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 nonlinear ordinary differential equations (ODE) with small parameter $\mu$ such that two variables $(x,y)$ are fast and the remaining variable $z$ is slow. In the limit as $\mu \to 0$, 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 with parameter $z$ in some interval. It is assumed that there exists a monotone function $\boldsymbol \rho (z)$ that, in the three-dimensional phase space of a complete dynamical system, defines a parametrization of some arc $\mathcal L$ of a slow curve consisting of the family of fixed points of the degenerate subsystems. Let $\mathcal L$ have two points of the Andronov–Hopf bifurcation in which some stable limit cycles arise and disappear in the two-dimensional subsystems. These bifurcation points divide $\mathcal L$ into the three arcs; two arcs are stable, and the third arc between them is unstable. For the complete dynamical system, we prove the existence of a trajectory that is located as close as possible to both the stable and unstable branches of the slow curve $\mathcal L$ as $\mu$ tends to zero for values of $z$ within a given interval.