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