Remark on Periodic Solutions of Non Linear Oscillators

Bifurcation of asymptotically stable periodic solutions in nearly impact oscillators

We use an averaging approach to prove bifurcation of asymptotically stable periodic solutions in a bi-linear oscillator whose one spring has nearly infinite stiffness. This leads to a singularly perturbed problem where the classical theory does not apply.

Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator

In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for a Lipschitz system with a small right hand side. Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our ones. By means of the results established we construct the curves of dependence of the amplitude of asymptotically stable $2π$--periodic solutions of the nonsmooth van der Pol oscillator on the detuning parameter and the amplitude of the perturbation. After, we compare the resonance curves obtained, with the resonance curves of the classical van der Pol oscillat…

Some remarks on almost periodic time scales and almost periodic functions on time scales

In this note we communicate some important remarks about the concepts of almost periodic time scales and almost periodic functions on time scales that are proposed by Wang and Agarwal in their recent papers (Adv. Difference Equ. (2015) 2015:312; Adv. Difference Equ. (2015) 2015:296; Math. Meth. Appl. Sci. 2015, DOI: 10.1002/mma.3590).

Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals

We discuss the asymptotic behaviour for the best constant in L^p-L^q estimates for trigonometric polinomials and for an integral operator which is related to the solution of inhomogeneous Schrodinger equations. This gives us an opportunity to review some basic facts about oscillatory integrals and the method of stationary phase, and also to make some remarks in connection with Strichartz estimates.

受迫振子周期解的反最大值原理

An antimaximum principle for periodic solutions of a forced oscillator

针对带正的2π周期强迫项的线性振子方程，研究其周期解与正解的存在性关系，通过建立该问题与凸分析分离问题的联系，证明了周期解存在则正解存在的反最大值原理。

Asymptotics for the Eigenvalues of the Harmonic Oscillator with a Quasi-Periodic Perturbation

We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form λ_n(H+V)=λ_n(H)+W(\sqrt n)n^{-1/4}+O(n^{-1/2}\ln(n)) as n\to+\infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.