2016
DOI: 10.3934/dcdss.2016044
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Normal forms <em>à la Moser</em> for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

Abstract: The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically timedependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.2010 Mathematics Subject Classification. Primary: 37J40. Secondary: 70H09.

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Cited by 5 publications
(10 citation statements)
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“…We shall suppose h(I) ∈ C ρ,· and f ∈ C ρ,σ while it is sufficient to assume that, for all I ∈ G ρ , f k (I, ·) ∈ C 1 (R + ). Similarly to [FW15b], we introduce the following Hypothesis 2.1 (Time decay). There exists M f > 0 and a ∈ (0, 1)…”
Section: R a F T 2 Setting And Main Resultsmentioning
confidence: 99%
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“…We shall suppose h(I) ∈ C ρ,· and f ∈ C ρ,σ while it is sufficient to assume that, for all I ∈ G ρ , f k (I, ·) ∈ C 1 (R + ). Similarly to [FW15b], we introduce the following Hypothesis 2.1 (Time decay). There exists M f > 0 and a ∈ (0, 1)…”
Section: R a F T 2 Setting And Main Resultsmentioning
confidence: 99%
“…Alternatively, those terms can be removed by including them into the homological equation, which turns out to be, in this way, a linear ODE in time. This has been profitably used in [FW14b], [FW15a] and in [FW15b] but requires (except for a particular case described in [FW15b]) an important assumption. More precisely, it is necessary to suppose that the perturbation, as a function of t, belongs to the class of summable functions over the real semi-axis 2 .…”
Section: Introductionmentioning
confidence: 99%
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“…A similar strategy is carried out in the nonautonomous case, but it is not as straightforward as in the autonomous case. There has been a recent extension of Moser's theorem to the nonautonomous case ( [Fortunati and Wiggins, 2016]), but the requirements on the time dependence are too stringent for all the applications that we will consider, and therefore we will not utilize this result in our arguments for this example. Another approach is to utilize a result like the Hartman-Grobman theorem ( [Hartman, 1960a[Hartman, ,b, 1963Grobman, 1959Grobman, , 1962) for nonautonomous systems.…”
Section: The Nonautonomous Nonlinear Saddle Pointmentioning
confidence: 99%
“…The strategy of this paper is to derive the integrability of the system of ODEs at hand, see (7), as a particular case of the existence of a normal form for a real-analytic Hamiltonian with aperiodic perturbation, see (1), by using the tools introduced in Ref. 7 for the one degree of freedom case.…”
Section: A Introductionmentioning
confidence: 99%