Singularly Perturbed Equations with Zeros and Poles and Solution Delay

Examines the asymptotic behavior of solutions to a system of singularly perturbed differential equations with logarithmic poles. The system under consideration possesses an equilibrium point whose stability is lost at a certain value of the slow variable. The main focus is on the phenomenon of solution delay after the loss of stability, in which the trajectory of the solution corresponding to the fast variables remains in the vicinity of the unstable equilibrium point for a finite time interval. For the analysis, the system is transformed into a complex form and reduced to a special type of integral equation. An appropriate domain is constructed in the complex plane, and the existence of a solution is proved using the methods of level curves and successive approximations. An asymptotic estimate of the solution on an interval, part of which corresponds to the unstable equilibrium, is obtained. The results clarify the influence of logarithmic poles on the system dynamics and describe the region in which the phenomenon of solution delay is observed.