Abstract:
In this article we study certain nonlinear autonomous systems of three nonlinear ordinary differential equations (ODEs) with small parameter $\mu$ such that the two variables $(x,y)$ are fast and the other $z$ is slow.
Taking the limit as $\mu \to 0$, this becomes the «degenerate system» that is included in the one-parameter family of the two-dimensional subsystems of fast motions with the parameter $z$ from some interval. It is assumed that
there is a monotonic function $\boldsymbol \rho(z)$, which 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 the 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: the two arcs are stable and the third arc between them is unstable. For the complete dynamical system we prove the existence of a trajectory which is located as close as possible to the both stable and unstable branches of the slow curve ${\mathcal L}$ as $\mu$ tends to zero and values of $z$ for the given interval.
Keywords:Andronov—Hopf bifurcation, nonlinear ordinary differential equations (ODEs), ODEs with a small parameter, asymptotic expansion, Lyapunov function.
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