Spectra of oscillation and rotatability exponents of solutions of homogeneous differential systems

It is known that all weak wandering exponents, as well as the lower strong wandering exponent, are equal to zero on the set of solutions of linear homogeneous triangular differential systems with continuous coefficients bounded on the positive semiaxis. At the same time, the upper strong wandering exponent of some solution from the specified set can take a positive value. In this paper, the exponents of oriented rotatability and the exponents of oscillation of signs, zeros, roots, and hyperroots of solutions of linear homogeneous triangular differential systems with continuous (optionally bounded) on the positive semiaxis by coefficients are fully studied. It has been established that for any solution of a triangular system of differential equations, its oscillation and rotatability exponents are exact, absolute and coincide with each other. It is also shown that the spectra of these exponents (i.e., the set of values on nonzero solutions) of triangular systems consist of one zero value. The results obtained enable us to conclude that, despite their simple and natural definitions, the oriented rotatability exponents and oscillation exponents are not analogues of the Perron exponent. In addition, the coincidence of the spectra of each (strong or weak, upper or lower) exponent of oriented rotatability and the exponent of oscillation of signs, zeros, roots and hyperroots of mutually conjugate linear homogeneous systems of differential equations with continuous coefficients on the positive semiaxis is established.