275, 324,

330–333, 338, 339Riccati equation, 196Riemann inversion problem, 228Riemann theta function, 228robustness, 412rotation number, 151, 152, 197, 202, 251roughness theorem, 373

Sacker–Sell spectrum, 147saddle, 617, 664saddle–node, 617, 642, 656Schwarzian derivative, 332Schwarzian differential equation, 271, 332–334Schwarzmann homomorphism, 141, 201sequence– equicontinuous, 474– uniformly bounded, 473shadow, 425shooting method, 567Siegel lemma, 350singular point of Poincaré type, 629singularity of the Fuchs type, 598small function, 324small meromorphic coefficients, 354smooth G-vector bundle, 20smoothness of the projections, 376SNA (strange nonchaotic attractor), 175, 176solutions of zero order, 344space, 314spectral matrix, 190, 191spectral measure, 189spectrum, 191Splitting lemma, 40, 41stable and unstable manifolds, 367, 386, 395, 396stable fibre, 402stochastic dynamical systems, 135Stokes cocycle, 608Stokes matrix, 607Stokes operator, 607Stokes sheaf, 607strictly ergodic flow, 145strong manifold, 643strongly elliptic case, 167Sturm–Liouville operator, 140, 184, 191subordinate solution, 191subrepresentation, 12subset, 51

702 Subject Index

successive approximations, 453, 457, 470, 486,488, 492, 498, 523, 540, 548, 557, 570

sufficient condition– for Hopf bifurcation, 79, 81– for symmetric Hopf bifurcation, 67– for the occurrence of Hopf bifurcation, 58Suspension Procedure, 40symmetric configuration of transmission lines, 105symmetric Hopf bifurcation, 66– problem, 4symmetric system of the Hutchinson model, 114

Takens prenormal form, 671, 681telegrapher’s equation, 103theorem– Arzelà–Ascoli, see Arzelà–Ascoli theorem– Borsuk, see Borsuk theorem– Krein–Rutman, see Krein–Rutman theorem– Picard–Lindelöf, see Picard–Lindelöf theoremtheorem due to L. Fuchs, 326topological support, 145, 193topological transformation group, 10transmission lines, 102transversal, 367, 405trichotomy, 367, 368, 405trivial solution, 87tubular map, 35twisted (by the homomorphism ϕ :K → S1)

l-folded subgroup, 30twisted conjugacy class, 31twisted equivariant degree, 33

twisted G-equivariant degree, 6twisted subgroups, 30

Uj -multiplicity, 77, 96unbounded mean motion, 177, 178unitary Hilbert G-representation, 17univalence of solutions, 307, 308univalent, 332unstable fibre, 402

Vi -multiplicity, 96Vj,l -isotypical crossing number, 72Valiron–Mohon’ko, 322, 352– theorem, 323Van der Pol equation, 481vector of Riemann constants, 228

W -singular point, 57weakly elliptic case, 167, 169weakly hyperbolic case, 167, 169Weierstraß ℘-function, 302Weyl group, 12Weyl m-function, 188, 190, 194Wiman–Valiron, 290– theory, 288Wronskian, 293, 294

X �� Y , 531

Z, 443zeros, 310