ON THE SPECTRA OF OSCILLATION EXPONENTS OF A TWO-DIMENSIONAL NONLINEAR SYSTEM AND ITS FIRST APPROXIMATION SYSTEM

The sets of values (spectra) of the exponents of oscillation of strict signs, non-strict signs, zeros, roots and hyperroots of solutions of differential systems are studied. Two-dimensional nonlinear systems are constructed, all of whose solutions are infinitely extendable to the right and any of the spectra of their oscillation exponents can coincide with both the segment [0, 1] and with any pre-defined nonempty subset of rational numbers of this segment, while the spectra of linear systems of their first approximation consist of only one element. Moreover, the spectra of the exponents of the original system coincide with the corresponding spectra of the exponents of oscillation of the narrowing of the constructed nonlinear two-dimensional systems to the direct product of any open neighborhood of the zero of the phase plane and the time semi-axis. In addition, the existence of a nonlinear system has been proven, the spectrum of any of the oscillation exponents under consideration of which coincides with an arbitrary predetermined interval of the segment [0, 1], and the corresponding spectra of the system of its first approximation consist of one non-negative number.