Comparing the spectra of wandering exponents of a nonlinear two-dimensional system and a first approximation

N. A. Loboda

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 146, pp. 176-187

Résumé

In this paper, we study various varieties of wandering exponents for solutions of linear homogeneous and nonlinear two-dimensional differential systems with coefficients continuous on the positive semiaxis. Moreover, all non-extendable solutions of the nonlinear system under consideration are defined on the entire positive time semi-axis. In 2010, I. N. Sergeev determined the wandering speed and wandering exponents (upper and lower, strong and weak) of a nonzero solution $x$ of a linear system. The wandering speed of the solution is the time-average velocity at which the central projection of the solution moves onto the unit sphere. Strong and weak exponents of wandering are the wandering speed of the solution, but minimized over all coordinate systems, and in the case of a weak exponent of wandering, minimization is performed at each moment of time. Therefore, strong and weak exponents of wandering take into account only the information about the solution that is not is suppressed by linear transformations: for example, they take into account the revolutions of the vector $x$ around zero, but do not take into account its local rotation around some other vector. In this work, a first approximation study of strong and weak wandering exponents was carried out. It is established that there is no dependence between the spectra (i. e., a set of different values on non-zero solutions) of strong and weak wandering exponents of a nonlinear system and the system of its first approximation. Namely, a two-dimensional nonlinear system is constructed such that the spectra of wandering exponents of its restriction to any open neighborhood of zero on the phase plane consist of all rational numbers in the interval $[0,1],$ and the spectra of the linear system of its first approximation consist of only one element.

Détail

Keywords: linear homogeneous differential system, nonlinear differential system, first approximation system, spectrum of a system exponent, variability of solution, exponent of oscillation, exponent of wandering, the wandering speed of a solution