D. Novikov
An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system alone.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+ψ'(x)=p(t)$ with the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $ψ'(x)$ are periodic and $g(x)$ is bounded. For the critical situation that $\big |\int_0^{2π}p(t)e^{int}dt \big|=2\big|g(+\infty)-g(-\infty)\big|$, we also prove a sufficient and necessary condition for the boundedness if $ψ'(x)\equiv0$.
In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian and an oscillating term $(ϕ_p(x'))'+aϕ_p(x^+)-bϕ_p(x^-)=G_x(x,t)+f(t)$, where$x^+=\max (x,0)$,$x^- =\max(-x,0)$,$ϕ_p(s)=|s|^{p-2}s$,$p\geq2$, $a $ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t)\in {\cal C}^{23}(\RR/2π_p \ZZ)$, the oscillating term $G\in {\cal C}^{21}(\RR\times\RR/2π_p \ZZ)$,where $π_p=\frac{2π(p-1)^{\frac{1}{p}}}{p\sin\fracπ{p}},$ and $G(x,t)$ satisfies $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21,$ and $\la…
Although detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply conne…
We construct a transcendental entire $f:\mathbb{C}\rightarrow\mathbb{C}$ such that (1) $f$ has bounded singular set, (2) $f$ has a wandering domain, and (3) each singular value of $f$ escapes to infinity under iteration by $f$.
A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor-repeller pair of hyperbolic solutions to the lack of bounded solutions. In this paper, a tool to analyze this phenomenon for asymptotically nonautonomous ODEs with bounded uniformly continuous or bounded piecewise uniformly continuous coefficients is described, and used to determine the occurrence of critical transitions for certain parametric equations. Some numerical exp…
Bounded and exponentially decaying solutions of almost linear dynamic systems on time scales
本文研究时间尺度上线性系统的小非线性扰动，通过Δ-微分算子在函数空间中的可解性方法，推广了指数二分性并引入Lyapunov正则性条件，证明了扰动系统有界解的存在性判据。
We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.
We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions by transforming the system of state-dependent delay equations into an abstract ordinary differential equation in a subspace of the sequence space $l^{\infty}(\mathbb{R}^{N+1})$ and prove the existence of complex extension of the bounded solutions. An example is given to illustrate the general results.
We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $Ω\subset \mathbb{C}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.
Ordinary differential equations defined by a trigonometric polynomial field: Behavior of the solutions
研究由三角多项式场定义的常微分方程，证明解具有旋转向量并具有弱几乎周期性质。
Given $ρ\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-ρ}g_{1}(t/\varepsilon ) $$ together with the {\it averaged} equation $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,τ)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors $…
Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star (x,t) = e^{-i ωt} V_\star(x - ct)$ with a profile $V_{\star}$ which decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = αU_{xx} + G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at $+ \infty$. The underlying assumptions guarantee that the operato…
By revisiting an asymptotic integration theory of nonlinear ordinary differential equations due to J.K. Hale and N. Onuchic [Contributions Differential Equations 2 (1963), 61--75], we improve and generalize several recent results in the literature. As an application, we study the existence of bounded positive solutions to a large class of semi-linear elliptic partial differential equations via the subsolution-supersolution approach.
We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time…
An estimate of Green's function of the problem of bounded solutions in the case of a triangular coefficient
本文针对矩阵系数为三角矩阵的常微分方程有界解问题，提出了Green函数的估计，该估计推广了Ch. F. Van Loan证明的矩阵指数估计。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。