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| Ïîèñê ïî óêàçàòåëÿì |
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| Tabor M. — Chaos and Integrability in Nonlinear Dynamics: An Introduction |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel, Niels 337
Abelian functions, integrals, varieties 337
Action integral 43
Action-angle variables, for many degrees of freedom 68—70 74—77
Action-angle variables, for one degree of freedom 65—68
Action-angle variables, in semiclassical quantization 234 235
Airy function 3
Airy function, and uniform approximations 274
Algebraic curves 336
Algebraically integrable systems 347
Analytic structure, and integrability 39
Analytic structure, of elliptic functions 11
Anosov systems 173
Anti-kink solution 306
Area preserving maps, and Hamiltonian systems 132
Area preserving maps, and surface of section 123
Area preserving maps, invertible 130
Area preserving maps, on the plane 128
Area preserving maps, twist maps 126
Arnold diffusion 74
Averaging, in canonical perturbation theory 98
Baker's transformation 171
Bernoulli equation 93
Bernoulli system 171
Bifurcation, Hopf see “Hopf bifurcation theory”
Bifurcation, period doubling see “Period doubling”
Bifurcation, pitchfork 225
Bifurcation, saddle-node 225
Bifurcation, subcritical 198
Bifurcation, supercritical 198
Bifurcation, tangent 224
Bions 311
Bohr — Sommerfeld quantization 233
Boutroux transformation 328
Branch points, logarithmic 335
Branch points, rational 342
Breathers 311
Burgers equation 352
C-systems 173
Canonical perturbation theory, and small divisors 104
Canonical perturbation theory, for many degrees of freedom 102—104
Canonical perturbation theory, for perturbed oscillator 101
Canonical perturbation theory, to first order 98—100
Canonical perturbation theory, to higher orders 100—101
Canonical transformations 54—63
Cantor set 202 203
Cantori 135
Cat map 168—171
Catastrophe theory 275
Cauchy — Kovaleskaya theorem 352
Caustics 233
Caustics, in quantum maps 260
Chaos, definition of 34
Chaotic advection 176
Characteristics 282
Clairaut's equation 333
Closed orbits 77
Closed orbits, and semiclassical quantization 264—272
Closed orbits, and spectral rigidity 246
cn function 9
Conservation laws 287—288
Conservation laws, and Gardner equation 289
Conservative systems, phase portraits of 17
Constant of motion 2
Constraints, holonomic 43 49
Continued fractions 110—111
Continued fractions, and Greene's method 164
Contravariant variables 47 81
Correspondence identities 234
Correspondence principle 241
Couette flow 192—194
Covariant variables 47 81
Cremona transformation 129
Crisis 221
Curry attractor 210
Cyclic coordinates 46
Damped oscillator 5
Damped oscillator, with driving 36
de Broglie wavelength 230
Deformation parameter 295
Density of states, and closed orbit quantization 264—272
Density of states, Thomas — Fermi formula 243
Differential forms, 1-form 61 83
Differential forms, 2-form 55 85
Differential forms, and Stokes' theorem 87
Differential forms, n-form 86
Direct scattering problem 291 293—294
Discrete Lagrangians 133
Discrete Lagrangians, and quantum maps 256
Dispersion relation, for KdV equation 284
Doubly periodic functions 9 10
Driven oscillator 34
Duffing oscillator 35
EBK quantization 236
Einstein, and semiclassical quantization 235
Elliptic curves 336
Elliptic fixed points 24 see
Elliptic fixed points, and Poincare — Birkhoff fixed point theorem 134—142
Elliptic fixed points, for area-preserving maps 137
Elliptic functions, analytic structure of 336—337
Elliptic functions, and algebraic geometry 7 336—337
Elliptic functions, and solution to pendulum equation 11—13
Elliptic functions, general properties of 39—41
Elliptic functions, of Jacobi 7—9
Elliptic functions, of Weierstrass 10
Elliptic functions, periodic structure of 10
Elliptic integrals 6 8
Elliptic integrals, general properties of 39—41
Energy shell 73
Entire functions 10
Entire functions, and integrability 347—348
Equilibrium points, of pendulum 16
Ergodicity 73 167—168
Essential singularities 326 332
Euler — Poisson equations 324
Eulerian description of fluid dynamics 174
Extended phase space 51
Exterior, differentiation 86
Exterior, product 85
Feigenbaum number 215
Fermi — Ulam — Pasta (FUP) experiment, and KdV equation 280
Fermi — Ulam — Pasta (FUP) experiment, and statistical sharing of energy 105
First integral 2
First variation, of action integral 43
Fixed points, classification of, elliptic 24
Fixed points, classification of, hyperbolic 23
Fixed points, classification of, improper node, stable and unstable 25
Fixed points, classification of, node, stable and unstable 22—23
Fixed points, classification of, spiral point, stable and unstable 23
Fixed points, classification of, star, stable and unstable 25
Fixed points, examples of, damped linear oscillator 25
Fixed points, examples of, pendulum 25
Fixed points, examples of, predator-prey equations 26—29
Fixed points, failure of analysis 29
Fourier transforms, and linear evolution equations 292
Fractals 203—204
Functional derivative 311—314
Functional iteration 216
Functional renormalization group 220
Fundamental problem of Poincare 105
Galilean invariance of KdV equation 290
Gardner equation 289
Gelfand — Levitan — Marchenko equation 295
General and singular solutions 333
Generalized coordinates 42
Generalized momenta, as covariant variables 47 82
Generalized momenta, as gradient of action 48
Generalized momenta, properties of 47—48
Generating functions 58—63
Golden mean, and continued fractions 111
| Golden mean, and Greene's method 167
Green function, in quantum mechanics 264
Greene's method, and onset of chaos 163—167
Hamilton — Jacobi equation, and canonical perturbation theory 97—98
Hamilton — Jacobi equation, for one degree of freedom 64—65
Hamilton — Jacobi equation, time dependent 63
Hamilton — Jacobi equation, time independent 63
Hamilton's principle 42—43
Hamiltonian 96
Hamiltonian structure, of KdV equation 314
Hamiltonian structure, of NLS equation 316
Hamiltonian, equations of motion 50—52
Hamiltonian, function 49—50
Hamiltonian, transformation to 48 80—81
Hausdorf — Besicovitch dimension 203
Henon — Heiles Hamiltonian, analytic structure of 337—344
Henon — Heiles Hamiltonian, eigenfunctions of 251—252
Henon — Heiles Hamiltonian, integrable cases of 332
Henon — Heiles Hamiltonian, surface of section of 121—122
Henon's mapping, area preserving 129—132
Henon's mapping, strange attractor 212—214
Heteroclinic points, of cat map 171
Heteroclinic points, properties of 142—146
Homoclinic orbit 143
Homoclinic oscillations 147
Homoclinic points, of cat map 171
Homoclinic points, properties of 142—146
Homogeneity of time 44
Hopf bifurcation theory 197—199
Hopf — Cole transformation 352
Hyperbolic fixed points 23 see
Hyperbolic fixed points, and Poincare — Birkhoff fixed point theorem 139—142
Hyperbolic fixed points, for area preserving maps 137
Hyperbolic-with-reflection fixed point 137
Hyperelliptic integrals 37 337
Inertial frame 44
Inertial manifolds 212
Integrable Hamiltonians, and partial differential equations 311—316
Integrable Hamiltonians, examples of 74—77
Integrable Hamiltonians, properties of 70—74
Integration by quadrature 2—4
Integration of differential equations, further remarks on 36—39
Intermittency 203 210
Internal crisis 221
Inverse scattering problem 294—295
Inverse scattering transform (IST), basic principles 290—295
Inverse scattering transform (IST), for KdV equation 295—303
Inverse scattering transform (IST), for mKdV equation 310—311
Inverse scattering transform (IST), for NLS equation 310
Inverse scattering transform (IST), for Sine — Gordon equation 310
Invertible mappings 130 213
Irrational numbers 111
Irregular spectrum, definition of 240
Irregular spectrum, general properties of eigenfunctions 246—254
Irregular spectrum, general properties of eigenvalues 240—246
Isoenergetic nondegeneracy 113—114
Isospectral deformation 295
Isotropy of space 44
Jacobi elliptic functions see “Elliptic functions”
Jacobi identity 53 315
Jacobian of transformation 56
Journal bearing, chaotic advection experiment 176
K-systems 173
KAM theorem, basic ideas 105—107
KAM theorem, for autonomous systems 113
KAM theorem, for mappings 114
KAM theorem, for periodically perturbed systems 115
KAM theorem, for stable equilibrium points 116
Kink solution 306
Koch snowflake 203
Kolmogorov entropy 151
Korteweg — de Vries (KdV) equation, and discovery of soli ton 280—281
Korteweg — de Vries (KdV) equation, basic properties of 282—290
Korteweg — de Vries (KdV) equation, first derivation of 279
Korteweg — de Vries (KdV) equation, Hamiltonian structure of 314—316
Korteweg — de Vries (KdV) equation, inverse scattering transform of 295—299
Korteweg — de Vries (KdV) equation, two-soliton solution of 299—302
Kovalevskaya, Sofya 324
Kovalevsky exponents 330
Lagrangian description of fluid dynamics 174
Lagrangian turbulence 176
Lagrangian, discrete 133
Lagrangian, equations of motion 44
Lagrangian, function and Hamilton's principle 43—45
Lagrangian, general properties of 45—47
Lagrangian, manifold 239
Lagrangian, transformation to Hamiltonian picture 48—50 80
Landau — Hopf theory of turbulence 196
Laser Doppler velocimetry 194
Laurent series 329—330
Lax pairs 303
Lax pairs, and Painlev6 property 327 355
Leading order analysis for singularities 328
Legendre transformations 48 79—81
Level spacing distributions 243—245
Lie algebra 53
Limit cycles 30—31 196
Linear stability analysis 20—31 29 see
Linearizing transformations 3
Liouville's theorem 52
Liouville's theorem, and canonical transformations 56
Liouville's theorem, differential geometry of 88
Local representations, of solutions to differential equations 328
Localization theorems 254
Logistic map, basic phenomenology of 215
Logistic map, bifurcation diagram of 220—222
Logistic map, crgodic behavior of 222—223
Logistic map, noninvertibility of 223
Logistic map, period doubling of 216—220
Lorenz system, analytic structure of 344—347
Lorenz system, general properties of 204—210
Lorenz system, integrable cases of 345
Lorenz system, strange attractor of 208—210
Lorenz system, time-dependent integral of 346—347
Lozi map 213
Lyapunov exponents, computation of 151
Lyapunov exponents, for flows 150
Lyapunov exponents, for mappings 149
Mappings, area-preserving, and Hamiltonians 132—133
Mappings, area-preserving, and surface of section 123—126
Mappings, area-preserving, Henon's 129—131
Mappings, area-preserving, on the plane 128—132
Mappings, area-preserving, twist 126—128
Maslov indices 238
Meromorphic functions 11
Method of stationary phase 267 272—275
microcanonical ensemble 73
Microwave ionization of hydrogen 254
Miura transformation 288—290
Mixing 168
Modified KdV (mKdV) equation, and Miura transformation 288
Modified KdV (mKdV) equation, inverse scattering transform of 310
Modified KdV (mKdV) equation, simple properties of 304—305
Movable singularities, definition of 325
Multiply periodic orbits 77
N-soliton solutions 302
Natural boundaries 349
Navier — Stokes equation 188
Ncwton — Raphson method, and superconvergent perturbation theory 107
Newell — Whitehead equation 319
Newtonian equations of motion 42
Nodal patterns of eigenfunctions 254
Noether's theorem 47
Nonautonomous systems 33—36
Nonautonomous systems, Nondegenerate Hamiltonians 75 78 See
Noninvertible mapping 223
Nonlinear feedback 35
Nonlinear Fourier transform 299 see
Nonlinear Schrodinger (NLS) equation 309 310
normalization constant 294 See also “Direct scattering problem”
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