Abstract

We consider nonautonomous linear systems of differential equations admitting a time-dependent first integral that is a quadratic form. The duality between mutually adjoint linear systems with quadratic integrals is established. Conditions for the spectrum of such linear systems to be symmetric about zero are indicated. We prove that a linear system is stable if and only if it admits a first integral that is a positive definite quadratic form. For linear systems with a quadratic integral, invariant measures whose densities are positive functions of time are studied. An explicit form of a series of quadratic integrals is specified if one of them is known. It is shown that the degree of instability of a regular linear system (the number, counting multiplicities, of positive points in the spectrum) is at most the maximum of the indices of inertia of a reducible quadratic integral.

中文翻译：

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具有二次积分的微分方程组的线性非自治系统

摘要

我们考虑微分方程的非自治线性系统，该系统允许采用与时间相关的第一积分，该积分是二次形式。建立了具有二次积分的相互邻接的线性系统之间的对偶性。指出了这样的线性系统的光谱关于零对称的条件。我们证明，线性系统只有当其接受为正定二次型的第一积分时，才是稳定的。对于具有二次积分的线性系统，研究了密度为时间的正函数的不变测度。如果知道其中一个，则指定一系列二次积分的显式形式。结果表明，规则线性系统的不稳定性程度（数量，重复数，