Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

Abstract

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.