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Tabor M. Ч Chaos and Integrability in Nonlinear Dynamics: An Introduction
Tabor M. Ч Chaos and Integrability in Nonlinear Dynamics: An Introduction

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Ќазвание: Chaos and Integrability in Nonlinear Dynamics: An Introduction

јвтор: Tabor M.

јннотаци€:

Presents the newer field of chaos in nonlinear dynamics as a natural extension of classical mechanics as treated by differential equations. Employs Hamiltonian systems as the link between classical and nonlinear dynamics, emphasizing the concept of integrability. Also discusses nonintegrable dynamics, the fundamental KAM theorem, integrable partial differential equations, and soliton dynamics.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1984

 оличество страниц: 384

ƒобавлена в каталог: 13.04.2008

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ѕредметный указатель
Abel, Niels      337
Abelian functions, integrals, varieties      337
Action integral      43
Action-angle variables, for many degrees of freedom      68Ч70 74Ч77
Action-angle variables, for one degree of freedom      65Ч68
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 УHopf bifurcation theoryФ
Bifurcation, period doubling      see УPeriod doublingФ
Bifurcation, pitchfork      225
Bifurcation, saddle-node      225
Bifurcation, subcritical      198
Bifurcation, supercritical      198
Bifurcation, tangent      224
Bions      311
Bohr Ч 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      102Ч104
Canonical perturbation theory, for perturbed oscillator      101
Canonical perturbation theory, to first order      98Ч100
Canonical perturbation theory, to higher orders      100Ч101
Canonical transformations      54Ч63
Cantor set      202 203
Cantori      135
Cat map      168Ч171
Catastrophe theory      275
Cauchy Ч 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      264Ч272
Closed orbits, and spectral rigidity      246
cn function      9
Conservation laws      287Ч288
Conservation laws, and Gardner equation      289
Conservative systems, phase portraits of      17
Constant of motion      2
Constraints, holonomic      43 49
Continued fractions      110Ч111
Continued fractions, and Greene's method      164
Contravariant variables      47 81
Correspondence identities      234
Correspondence principle      241
Couette flow      192Ч194
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      264Ч272
Density of states, Thomas Ч 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 293Ч294
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 Ч Birkhoff fixed point theorem      134Ч142
Elliptic fixed points, for area-preserving maps      137
Elliptic functions, analytic structure of      336Ч337
Elliptic functions, and algebraic geometry      7 336Ч337
Elliptic functions, and solution to pendulum equation      11Ч13
Elliptic functions, general properties of      39Ч41
Elliptic functions, of Jacobi      7Ч9
Elliptic functions, of Weierstrass      10
Elliptic functions, periodic structure of      10
Elliptic integrals      6 8
Elliptic integrals, general properties of      39Ч41
Energy shell      73
Entire functions      10
Entire functions, and integrability      347Ч348
Equilibrium points, of pendulum      16
Ergodicity      73 167Ч168
Essential singularities      326 332
Euler Ч Poisson equations      324
Eulerian description of fluid dynamics      174
Extended phase space      51
Exterior, differentiation      86
Exterior, product      85
Feigenbaum number      215
Fermi Ч Ulam Ч Pasta (FUP) experiment, and KdV equation      280
Fermi Ч Ulam Ч 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      22Ч23
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      26Ч29
Fixed points, failure of analysis      29
Fourier transforms, and linear evolution equations      292
Fractals      203Ч204
Functional derivative      311Ч314
Functional iteration      216
Functional renormalization group      220
Fundamental problem of Poincare      105
Galilean invariance of KdV equation      290
Gardner equation      289
Gelfand Ч Levitan Ч 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      47Ч48
Generating functions      58Ч63
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      163Ч167
Hamilton Ч Jacobi equation, and canonical perturbation theory      97Ч98
Hamilton Ч Jacobi equation, for one degree of freedom      64Ч65
Hamilton Ч Jacobi equation, time dependent      63
Hamilton Ч Jacobi equation, time independent      63
Hamilton's principle      42Ч43
Hamiltonian      96
Hamiltonian structure, of KdV equation      314
Hamiltonian structure, of NLS equation      316
Hamiltonian, equations of motion      50Ч52
Hamiltonian, function      49Ч50
Hamiltonian, transformation to      48 80Ч81
Hausdorf Ч Besicovitch dimension      203
Henon Ч Heiles Hamiltonian, analytic structure of      337Ч344
Henon Ч Heiles Hamiltonian, eigenfunctions of      251Ч252
Henon Ч Heiles Hamiltonian, integrable cases of      332
Henon Ч Heiles Hamiltonian, surface of section of      121Ч122
Henon's mapping, area preserving      129Ч132
Henon's mapping, strange attractor      212Ч214
Heteroclinic points, of cat map      171
Heteroclinic points, properties of      142Ч146
Homoclinic orbit      143
Homoclinic oscillations      147
Homoclinic points, of cat map      171
Homoclinic points, properties of      142Ч146
Homogeneity of time      44
Hopf bifurcation theory      197Ч199
Hopf Ч Cole transformation      352
Hyperbolic fixed points      23 see
Hyperbolic fixed points, and Poincare Ч Birkhoff fixed point theorem      139Ч142
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      311Ч316
Integrable Hamiltonians, examples of      74Ч77
Integrable Hamiltonians, properties of      70Ч74
Integration by quadrature      2Ч4
Integration of differential equations, further remarks on      36Ч39
Intermittency      203 210
Internal crisis      221
Inverse scattering problem      294Ч295
Inverse scattering transform (IST), basic principles      290Ч295
Inverse scattering transform (IST), for KdV equation      295Ч303
Inverse scattering transform (IST), for mKdV equation      310Ч311
Inverse scattering transform (IST), for NLS equation      310
Inverse scattering transform (IST), for Sine Ч Gordon equation      310
Invertible mappings      130 213
Irrational numbers      111
Irregular spectrum, definition of      240
Irregular spectrum, general properties of eigenfunctions      246Ч254
Irregular spectrum, general properties of eigenvalues      240Ч246
Isoenergetic nondegeneracy      113Ч114
Isospectral deformation      295
Isotropy of space      44
Jacobi elliptic functions      see УElliptic functionsФ
Jacobi identity      53 315
Jacobian of transformation      56
Journal bearing, chaotic advection experiment      176
K-systems      173
KAM theorem, basic ideas      105Ч107
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 Ч de Vries (KdV) equation, and discovery of soli ton      280Ч281
Korteweg Ч de Vries (KdV) equation, basic properties of      282Ч290
Korteweg Ч de Vries (KdV) equation, first derivation of      279
Korteweg Ч de Vries (KdV) equation, Hamiltonian structure of      314Ч316
Korteweg Ч de Vries (KdV) equation, inverse scattering transform of      295Ч299
Korteweg Ч de Vries (KdV) equation, two-soliton solution of      299Ч302
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      43Ч45
Lagrangian, general properties of      45Ч47
Lagrangian, manifold      239
Lagrangian, transformation to Hamiltonian picture      48Ч50 80
Landau Ч Hopf theory of turbulence      196
Laser Doppler velocimetry      194
Laurent series      329Ч330
Lax pairs      303
Lax pairs, and Painlev6 property      327 355
Leading order analysis for singularities      328
Legendre transformations      48 79Ч81
Level spacing distributions      243Ч245
Lie algebra      53
Limit cycles      30Ч31 196
Linear stability analysis      20Ч31 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      220Ч222
Logistic map, crgodic behavior of      222Ч223
Logistic map, noninvertibility of      223
Logistic map, period doubling of      216Ч220
Lorenz system, analytic structure of      344Ч347
Lorenz system, general properties of      204Ч210
Lorenz system, integrable cases of      345
Lorenz system, strange attractor of      208Ч210
Lorenz system, time-dependent integral of      346Ч347
Lozi map      213
Lyapunov exponents, computation of      151
Lyapunov exponents, for flows      150
Lyapunov exponents, for mappings      149
Mappings, area-preserving, and Hamiltonians      132Ч133
Mappings, area-preserving, and surface of section      123Ч126
Mappings, area-preserving, Henon's      129Ч131
Mappings, area-preserving, on the plane      128Ч132
Mappings, area-preserving, twist      126Ч128
Maslov indices      238
Meromorphic functions      11
Method of stationary phase      267 272Ч275
microcanonical ensemble      73
Microwave ionization of hydrogen      254
Miura transformation      288Ч290
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      304Ч305
Movable singularities, definition of      325
Multiply periodic orbits      77
N-soliton solutions      302
Natural boundaries      349
Navier Ч Stokes equation      188
Ncwton Ч Raphson method, and superconvergent perturbation theory      107
Newell Ч Whitehead equation      319
Newtonian equations of motion      42
Nodal patterns of eigenfunctions      254
Noether's theorem      47
Nonautonomous systems      33Ч36
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 УDirect scattering problemФ
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