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Haller G. — Chaos Near Resonance
Haller G. — Chaos Near Resonance

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Название: Chaos Near Resonance

Автор: Haller G.

Аннотация:

Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled with instabilities, can result in highly irregular behavior. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrvdinger equation and coupled NLS equations.


Язык: en

Рубрика: Физика/Динамические системы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 443

Добавлена в каталог: 10.09.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$D_4$ symmetry      223
$H^{h}$ norm      398
$L^{2}$ norm      393
$L^{p}$ norm      394
$L^{\infty}$ norm      394
$\alpha$-limit set      2
$\omega$-limit set      2
A priori, stable system      277
A priori, unstable system      277
Abelian Lie groups      14
Action-angle variables      16
Arnold diffusion      256 277
Averaging      36
Averaging, multiphase      38 183
Averaging, partial      39
Averaging, second-order      183
Baecklund transformation      296
Banach space      393
Base point      65 150 297 357
Bernoulli shift map      44
Bi-infinite binary symbols      43
Bifurcation in PDEs      342
Bifurcation of whiskers      266 284
Bifurcation, global      7
Bifurcation, local      7
Bifurcation, point      7
Bifurcation, universal homoclinic      136 166
Blowing up a singularity      237
Boundary of a manifold      372
Boundary, orientation      386
Bump function      398
Cantor set      43
Cauchy — Schwarz inequality      393
Cauchy’s inequality      392
Chaos      45
Chaos, definition of      46
Chaos, Hamiltonian      123
Chaos, transient      48
Chaotic, attractor      48
Chaotic, dynamics      46
Chaotic, orbit      46
Chaotic, set      46 118
Chart      380
Classical water molecule      248
Codimension      372
Compact support      396
Coordinate system      371
Coordinate system, global      374
Cotangent, bundle      373
Cotangent, critical point      377
Cotangent, space      373
Degrees of freedom      7
Derivative on manifolds      376
Derivative, distributional      396
Derivative, exterior      384
Derivative, Frechet      394
Derivative, Gateaux      395
Derivative, Lie      377 385
Derivative, notation for      376
Derivative, partial, on Banach spaces      395
Diameter sequence      138
Differential equation as a dynamical system      3
Differential equation on a manifold      5
Differential form      382
Differential form, closed      385
Differential form, exact      385
Diffusion, Arnold      257 277
Diffusion, barriers to      263
Diffusion, cross-resonance      257 258 284
Diffusion, path      278
Disintegration of invariant manifolds      114 337
Distributions      396
Distributions in the sense of      13 305 396
Distributions, derivative of      396
Divergence      5
Dynamical system      1
Embedding      376
Energetic uniqueness      137
Energy as a coordinate      91 322
Energy function      103 329
Energy function, chain-independent      110 336
Energy function, resonant      131 242
Energy surface      8
Energy surface, regular      8
Energy transfer      204 205 232 251 253
Energy vector      359
Entry conditions      73 314
entry point      92 96 323
Euler variables      236
Exchange Lemma      31 34
Exchange lemma, exponentially small      33
Exchange lemma, generalized      33
Fenichel fiber      25 306
Fiber, Hamiltonian      98
Fiber, infinite-dimensional      306
Fiber, stable, unstable      25
First integral      9
Fitzhugh — Nagumo equations      31
Fixed point      1
Flow      1
Flow of an ODE      3
Flow, semi      2
Foliation      24
Foliation, infinite-dimensional      306
Fourier series      390
Fourier series, estimates on remainder terms of      39
Free group action      379
Generating function      11
Generating function, time-dependent      11
Global map      88 321
Gronwall estimate      76
Gronwall inequality      4 392
Hamiltonian      7
Hamiltonian, canonical      7
Hamiltonian, locally      10
Hamiltonian, Poincare maps      9
Hamiltonian, reduced      15
Hamiltonian, slow      92
Hamiltonian, system conservation properties of      9 12
Hamiltonian, system generalized      8
Hamiltonian, system infinite-dimensional      12
Hamiltonian, vector field      8 12
Heteroclinic      143
Heteroclinic, cycles, attracting      227
Heteroclinic, cycles, fast      225
Heteroclinic, jumping      141
Heteroclinic, orbit      28
Heteroclinic, structure, partially broken      142
Heteroclinic, structure, “Cat’s eyes”      145
Hilbert space      393
Homoclinic, excursions      57
Homoclinic, manifold      29
Homoclinic, orbit      28
Homoclinic, orbit to a sphere      125
Homoclinic, orbit, N-pulse      66 310
Homoclinic, orbit, primary      138
Homoclinic, point, transverse      40
Homoclinic, tangle      40 123 128
Homoclinic, tree      139
Homoclinic, tree in an atmospheric problem      214
Homoclinic, tree in the NLS equation      342
Homoclinic, tree, breakup of      172
Homological equation      35
Hyperbolic set      47
Hyperbolicity, normal      23
Hyperbolicity, weak      27
Hyperbolicity, weak in resonant normal forms      245
Hyperbolicity, weak in surface waves      192
Hypersurface      372
Immersion      376
Infinitesimal generator      379
Integrability by quadratures      15
Integrability of a system      15
Integrability, complete      16 290
Integrability, complete, on a resonant manifold      59
Integration on manifolds      388
Intersecting resonances      255
Invariant      9
Invariant, foliation      24
Invariant, locally      19
Invariant, manifold      19
Invariant, overflowing (inflowing)      19
Invariant, positively (negatively)      19
Invariant, torus      17
Involution      15 387
Isoenergetic, nondegeneracy      17
Isoenergetic, set      8
Jump sequence      65 66 114 150 298 310 337 357 358
KAM theory      16
KAM theory for whiskered tori      17
Lambda lemma      40
Landing point      66 150 310 358
Lie, algebra      378
Lie, bracket      378
Lie, derivative      377 385
Lie, group      378
Lie, group, action of      379
Lie, theorem on volume      5
Linear wave equation      12
Liouville, -Arnold-Jost theorem on integrable systems      16
Lipschitz constant      4
Local diffeomorphism      376
Local map      87 319
Lorenz equations      45
Lyapunov-type numbers      23 25 27
Manifold      371
Manifold with boundary      372
Manifold, center      22
Manifold, class C’      371
Manifold, codimension of      372
Manifold, critical      28 30 56
Manifold, dimension of      371
Manifold, infinite-dimensional      380
Manifold, interior of      372
Manifold, invariant infinite-dimensional      24
Manifold, invariant normally hyperbolic      23
Manifold, invariant persistence of      23
Manifold, modeled over a Banach space      380
Manifold, onrientable      385
Manifold, parametrization of      371
Manifold, partially slow      56 298
Manifold, partially slow of higher codimension      147 354
Manifold, partially slow, infinite-dimensional      286
Manifold, Poisson      387
Manifold, resonant      28 56 240
Manifold, slow      30 56 294
Manifold, stable global      22 24
Manifold, stable local      22
Manifold, stable strong      26
Manifold, symplectic      386
Manifold, symplectic modeled over a Banach space      12
Maximal atlas      380
Mean value, inequality      396
Mean value, theorem      395
Melnikov, function      50
Melnikov, method      48
Miles’s amplitude equations      179
Mobius strip      374
Momentum mapping      14
Momentum mapping for resonances      235
Morse lemma      377
n-chain      64 295 297 357
N-chain, admissible      103 330
N-chain, high-codimension case      150
N-jump orbit      114 337
N-pulse homoclinic orbit      66 150 310 358
N-pulse homoclinic orbit, existence of      98 103 330 358
Nondegenerate form      386
Nonlinear      232 237
Nonlinear Schrodinger equation      287
Nonlinear Schrodinger equation, coupled      288
Nonlinear Schrodinger equation, discretized      58
Nonlinear Schrodinger equation, truncated      57
Normal bundle      373
Normal form      34
Normal form, Birkhoff      36 232
Normal form, Fenichel      36 68
Normal form, Fenichel, infinite-dimensional      311
Normal form, Hamiltonian      35
Normal form, preliminary      304
Normal mode      232
Normal space      373 0 2
O(2) symmetry      222
Optical cavity      215
Orbit      1
Orbit of a map      2
Orbit space      379
Orbit, N-jump      114
Orbit, periodic      2
Orientation      385
Orientation-preserving map      386
Orientation-reversing map      386
Parametrically forced beam      160
Particle motion in the atmosphere      210
Partition of unity      389
Passage lemmas      68 310
Passing orbit      146
Pendulum-type Hamiltonian      262
Phase shift      63 96
Phase shift, resonant      131
Phase shifts of opposite sign      135
Phase space      1 3
Phase space, extended      3
Phase space, reduced      14
Poincare — Cartan integral invariant      10 90
Poincare, map      2
Poincare, section      2
Poisson, bracket      387
Poisson, bracket canonical      387
Poisson, manifold      387
Poisson, structure      387
Preimage theorem      377
Proper group action      379
Pullback      383
Pulse number      138
Pulse sequence      138
Pushforward      383
Quotient space      379
Regular point      377
Regular value      377
Resonance on an invariant manifold      282
Resonance, boundary of      262
Resonance, double      258
Resonance, generator of      232
Resonance, junction      258
Resonance, multiple      258
Resonance, parabolic      213
Resonance, weak      258
Resonance, weak-strong      258
Resonance, web      256
Resonant, equilibria      231
Resonant, manifold      56
Resonant, module      234
Resonant, normal form terms      35 36
Resonant, torus      16 256 282
Resonant, torus, fully      60
Rigid body dynamics      271 278
Rotational orbit      146 277
Sard’s theorem      377
Sensitive dependence on initial, conditions      45
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