Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

We fully study the oriented rotatability exponents of solutions tohomogeneous autonomous linear differential systems andestablish that the strong and weak orientedrotatability exponents coincide for each solution to an autonomous systemof differential equations. We also show that thespectrum of this exponent (i.e., the set of values of nonzerosolutions) is naturally determined by the number-theoreticproperties of the set of imaginary parts of the eigenvalues of thematrix of a system. This set (in contrast to the oscillationand wandering exponents) can contain other than zero values and theimaginary parts of the eigenvalues of the system matrix; moreover,the power of this spectrum can be exponentially large incomparison 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 exponentsof autonomous systems with a symmetricalmatrix consist of a single zero value.We also establish relationshipsbetween the main values of the exponents on a set of autonomous systems.The obtained results allow us to conclude that the exponents oforiented rotatability, despite their simple and natural definitions,are not analogs of the Lyapunov exponent in oscillation theory.