Abstract

This paper considers an autonomous system of singularly perturbed equations of fast variables, consisting of \(2n\) first-order equations and one equation of a slow variable. The first approximation matrix of singularly perturbed equations has pairwise complex conjugate eigenvalues. The system has an equilibrium position, and the stability of the equilibrium position is lost by all eigenvalues at some value of the slow variable. It is proven that the solution of a singularly perturbed equation remains near an unstable equilibrium position during a finite time. Thus, the solution is delayed near the unstable equilibrium position. Early works considered cases when the stability of the equilibrium position is lost by one pair of complex conjugate eigenvalues.

中文翻译：

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求解不稳定平衡位置附近的自主奇异摄动方程的延迟

 抽象的

本文考虑快速变量奇异摄动方程的自治系统，由 \(2n\) 一阶方程和一个慢变量方程组成。奇异摄动方程的第一近似矩阵具有成对的复共轭特征值。系统有一个平衡位置，并且在慢变量的某个值处，所有特征值都失去了平衡位置的稳定性。证明了奇异摄动方程的解在有限时间内保持在不稳定平衡位置附近。因此，解在不稳定平衡位置附近被延迟。早期的工作考虑了一对复共轭特征值失去平衡位置稳定性的情况。