Establishing conditions for the existence of bounded solutions to the weakly nonlinear pulse systems

Abstract

Processes that involve jump-like changes are observed in mechanics (the movement of a spring under an impact; clockwork), in radio engineering (pulse generation), in biology (heart function, cell division). Therefore, high-quality research of pulse systems is a relevant task in the modern theory of mathematical modeling. This paper considers the issue related to the existence of bounded solutions along the entire real axis (semi-axis) of the weakly nonlinear systems of differential equations with pulse perturbation at fixed time moments. A concept of the regular and weakly regular system of equations for the class of the weakly nonlinear pulse systems of differential equations has been introduced. Sufficient conditions for the existence of a bounded solution to the heterogeneous system of differential equations have been established for the case of poorly regularity of the corresponding homogeneous system of equations. The conditions for the existence of singleness of the bounded solution along the entire axis have been defined for the weakly nonlinear pulse systems. The results were applied to study bounded solutions to the systems with pulse action of a more general form. The established conditions make it possible to use the classical methods of differential equations to obtain statements about solvability and the continuous dependence of solutions on the parameters of a pulse system. It has been shown that classical qualitative methods for studying differential equations are mainly naturally transferred to dynamic systems with discontinuous trajectories. However, the presence of a pulse action gives rise to a series of new specific problems. The theory of systems with pulse influence has a wide range of applications. Such systems arise when studying pulsed automatic control systems, in the mathematical modeling of various mechanical, physical, biological, and other processes.