Lombardi E. — Oscillatory Integrals and Phenomena Beyond all Algebraic Orders: with Applications to Homoclinic Orbits in Reversible Systems :: Электронная библиотека попечительского совета мехмата МГУ

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Lombardi E. — Oscillatory Integrals and Phenomena Beyond all Algebraic Orders: with Applications to Homoclinic Orbits in Reversible Systems

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Название: Oscillatory Integrals and Phenomena Beyond all Algebraic Orders: with Applications to Homoclinic Orbits in Reversible Systems

Автор: Lombardi E.

Аннотация:

During the last two decades, in several branches of science (water waves, crystal growth, travelling waves in one dimensional lattices, splitting of separatrices,...) different problems appeared in which the key point is the computation of exponentially small terms. This self-contained monograph gives new and rigorous mathematical tools which enable a systematic study of such problems. Starting with elementary illuminating examples, the book contains (i) new asymptotical tools for obtaining exponentially small equivalents of oscillatory integrals involving solutions of nonlinear differential equations; (ii) implementation of these tools for solving old open problems of bifurcation theory such as existence of homoclinic connections near resonances in reversible systems.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 412

Добавлена в каталог: 05.12.2009

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Предметный указатель

$C_{\lambda}^{k}(I, \mathbb{R}^{n})$       149 (Definition 6.3.1)
$E^{\gamma, \lambda}_{\sigma, \delta}$       37 (Definition 2.1.17)
$GL_{n}(\mathbb{R})$       79
$H_{\ell}^{\lambda}$       23 (Lemma 2.1.1)
$H_{\ell}^{\lambda}(\mathbb{C}^{n})$       155 (Definition 6.3.10)
$H_{\ell}^{\lambda}|_{AR, D}$       386 (Definition 9.3.3)
$H_{\ell}^{\lambda}|_{R, D}$       386 (Definition 9.3.3)
$H_{\sigma, \delta}^{\gamma, \lambda}$       30 (Lemma 2.1.9)
$P^{\pm}_{r}$       38 (Lemma 2.1.19)
$x_{1} \wedge...\wedge x_{n-1}$       151 (Lemma 6.3.6)
$\langle\cdot,\cdot\rangle$       151 (Definition 6.3.5)
$\langle\cdot,\cdot\rangle_{*}$       151 (Definition 6.3.5)
$\mathcal{B}_{\ell, \nu}^{S}$       235 (Definition 7.4.3)
$\mathcal{B}_{\ell}$       23 (Lemma 2.1.1)
$\mathcal{H}^{\lambda}_{\ell}(\mathbb{C}^{n})$       149 (Definition 6.3.1)
$\mathcal{M}_{n}(\mathbb{R})$       79
$\widehat{H}^{m_{0}}_{\delta}$       37 (Definition 2.1.17)
$\widehat{H}^{m_{1}}_{\delta, r}$       37 (Definition 2.1.17)
$^{b}E^{\gamma, \lambda}_{\sigma, \delta}$       66 (Definition 2.2.6)
$^{b}H_{\ell}^{\lambda}$       64 (Lemma 2.2.1)
$^{b}H_{\sigma, \delta}^{\gamma, \lambda}$       65 (Lemma 2.2.4)
$^{b}\mathcal{H}_{\ell}^{\lambda}(\mathbb{C}^{2})|_{AR, R}^{\perp}$       169 (Proposition 6.5.7)
$^{b}\mathcal{H}_{\ell}^{\lambda}(\mathbb{C}^{n})$       167 (Definition 6.5.1)
$^{b}\mathcal{H}_{\ell}^{\lambda}(\mathbb{C}^{n})|_{AR, R}$       167 (Definition 6.5.1)
$^{b}\mathcal{H}_{\ell}^{\lambda}(\mathbb{C}^{n})|_{AR^{*}, R}$       167 (Definition 6.5.1)
$^{b}\mathcal{H}_{\ell}^{\lambda}(\mathbb{C}^{n})|_{R, R}$       167 (Definition 6.5.1)
$^{b}\mathcal{H}_{\ell}^{\lambda}(\mathbb{C}^{n})|_{R^{*}, R}$       167 (Definition 6.5.1)
$^{s}E^{\gamma, \lambda}_{\sigma, \delta}$       236 (Definition 7.4.4)
$^{s}H_{\sigma, \delta}^{\gamma, \lambda}$       235 (Definition 7.4.3)
$^{s}\widehat{H}^{m_{0}}_{\delta}$       236 (Definition 7.4.4)
$^{s}\widehat{H}^{m_{1}}_{\delta, r}$       236 (Definition 7.4.4)
$|_{AR^{*}}$       149 (Definition 6.3.1)
$|_{AR}$       149 (Definition 6.3.1)
$|_{R^{*}}$       149 (Definition 6.3.1)
$|_{R}$       149 (Definition 6.3.1)
Bi-oscillatory integral      64
Critical size      5 189 362
Crystal growth model      7
Devaney’s theorem      101
Dual basis      151—152
Eigenvalue of reversible fixed point      80 (Lemma 3.1.5)

Exponential lemma, first Bi-frequency      64
Exponential lemma, first Mono-frequency      23
Exponential lemma, second Bi-frequency      65
Exponential lemma, second Mono-frequency      30
Exponential lemma, third Bi-frequency      67
Exponential lemma, third Mono-frequency      39
Exponential splitting of separatrices      12 15
Inner system      46 48 239
Inner system of coordinates      44 48 239
Jor(L)      83
Jordan normal form      83
Jordan normal form of reversible matrix      84 (Theorem 3.1.10)
Korteweg de Vries's equation      8
Matched asymptotic expansions theory      16 48
Melnikov function      27 52 232—233 397—398
Melnikov theory      231 396
Nomenclature of reversible fixed point      87 (Remark 3.1.16)
Nomenclature of reversible pair of matrices      85
Normal Form Theorem      93
Normal form theorem for critical spectrum      88
Oscillatory integral, Bi-frequency      64
Oscillatory integral, Mono-frequency      23
Outer system of coordinates      44 48 239
Principal part      38 (Remark 2.1.20)
Rapidly forced pendulum      12 15 16
Resonance      87 (Remark 3.1.16)
Resonant, reversible fixed point      87
Reversible family of vector fields      86 (Definition 3.1.13)
Reversible fixed point      80 (Lemma 3.1.5)
Reversible homoclinic connection to fixed point      80 (Lemma 3.1.5)
Reversible homoclinic connection to periodic orbits      81 (Lemma 3.1.8)
Reversible pair of matrices      82—86
Reversible periodic orbit      81 (Lemma 3.1.8)
Reversible solution      79 (Lemma 3.1.2)
Reversible vector field      79 (Definition 3.1.1)
Solvability condition      158—162 170 210 231 338 380—382 397
Sor(L, S)      84 (Theorem 3.1.10)
Stable manifold of reversible fixed point      80 (Lemma 3.1.5)
Stable manifold of reversible periodic orbit      81 (Lemma 3.1.8)
Unstable manifold of reversible fixed point      80 (Lemma 3.1.5)
Unstable manifold of reversible periodic orbit      81 (Lemma 3.1.8)

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