We fully study the oriented rotatability exponents of solutions to homogeneous autonomous linear differential systems and establish that the strong and weak oriented rotatability exponents coincide for each solution to an autonomous system of differential equations. We also show that the spectrum of this exponent (i.e., the set of values of nonzero solutions) is naturally determined by the number-theoretic properties of the set of imaginary parts of the eigenvalues of the matrix of a system. This set (in contrast to the oscillation and wandering exponents) can contain other than zero values and the imaginary parts of the eigenvalues of the system matrix; moreover, the power of this spectrum can be exponentially large in comparison with the dimension of the space. In demonstration we use the basics of ergodic theory, in particular, Weyl’s Theorem. We prove that the spectra of all oriented rotatability exponents of autonomous systems with a symmetrical matrix consist of a single zero value. We also establish relationships between the main values of the exponents on a set of autonomous systems. The obtained results allow us to conclude that the exponents of oriented rotatability, despite their simple and natural definitions, are not analogs of the Lyapunov exponent in oscillation theory.

我们充分研究了齐次自治线性微分系统解的定向旋转性指数，并确定了微分方程自治系统的每个解的强和弱定向旋转性指数是一致的。我们还表明，该指数的谱（即非零解的值集）自然地由系统矩阵特征值的虚部集的数论性质决定。该集合（与振荡和漂移指数相反）可以包含零值和系统矩阵特征值的虚部以外的值；此外，与空间维度相比，该频谱的功率可以呈指数级增长。在演示中，我们使用遍历理论的基础知识，特别是韦尔定理。我们证明具有对称矩阵的自治系统的所有定向旋转指数的谱由单个零值组成。我们还建立了一组自治系统上指数的主要值之间的关系。所获得的结果使我们得出结论，定向旋转性的指数尽管定义简单自然，但与振荡理论中的李雅普诺夫指数并不相似。