Abstract

The problem of stability of the zero solution of a system with a “center”-type critical point at the origin of coordinates is considered. For the first time, such a problem for autonomous systems was investigated by A.M. Lyapunov. We continued Lyapunov’s investigations for systems with a periodic dependence on time. In this paper, systems with a quasi-periodic time dependence are considered. It is assumed that the basic frequencies of quasi-periodic functions satisfy the standard Diophantine-type condition. The problem under consideration can be interpreted as the problem of stability of the state of equilibrium of the oscillator \(\ddot {x} + {{x}^{{2n - 1}}}\) = 0, n is an integer number, n ≥ 2, under “small” quasi-periodic perturbations.

中文翻译：

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准周期扰动下非线性中心的稳定性

摘要

考虑了在坐标原点具有“中心”型临界点的系统的零解的稳定性问题。AM Lyapunov首次研究了这种自治系统的问题。我们继续进行Lyapunov对周期性依赖时间的系统的研究。在本文中，考虑了具有准周期时间依赖性的系统。假设准周期函数的基本频率满足标准Diophantine型条件。所考虑的问题可以解释为振荡器平衡状态的稳定性\（\ ddot {x} + {{x} ^ {{2n-1}}} \） = 0，n是整数在“小”准周期扰动下，n≥2。