Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.4462

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

Adriana Buica·Jaume Llibre·Oleg Makarenkov
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 oscillator which were first constructed by Andronov and Witt.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Adriana Buica, Jaume Llibre, Oleg Makarenkov. 2007-09-28. Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator. https://arxiv.org/abs/0709.4462

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reverse square function estimates for degenerate curves and their applications

Building on the classical work of Córdoba--Fefferman and the recent work of Schippa, we establish $L^4$ reverse square function estimates for functions whose Fourier support is contained in a $δ$-neighborhood of the curve $\{(ξ,|ξ|^a): |ξ|\leq 1\}$ in $\mathbb{R}^2$, for all exponents $a\in(0,\infty)\backslash\{1\}$. As applications, we derive sharp $L^4$ Strichartz estimates on the one-dimensional torus for fractional Schrödinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.

math.CA↗
arXiv2026

Gaussian rigidity and width of measures

Given $0\le m<n$ and a positive Radon measure $μ$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal D(μ):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}μ}|G_R^μ(x)-1|\Bigr],$$ where $G_R^μ(x):=(4πR^2)^{-m/2}\int_{\mathbb R^n} e^{-|z-x|^2/(4R^2)}\,dμ(z)$. Provided that $μ$ admits a flat blow-down $\mathcal H^m\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\leδ<\infty$, the so-called width of $μ$, such that $$\mathrm{spt}μ\subseteq \bar B_δ(V+x_0),\quad δ^2\le C(m,n)\mathcal{D}(μ),$$ where $\bar B_δ(V+x_0)$ denotes the closed $δ$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal D(μ)=0$, the support of $μ$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $μ$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $μ$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.

math.CA↗
arXiv2026

Fourier decay and endpoint estimates for lacunary maximal operators

We study endpoint estimates for lacunary maximal operators associated with compactly supported measures. For every $d\geq1$, we construct a compactly supported, absolutely continuous probability measure on $\mathbb R^d$ with polynomial Fourier decay whose associated lacunary maximal operator fails to map $H^1(\mathbb R^d)$ boundedly into $L^{1,\infty}(\mathbb R^d)$. In particular, it fails to be of weak type $(1,1)$. This gives a negative answer to the weak type $(1,1)$ question posed by Seeger and Wright in 2011 and to Open Problems 14 and 15, contributed by Wright at the 2016 Conference in Harmonic Analysis, held at the University of Wisconsin--Madison in honor of Professor Michael Christ.

math.CA↗