There are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Therefore, it can be assumed that these external effects are "instantaneous", i.e. they are in the form of impulses. The investigation of such "leaps and bounds" developing dynamical states is a subject of different sciences: mechanics, control theory, pharmacokinetics, epidemiology, population dynamics, economics, ecology, etc., [1][2][3][4][5].
There Are Many Examples-Operation of a damper, subjected to the percussive effects.-Change of the valve shutter speed in its transition from open to closed state.-Fluctuations of pendulum system in the case of external impulsive effects.-Percussive model of a clock mechanism.-Percussive systems with vibrations.-Relaxational oscillations of the electromechanical systems.-Electronic schemes.-Remittent oscillator, subjected to the impulsive effects.-Dynamic of a system with automatic regulation.-The passage of the solid body from a given fluid density to another fluid density.-Control of the satellite orbit, using the radial acceleration.-Change of the speed of a chemical reaction in the addition or removal of a catalyst.-Disturbances in cellular neural networks.-Impulsive external intervention and optimization problems in the dynamics of isolated populations.-Death in the populations as a result of impulsive effects.-Impulsive external interference and the optimization problems in population dynamics of the predator-prey types.-"Shock" changes of the prices in the closed markets etc.The use of mathematical apparatus in the form of modeling systems impulsive differential equations in all these cases is natural and binding as a rule.
Brief Scientific Historical ReferenceThe mathematical theory of impulsive differential equations develops in two main directions.
Common Form of Impulsive Systems Differential EquationsIn general, the impulsive equations consist of three parts: -Differential equation which describes the differentiable part of the solution and usually has the form
Autonomous systems of differential equations with variable structure and impulsive effects are the main objects of our study. The structure changes and the impulsive effects realize at the switching moments, which are specific to each different solution of the system. In these moments, the trajectory of the corresponding initial value problem meets successively the switching sets. For this class of equations, sufficient conditions for the existence of periodic solutions are found. The results are based on the Brouwer's fixed point theorem. We investigate the question of existence of periodic solutions of a generalized model of predator-prey community.
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