Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2270310.22363/2658-4670-2019-27-3-242-262Research ArticleOn the properties of numerical solutions of dynamical systems obtained using the midpoint methodGerdtVladimir P.<p>Doctor of Physical and Mathematical Sciences, Full Professor at the Joint Institute for Nuclear Research (JINR) where he is the head of the Group of Algebraic and Quantum Computations</p>gerdt@jinr.ruMalykhMikhail D.<p>Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia</p>malykh-md@rudn.ruSevastianovLeonid A.<p>professor, Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)</p>sevastianov-la@rudn.ruYingYu<p>postgraduate student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); assistant professor of Department of Algebra and Geometry, Kaili University</p>yingy6165@gmail.comJoint Institute for Nuclear ResearchPeoples’ Friendship University of Russia (RUDN University)Kaili University1512201927324226222012020Copyright © 2019, Gerdt V.P., Malykh M.D., Sevastianov L.A., Ying Y.2019<p>The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Coopers theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. 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