Oriented rotatability exponents of solution of autonomous differential systems

In this paper, the exponents of oriented rotatability of solutions of linear homogeneous autonomous differential systems are fully studied. It is found that for any solution of an autonomous system of differential equations, its strong exponents of oriented rotatability coincide with weak ones. It is also shown that the spectrum of this exponent (i.e., the set of values on nonzero solutions) is naturally determined by the number-theoretic properties of the set of imaginary parts of the eigenvalues of the matrix of the system. This set can contain (unlike the oscillation and wandering exponents) values other than zero and from 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. As a consequence, it is deduced that the spectra of all indicators of the oriented rotatability of autonomous systems with a symmetric matrix consist of a single zero value. In addition, on a set of autonomous systems, relationships were established between the main values of the studied exponents. The obtained results allow us to conclude that the exponents of oriented rotatability, despite its simple and natural definitions, is not an analogs of the Lyapunov exponent in the theory of oscillations.