Introduction to the Perturbation Theory of Hamiltonian Systems

Treschev, Dmitry; Zubelevich, Oleg

doi:10.1007/978-3-642-03028-4

Cited by 81 publications

(75 citation statements)

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“…and there is no need of the sign s. For the classical and very popular separatrix map (see, e.g., [6,38,39,41]) one takes ξ(t) = sin(t). As said, the function ξ can also depend on ε and it is clear that both a and ξ can depend on additional parameters.…”

Section: General Return Map Modelsmentioning

confidence: 99%

“…See, e.g., [4]. There is another technique of inclusion of a map into a flow generated by a vector field periodic in time which preserves analyticity, see [25] and [38].…”

Section: On the Splitting Functionmentioning

confidence: 99%

“…The problem can be reduced to an autonomous 1 dof Hamiltonian, say εH, plus a time-depending remainder which has a bound of the form O(exp(−a/ε)), where the value of a depends on the analyticity width wrt the variables (q, p) of the domain where H(q, p, t) is defined. More concretely, the value of a is related to the complex singularities of the averaged system, see [36,38] for exact results. The role of the singularities of the limit flow in the splitting is discussed later.…”

Section: On the Splitting Functionmentioning

confidence: 99%

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Return Maps, Dynamical Consequences and Applications

Martínez

Simó

2015

Qual. Theory Dyn. Syst.

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After reviewing some general settings for return maps in problems reducible to 2D symplectic maps, details on the construction of return maps are presented. Different forms of such maps close to splitted separatrices (separatrix maps) are introduced, taking into account the size and shape of the splitting function and also the return time to the domains of interest. Then it is shown how to derive approximations by suitable standard-like maps. Dynamical consequences concerning the existence of invariant rotational curves (IRC) are derived. An application is made to theoretically estimate the location of the outermost IRC in the Sitnikov problem, which is in good agreement with numerical data. To compare with the cases which are approximated by the classical standard map, some details on the properties of the standard-like map with two harmonic terms are included. Finally a method to estimate the amount of chaos depending on the form of the separatrix map is introduced. Except otherwise stated all the systems we consider are assumed to be analytic, despite several of the properties we study are no longer analytic.

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Section: General Return Map Modelsmentioning

confidence: 99%

“…See, e.g., [4]. There is another technique of inclusion of a map into a flow generated by a vector field periodic in time which preserves analyticity, see [25] and [38].…”

Section: On the Splitting Functionmentioning

confidence: 99%

Section: On the Splitting Functionmentioning

confidence: 99%

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Return Maps, Dynamical Consequences and Applications

Martínez

Simó

2015

Qual. Theory Dyn. Syst.

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“…The KAM theory [21], [49] showed that nonresonance dynamical systems are stable under small perturbations in the sense that a majority of its invariant tori are not destroyed. The works of Elliason, Gallavotti, and others [50], [51] demonstrated the convergence of the Poincaré-Lindstedt series on a nonresonant set and the systematic cancellations of small denominators.…”

Section: Gustavson Integralsmentioning

confidence: 99%

Kato perturbative expansion in classical mechanics and an explicit expression for the Deprit generator

Nikolaev¹

2015

Theor Math Phys

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We study the structure of the canonical Poincaré-Lindstedt perturbation series in the Deprit operator formalism and establish its connection to the Kato resolvent expansion. A discussion of invariant definitions for averaging and integrating perturbation operators and their canonical identities reveals a regular pattern in the series for the Deprit generator. This regularity is explained using Kato series and the relation of the perturbation operators to the Laurent coefficients for the resolvent of the Liouville operator. This purely canonical approach systematizes the series and leads to an explicit expression for the Deprit generator in any order of the perturbation theory: G = −Ŝ S SH Hj, whereŜ S SH is the partial pseudoinverse of the perturbed Liouville operator. The corresponding Kato series provides a reasonably effective computational algorithm. The canonical connection of the perturbed and unperturbed averaging operators allows describing ambiguities in the generator and transformed Hamiltonian, while Gustavson integrals turn out to be insensitive to the normalization style. We use nonperturbative examples for illustration.

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“…Generally, being applied to Cauchy-Kovalevskaya problem, this modification does not give anything di¤erent from the results of Nirenberg and Nishida [8]. Nevertheless, in some cases this method allows to obtain global in time existence theorems or at least e¤ective estimates for the solution's existence time [13].…”

Section: Introductionmentioning

confidence: 94%

Evolution Differential Equations in Fréchet Space with Schauder Basis

Zubelevich

2017

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Introduction to the Perturbation Theory of Hamiltonian Systems

Cited by 81 publications

References 0 publications

Return Maps, Dynamical Consequences and Applications

Return Maps, Dynamical Consequences and Applications

Kato perturbative expansion in classical mechanics and an explicit expression for the Deprit generator

Evolution Differential Equations in Fréchet Space with Schauder Basis

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Introduction to the Perturbation Theory of Hamiltonian Systems

Cited by 81 publications

References 0 publications

Return Maps, Dynamical Consequences and Applications

Return Maps, Dynamical Consequences and Applications

Kato perturbative expansion in classical mechanics and an explicit expression for the Deprit generator

Evolution Differential Equations in Fr&eacute;chet Space with Schauder Basis

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Evolution Differential Equations in Fréchet Space with Schauder Basis