On the existence of an isolated periodic solution of a nonlinear differential equation of the Lienar type

Abstract

The Lienar equation is a second-order differential equation used in modeling self-oscillating systems that maintain a stable rhythm without external intervention. The main feature is that under certain conditions the equation has a single stable limit cycle. This means that the system always reaches the same rhythm of oscillations, regardless of the initial conditions, that is, it returns to its working rhythm, even if it is strongly "pushed". This makes complex devices and living organisms predictable and reliable. The objective of the article is to obtain the conditions for the existence of an isolated periodic solution of a second-order nonlinear differential equation of the Lienar type, which is one of the current problems of the qualitative theory of nonlinear differential equations. The equation under study describes the nature of stationary processes that have a significant impact on the structure of many movements in real life. Research methods of proof are direct (analytical) methods, in particular – substitution of variables, integration, use of integral transformations. The results obtained in the process of studying the properties of a generalized nonlinear differential equation of the Lienar type describe the occurrence of nonlinear oscillations. Conclusions: the conditions for the existence of a single periodic solution, which corresponds to a stable limit cycle in the phase diagram, are formulated.