Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter

Abstract: In the present paper, we study some nonlinear autonomous systems of three nonlinearordinary differential equations (ODE) with small parameter (Formula presented.) such that two variables (x, y) are fast and the remaining variable z is slow. In the limit as (Formula presented.), from this “complete dynamical system” we obtain the “degenerate system,”which is included in a one-parameter family of two-dimensional subsystems of fast motions withparameter z in some interval. It is assumed that there exists a monotone function (Formula presented.) that, in the three-dimensional phase space of a complete dynamical system,defines a parametrization of some arc L of a slow curve consisting of thefamily of fixed points of the degenerate subsystems. Let L have two points of the Andronov–Hopf bifurcation in which some stable limitcycles arise and disappear in the two-dimensional subsystems. These bifurcation points divide L into the three arcs; two arcs are stable, and the third arc between them isunstable. For the complete dynamical system, we prove the existence of a trajectory that islocated as close as possible to both the stable and unstable branches of the slow curve L as (Formula presented.) tends to zero for values of z within a given interval. © 2022, Pleiades Publishing, Ltd.