Asymptotic solutions of forced nonlinear second order differential equations and their extensions

Asymptotic stability of forced oscillations emanating from a limit cycle

Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the derivative of the corresponding bifurcation function F at the bifurcation point t. We show that for analytic systems this result is the one of topological nature, namely the topological index ind(t,F) is conclusive versus the sign of the derivative (d/dt)F(t). This allows to detect asymptotic stability also in the case when (d/dt)F(t)=0.

受扰动的确定性与随机伸缩方程渐近行为的刻画

Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations

本文刻画了受确定性与随机强迫的线性伸缩方程的渐近行为，给出强迫项使所有解收敛到无强迫方程平衡态并保持有界的充要条件，并推广到一般无界时滞和有限维方程。

An extension of an asymptotic result of Tricomi concerning a definite integral

We consider the expansion of an integral considered by F.G. Tricomi given by \[\int_{-\infty}^\infty x e^{-x^2}(\frac{1}{2}+\frac{1}{2}\mbox{erf}\,x)^{m} dx\] as $m\to\infty$. The procedure involves a suitable change of variable and the inversion of the complementary error function $\mbox{erfc}\,x$. Numerical results are presented to demonstrate the accuracy of the expansion. A second part examines an extension of an integral arising in airfoil theory.

弱阻尼二阶线性微分方程稳定性与衰减率的特征分析

Characterisation of Stability and Decay Rates in a Weakly Damped Second Order Linear Differential Equation

本文研究了弱阻尼二阶线性微分方程在外部激励下的解收敛性，给出了解及其导数趋于零的条件，并提供了解在主导项下与无激励方程相同渐进行为的充分条件。

The asymptotic expansion of Kratzel's integral and an integral related to an extension of the Whittaker function

We consider the asymptotic expansion of Krätzel's integral \[F_{p,ν}(x)=\int_0^\infty t^{ν-1} e^{-t^p-x/t}\,dt\qquad (|\arg\,x|<π/2),\] for $p>0$ as $|x|\to \infty$ in the sector $|\arg\,x|<π/2$ employing the method of steepest descents. An alternative derivation of this expansion is given using a Mellin-Barnes integral approach. The cases $p<0$, $\Re (ν)<0$ and when $x$ and $ν$ ($p>0$) are both large are also considered. A second section discusses the asymptotic expansion of an integral involving a modified Bessel function that has recently been introduced as an extension of the Whittaker fun…

受扰自治线性常微分方程与随机微分方程的加权 $L_\infty$ 渐近刻画：第一部分——常微分方程

Weighted $L_\infty$ Asymptotic Characterisation of Perturbed Autonomous Linear Ordinary and Stochastic Differential Equations: Part I -- ODEs

本文研究加性扰动下自治线性常微分方程解的渐近增长与衰减，建立强迫函数时间平均渐近行为与解速率之间的充要条件，并概述向次主导扰动情形的扩展。