二阶振动线性微分方程的主对

关于某些常微分方程的振动积分

On the oscillatory integration of some ordinary differential equations

给出了一类非线性常微分方程x“（t）+a（t）w（x）=0，t>=1的条件，其中包含了线性方程的解x（t）具有给定的斜渐近线，且解x（t）具有振动的伪wronskian x”（t）-x（t）/t.

Oscillatory solutions of some perturbed second order differential equations

We discuss the occurrence of oscillatory solutions which decay to 0 as $s\to+\infty$ for a class of perturbed second order ordinary differential equations. As opposed to other results in the recent literature, the perturbation is as small as desired in terms of its improper integrals and it is independent of the coefficients of the non-oscillatory unperturbed equation. This class of equations reveals thus a new pathology in the theory of perturbed oscillations.

循环反馈环的双稳态与振荡行为

Bistability and oscillatory behaviours of cyclic feedback loops

本文分析循环反馈环 ODE 在仿射与 Hill 等函数下的稳定性，给出双稳态和振荡行为产生的参数条件，并统一涵盖已有基因调控网络研究。

An oscillation criterion for delay differential equations with several non-monotone arguments

The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup, is obtained. An example illustrating the significance of the result is also given.

Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear ra…

On the sharp estimates for convolution operators with oscillatory kernel

In this article, we study the convolution operators $M_k$ with oscillatory kernel, which are related to solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurfaces $Σ\subset \mathbb{R}^3$ of a hyperbolic equation and smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which exactly one of the principal c…

Oscillation and nonoscillation criteria for impulsive delay differential equations with Perron integrable righthand sides

We present new criteria for the existence of oscillatory and nonoscillatory solutions of measure delay differential equations with impulses. We deal with the integral forms of the differential equations using the Perron and the Perron-Stieltjes integrals. Thus the functions involved can have many discontinuities and be of unbounded variation and yet we obtain good results which encompass those in the literature. Examples are given to illustrate the main results.

具有广义型分段常数变元的二阶微分方程的一个简单振动准则

A simple oscillation criteria for second-order differential equations with piecewise constant argument of generalized type

本文针对具有广义型分段常数变元的二阶微分方程，提出了两个简单的振动性判定准则，推广了已有的Leighton-Wintner型结果，并通过实例加以说明。

Estimates for principal Lyapunov exponents: A survey

This is a survey of known results on estimating the principal Lyapunov exponent of a time-dependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself. Extensions to other differential equations are considered. Possible directions of further researc…

Oscillation criteria for second order two dimensional linear systems of ordinary differential equations

Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of ordinary differential equations.

Boundary Triples and Weyl $m$-functions for Powers of the Jacobi Differential Operator

The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl $m$-function for several self-adjoint extensions with interesting boundary conditions: separated, periodic and those that yield the Friedrichs extension. These matrix-valued Nevanlinna--Herglotz $m$-functions are, to the best knowledge of the author, the first explicit examples to stem from singular higher-order differential equations. The creation of the boundary triples involves taking pieces, determined in a previous paper, of the principal and non-prin…

具有阻尼非共振扰动的振荡系统中的渐近状态

Asymptotic regimes in oscillatory systems with damped non-resonant perturbations

本文研究平面上非线性振荡系统在满足非共振条件且随时间衰减的扰动下的长期渐近行为，结合平均化方法与Lyapunov函数构造，证明受扰系统既可保持极限系统行为，也可能产生新的渐近稳定状态。