Normal forms &lt;em&gt;à la Moser&lt;/em&gt; for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

Fortunati, Alessandro; Wiggins, Stephen

doi:10.3934/dcdss.2016044

Cited by 5 publications

(10 citation statements)

References 4 publications

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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%

“…[Gio03]) degenerates to a point. The paper uses in a concise but self-contained form, the tools developed in the above mentioned papers of the same authors, especially of [FW15b] in which the concept of "family" of canonical transformations parametrised by t is introduced. The proofs are entirely constructed by using the language and the tools of the Lie series and Lie transform methods developed by Giorgilli et al, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

Fortunati

Wiggins

2016

Celest Mech Dyn Astr

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The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called "geometric part" is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by A.Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems. 1 It is easy to see that any attempt to consider the limit r → ∞ would imply the degeneration into a trivial problem, (i.e. in which the allowed perturbation size reduces to zero, see also [GG85, formula (46), Pag. 105]).

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Section: R a F T 2 Setting And Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

Fortunati

Wiggins

2016

Celest Mech Dyn Astr

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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%

A Theoretical Framework for Lagrangian Descriptors

Lopesino

Balibrea-Iniesta

García-Garrido

et al. 2017

Int. J. Bifurcation Chaos

103

109

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This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for $n$-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as $n$-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors

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“…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%

Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

Fortunati

Wiggins

2016

Journal of Mathematical Physics

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium The paper deals with the problem of existence of a convergent "strong" normal form in the neighbourhood of an equilibrium, for a finite dimensional system of differential equations with analytic and time-dependent non-linear terms. The problem can be solved either under some non-resonance hypotheses on the spectrum of the linear part or if the non-linear term is assumed to be (slowly) decaying in time. This paper "completes" a pioneering work of Pustyl'nikov in which, despite under weaker non-resonance hypotheses, the nonlinearity is required to be asymptotically autonomous. The result is obtained as a consequence of the existence of a strong normal form for a suitable class of real-analytic Hamiltonians with non-autonomous perturbations. Published by AIP Publishing. [http://dx

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Normal forms <em>à la Moser</em> for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

Cited by 5 publications

References 4 publications

Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

A Theoretical Framework for Lagrangian Descriptors

Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

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