Measures of Fermi surfaces and absence of singular continuous spectrum for magnetic Schroedinger operators

Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of singular continuous spectrum and of singular continuous components in the density of states for symmetric periodic elliptic differential operators acting on vector bundles. This includes Schroedinger operators with periodic magnetic field and rational flux, as well as the corresponding Pauli and Dirac-type operators.Comment: 19 page

Subharmonic bifurcation from infinity

AbstractWe are concerned with a subharmonic bifurcation from infinity for scalar higher order ordinary differential equations. The equations contain principal linear parts depending on a scalar parameter, 2π-periodic forcing terms, and continuous nonlinearities with saturation. We suggest sufficient conditions for the existence of subharmonics (i.e., periodic solutions of multiple periods 2πn) with arbitrarily large amplitudes and periods. We prove that this type of the subharmonic bifurcation occurs whenever a pair of simple roots of the characteristic polynomial crosses the imaginary axis at the points ±αi with an irrational α. Under some further assumptions, we estimate asymptotically the parameter intervals, where large subharmonics of periods 2πn exist. These assumptions relate the quality of the Diophantine approximations of α, the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term

Twice degenerate equations in the space of vector-valued functions

New results are suggested which allow to calculate an index at infinity for asymptotically linear and asymptotically homogeneous vector fields in spaces of vector-valued functions. The case is considered where both linear approximation at infinity and "linear + homogeneous" approximation are degenerate. Applications are given to the 2π-periodic problem for a system of two nonlinear first order ODE's and to the two-point BVP for a system of two nonlinear second order ODE's

A nonradial bifurcation result with applications to supercritical problems

In this paper we consider the problem $-\Delta u=|x|^{\alpha} F(u)$ in $R^N$, with $\alpha>0$ and $N\ge3$. Under some assumptions on $F$ we deduce the existence of nonradial solutions which bifurcate from the radial one when $\alpha$ is an even integer.Comment: 20 page

An Extension of Popov Criterion to Multivariable Time-Varying Nonlinear Systems. Application to Criterion for Existence of Stable Limit Cycles

Projet SOSSOThis report deals with sufficient conditions for absolute stability of multivariable control systems. The proposed conditions extend in a simple way the classical Popov criterion to time-varying memoryless nonlinearities. They are expressed in terms of some Linear Matrix Inequalities (LMIs). A weaker frequency domain criterion is deduced, leading to a simple graphical interpretation. The results ensure in general local stability, however, the stability is global for example for linear time-varying systems. As an application, a Popov-like criterion for existence of stable periodic solutions for periodic systems is proposed, using previous results by the authors

On the gravitational potential of modified Newtonian dynamics

Producción CientíficaThe mathematical structure of the Poisson equation of Modified Newtonian Dynamics (MOND) is studied. The appropriate setting turns out to be an Orlicz-Sobolev space whose Orlicz function is related to Milgrom’s μ-function, where there exists existence and uniqueness of weak solutions. Since these do not have in principle much regularity, a further study is performed where the gravitational field is not too large, where MOND is most relevant. In that case the field turns out to be H¨older continuous. Quasilinear MOND is also analyzed

Dither in Systems with Hysteresis

Projet SOSSOThis paper deals with differential inclusion containing an hysteresis nonlinearity and two inputs: a control input and a dither input of high frequency. Conditions are introduced under which its solution admits asymptotic behavior when the dither frequency goes to infinity. According to asymptotic growth of the dither amplitude, two different behaviors appear: the nonlinearity is smoothed (resp. quenched) if the velocities induced by the dither are asymptotically bounded (resp. unbo- unded). Convergence results for finite and infinite time intervals are given, and linked with the averaging principle. The case of bounded dithering velocities is of interest in a mechanical context, where hysteresis is used to model dry friction. A very interesting feature is that the averaged hysteresis operator may be linearized for small velocities. The hypotheses on the dither include periodicity, $F$-repetitiveness and (asymptotic) almost-periodicity

On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe