On quantum averaging, quantum KAM and quantum diffusion

For nonautonomous Hamiltonian systems and their quantisations we discuss properties of the quantised systems, related to those of the corresponding classical systems, described by the KAM-related theories: the proper KAM, the averaging theory, the Nekhoroshev stability and the diffusion.Comment: 15 page

New results on averaging theory and applications

Agraïments: The first author is supported by CNPq 248501/2013-5. CAPES grant 88881.030454 /2013-01 from the Program CSF-PVEThe usual averaging theory reduces the computation of some periodic solutions of a system of ordinary differential equations, to find the simple zeros of an associated averaged function. When one of these zeros is not simple, i.e. the Jacobian of the averaged function in it is zero, the classical averaging theory does not provide information about the periodic solution associated to a non simple zero. Here we provide sufficient conditions in order that the averaging theory can be applied also to non simple zeros for studying their associated periodic solutions. Additionally we do two applications of this new result for studying the zero--Hopf bifurcation in the Lorenz system and in the Fitzhugh--Nagumo system

Periodic solutions of continuous third-order differential equations with piecewise polynomial nonlinearities

We consider third-order autonomous continuous piecewise differential equations in the variable x. For such differential equations with nonlinearities of the form xm, we investigate their periodic solutions using the averaging theory. We remark that since the differential system is only continuous we cannot apply to it the classical averaging theory, that needs that the differential system be at least of class C2

Averaging theory for the structure of hydraulic jumps and separation in laminar free-surface flows

We present a simple viscous theory of free-surface flows in boundary layers, which can accommodate regions of separated flow. In particular this yields the structure of stationary hydraulic jumps, both in their circular and linear versions, as well as structures moving with a constant speed. Finally we show how the fundamental hydraulic concepts of subcritical and supercritical flow, originating from inviscid theory, emerge at intermediate length scales in our model.Comment: 6 EPSI figs included by psfig; 4 pages; to appear in PRL, vol.79, 1038 (Aug.11, 1997

Asymptotic behavior of periodic solutions in one-parameter families of Li\'{e}nard equations

In this paper, we consider one--parameter ($\lambda>0$) families of Li\'enard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of $\lambda>0$. To prove our main result we use the relaxation oscillation theory and a topological version of the averaging theory. More specifically, the first one is appropriate for studying the periodic solutions for large values of $\lambda$ and the second one for small values of $\lambda$. In particular, our hypotheses allow us to establish a link between these two theories