We study nonlinear systems of ordinary differential equations, ordinary differential equations with impulse perturbations, differential equations with a lagging argument and non-differential constraints on solutions and difference equations, whose operator coefficients commute with rotation operators. Based on this requirement, the sets of solutions of the studied systems of equations are invariant with respect to the mappings generated by three-dimensional rotations. This property of solutions of the studied systems of equations allows us to prove the instability of their unbounded solutions. As a consequence, the conditions for boundedness of solutions of the studied systems of equations are obtained. In the study of systems of equations, some found properties of rotations of three-dimensional space are used. An example of applying research results to celestial mechanics with a finite gravity speed is also given.

[1] Bakhvalov, S.V., Babushkin, L.I., Ivanitskaya, V.P. Analytical Geometry. Enlightenment, Moscow, 1965. (in Russian)

[2] Bellman, R., Cooke, K. L. Differential-Difference Equations. Academic Press, New York London, 1963.

[3] Kopeikin, S. V., Fomalont, E. The fundamental limit of the speed of gravity and its measurement. Earth and the Universe 2004, (3). http: //ziv.telescopes.ru/rubric/hypothesis/?pub=1 (in Russian)

[4] Korn, G., Korn, T. Mathematical Handbook for Scientists and Engineers, Definitions, Theorems and Formulas for Reference and Review. McGraw-Hill Book Company, INS, New York Toronto London, 1961.

[5] Kurosh, A.G. Course of Higher Algebra, Nauka, Moscow, 1971. (in Russian)

[6] Maurin, K. Metody Przestrzeni Hilberta. Panstwowe Wydawnictwo Naukowe, Warsawa, 1959.

[7] Miškis, A. D. Lectures in Higher Mathematics. Nauka, Moskva, 1969. (in Russian)

[8] Pontryagin, L.S. Continuous groups. Gostekhizdat, Moscow, 1954. (in Russian)

[9] Rubanik, V.P. Oscillations of quasilinear systems with delay. Nauka, Moscow, 1971. (in Russian)

[10] Samoylenko, A. M., Perestyuk, M. O., Parasyuk, I. A. Differential equations. Libid, Kyiv, 2003. (in Ukrainian)

[11] Slyusarchuk, V. Y. Absolute stability of dynamical systems with aftereffect. Publishing house of the National University of Water and Environmental Engineering, Rivne, 2003. (in Ukrainian)

[12] Slyusarchuk, V. Y. Mathematical model of the Solar system with account of gravitation velocity. Neliniini Koliv. 2018, 21 (2), 238–261. (in Ukrainian)

[13] Slyusarchuk, V. Y. Non-Keplerian behavior and instability of motion of two bodies caused by a finite velocity of gravity. Neliniini Koliv. 2018, 21 (3), 397–419. (in Ukrainian)

[14] Slyusarchuk, V. Y. Mathematical model of the Solar system with account of gravitation velocity. Proceedings of the international scientific conference "Modern problems of mathematics and its applications in the natural sciences and information technologies dedicated to the 50th anniversary of the Faculty of Mathematics and Informatics, Chernivtsi National University named after Yuriy Fedkovich, Chernivtsi, Ukraine, September 17-19, 2018, Chernivtsi National University, Chernivtsi, 2018, 98. (in Ukrainian)

[15] Slyusarchuk, V. Y. Kepler’s laws and the two-body problem with finite speed of gravity. Bukovinian Math. Journal 2018, 6 (3–4), 134–151. (in Ukrainian)

[16] Hale, J. Theory of Functional Differential Equations, Springer-Verlag, New York Heidelberg Berlin, 1977.

[17] Elsgolts, L.E., Norkin, S. B. Introduction to the theory of differential equations with deviating argument. Nauka, Moscow, 1971. (in Russian)