Abstract

We consider a system of \(n\)th-order ordinary differential equations whose right-hand side is the sum of a linear function of the solution with a constant matrix, an essential nonlinearity of the relay type with hysteresis, and a perturbing continuous periodic function. The matrix of the linear function has only real simple nonzero eigenvalues, of which at least one is positive. We study the question of whether such systems have continuous solutions with two switching points in the state space (two-point oscillatory solutions) such that the time in which the solution returns to each of these points coincides with the period of the perturbing function or is an integer fraction of the latter. A sufficient condition for the nonexistence of such solutions is established, and a theorem is proved that gives sufficient conditions for the existence of a two-point oscillatory solution with return time equal to the period of the perturbing function. A corroborating example is given.

中文翻译：

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迟滞扰动继电器系统两点振动解的存在性

摘要

我们考虑一个\（n \）的系统三阶常微分方程，其右手边是具有常数矩阵的解的线性函数，具有滞后作用的继电类型的本质非线性和扰动的连续周期函数的和。线性函数的矩阵仅具有实数简单的非零特征值，其中至少一个为正。我们研究的问题是，此类系统是否具有在状态空间中具有两个切换点的连续解（两点振荡解）​​，以使解返回到这些点中的每个点的时间与扰动函数的周期一致或为后者的整数分数。建立了不存在此类解决方案的充分条件，并证明了一个定理，它为返回时间等于扰动函数周期的两点振动解的存在提供了充分的条件。给出了一个佐证的例子。