Oscillation and wandering characteristics of solutions of a linear differential system

Abstract

We introduce new Lyapunov characteristics for the oscillation and wandering of solutions of linear differential equations or systems, namely, the frequency of a solution (the mean number of zeros on the time axis), of some coordinate of the solution, or of all possible linear combinations of these coordinates, and also the mean angular velocity of the rotation of a solution (about the origin in the phase space) and various wandering exponents (derived from the mean angular velocity). We shall show that the sets of values of all these quantities on the solutions of a linear autonomous system coincide with the set of absolute values of the imaginary parts of eigenvalues of the matrix of the system. We shall see that the frequencies of solutions are bounded above by their wandering exponents, and the frequencies and wandering exponents of all solutions of an arbitrary second-order equation coincide.