433 research outputs found
On the topology of nearly-integrable Hamiltonians at simple resonances
We show that, in general, averaging at simple resonances a real--analytic,
nearly--integrable Hamiltonian, one obtains a one--dimensional system with a
cosine--like potential; ``in general'' means for a generic class of holomorphic
perturbations and apart from a finite number of simple resonances with small
Fourier modes;
``cosine--like'' means that the potential depends only on the resonant angle,
with respect to which it is a Morse function with one maximum and one minimum.
\\ Furthermore, the (full) transformed Hamiltonian is the sum of an effective
one--dimen\-sio\-nal Hamiltonian (which is, in turn, the sum of the unperturbed
Hamiltonian plus the cosine--like potential) and a perturbation, which is
exponentially small with respect to the oscillation of the potential. \\ As a
corollary, under the above hypotheses, if the unperturbed Hamiltonian is also
strictly convex, the effective Hamiltonian at {\sl any simple resonance} (apart
a finite number of low--mode resonances) has the phase portrait of a pendulum.
\\ The results presented in this paper are an essential step in the proof (in
the ``mechanical'' case)
of a conjecture by Arnold--Kozlov--Neishdadt (\cite[Remark~6.8, p.
285]{AKN}), claiming that the measure of the ``non--torus set'' in general
nearly--integrable Hamiltonian systems has the same
size of the perturbation; compare \cite{BClin}, \cite{BC}
Pendulum Integration and Elliptic Functions
Revisiting canonical integration of the classical pendulum around its
unstable equilibrium, normal hyperbolic canonical coordinates are constructe
Persistence of Diophantine flows for quadratic nearly-integrable Hamiltonians under slowly decaying aperiodic time dependence
The aim of this paper is to prove a Kolmogorov-type result for a
nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time
dependence. The existence of a torus with a prefixed Diophantine frequency is
shown in the forced system, provided that the perturbation is real-analytic and
(exponentially) decaying with time. The advantage consists of the possibility
to choose an arbitrarily small decaying coefficient, consistently with the
perturbation size.Comment: Several corrections in the proof with respect to the previous
version. Main statement unchange
Fractional Lindstedt series
The parametric equations of the surfaces on which highly resonant
quasi-periodic motions develop (lower-dimensional tori) cannot be analytically
continued, in general, in the perturbation parameter, i.e. they are not
analytic functions of the perturbation parameter. However rather generally
quasi-periodic motions whose frequencies satisfy only one rational relation
("resonances of order 1") admit formal perturbation expansions in terms of a
fractional power of the perturbation parameter, depending on the degeneration
of the resonance. We find conditions for this to happen, and in such a case we
prove that the formal expansion is convergent after suitable resummation.Comment: 40 pages, 6 figure
A rigorous implementation of the Jeans--Landau--Teller approximation
Rigorous bounds on the rate of energy exchanges between vibrational and
translational degrees of freedom are established in simple classical models of
diatomic molecules. The results are in agreement with an elementary
approximation introduced by Landau and Teller. The method is perturbative
theory ``beyond all orders'', with diagrammatic techniques (tree expansions) to
organize and manipulate terms, and look for compensations, like in recent
studies on KAM theorem homoclinic splitting.Comment: 23 pages, postscrip
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 term. 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 Pustil'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.Comment: 10 page
1000–32 Spontaneous Evolution of Nonocclusive Coronary Dissection After PTCA: A 6 Month Angiographic Follow-up Study
We have previously shown that, when good distal flow is maintained, dissection after PTCA has a favourable short term (24 hrs) evolution and does not require bail-out interventions or CABG.To evaluate the long term (6 months) clinical and angiographic evolution of non occlusive dissection, we submitted 129 consecutive patients (103 male, mean age 53±11 yrs) undergoing elective PTCA (147 lesions, 66 LAD, 49 CX, 32 DX) to repeat angiography 24 hrs and 6 months after the procedure. Lesions were measured by QCA and coronary dissection was graded using the NHLBI classification (types A-E; Huber Am J Cardiol 1991;68: 467). Mean stenosis was 85±11% before and 25±7% immediately after PTCA (p<0.001). Residual stenosis was not significantly different at the 24 hrs restudy (24±9%). Non occlusive coronary dissection (flow TIMI grade 3 in all pts) was seen in 49/147 lesions (33%) and evolved as follows:Dissection (tot)Immediate 49 (33%)24 hrs 41 (28%)6 months 18 (12%)A332710B1085C442D221At the 6 month follow-up study, restenosis was seen in 51/147 lesions (34%), of which 5/49 (10%) had dissection and 46/106 (43%) did not. No cardiovascular events or recurrence of symptoms were recorded in the absence of restenosis.Therefore 1) nonocclusive dissection after PTCA usually improves after 6 month; 2) in the absence of flow impairment and ischemia this complication does not require any further intervention; 3) non occlusive dissection is not associated with increased incidence of restenosis
A Renormalization Proof of the KAM Theorem for Non-Analytic Perturbations
We shall use a Renormalization Group (RG) scheme in order to prove the
classical KAM result in the case of a non-analytic perturbation (the latter
will be assumed to have continuous derivatives up to a sufficiently large
order). We shall proceed by solving a sequence of problems in which the
perturbations are analytic approximations of the original one. We shall finally
show that the sequence of the approximate solutions will converge to a
differentiable solution of the original problem.Comment: 33 pages, no figure
- …