Singularly Perturbed Equations with Logarithmic Poles on a Single Line

Examines a system of singularly perturbed differential equations with logarithmic poles that possesses an equilibrium point. The stability of this equilibrium is lost at a certain value of the slow variable. The phenomenon of delay of the solution after the loss of stability is investigated—namely, a situation in which the trajectory remains close to the unstable equilibrium for a finite time interval. To carry out the analysis, the system is transformed into a complex form, which makes it possible to reduce it to an integral equation of a special type. Based on constructing a domain in the complex plane and applying appropriate estimates through the method of successive approximations, the existence of a solution is established and an asymptotic estimate is obtained on a segment where the equilibrium is partially unstable. The obtained estimate confirms the presence of the delay phenomenon of the solution near the unstable equilibrium. The results refine the understanding of how logarithmic poles influence the system's dynamics and describe the region in which the delay of the solution occurs.