PERIODIC MODES IN THE MODEL OF A MATHEMATICAL PENDULUM WITH IMPULSE EFFECT

Abstract

The article is devoted to establishing the conditions for the existence of periodic regimes of the model of a mathematical pendulum with impulse effect. An important aspect is that such a system is subjected to the action of instantaneous forces at the moments when the moving point passes some fixed position. In addition, it is subjected to impulse action at unfixed moments of time, whereby its amount of movement is increased by a certain amount. The paper presents some theoretical aspects, as well as a description of the sequence of moments of time, which describes the mechanism of reducing the problem with impulse action at unfixed moments of time to the problem of finding fixed points of the interval within itself. The relevance and degree of research of the problem is revealed by comparing existing solutions to the problem and finding and adding new ones. Actually, this is the main task of this work. The resulting problem involves investigating the existence of conditions that ensure the existence of cycles to which periodic solutions correspond. In the work, a second-order differential equation with impulse action is investigated in a specific case with a fixed value of the position of the impulse action depending on the values of the parameters in the impulse action function. As a result, two cycles of period three were found. It is also demonstrated by verification that the given cycles form periodic solutions. The obtained results are recorded as detailed and informative as possible. In the work, a clear view of the points that guarantee the existence of periodic regimes in the system of the mathematical pendulum with impulse action is obtained and two graphs are given, that demonstrate the results of the experiment. It is illustrated in colors how the trajectories change after each action of impulse forces. Using a corollary from Sharkovsky's theorem, it is shown that if a function is continuous and has a periodic point of period three, then it has periodic points of any natural period. Therefore, in the system there are such periodic regimes in which the phase point undergoes impulse action exactly n times per period, where n is a natural number.