A study of the boundaries of stability regions in two-parameter dynamical systems

We consider dynamical systems defined by autonomous and periodic differential equations that depend on two scalar parameters. We study the problems of constructing boundaries of stability regions for equilibrium points in the plane of parameters. We identify conditions under which a point on the boundary of a stability region has one or more smooth boundary curves coming through it. We show schemes to find the basic scenarios of bifurcations when parameters transition over the boundaries of stability regions. We distinguish types of boundaries (dangerous or safe). The main formulas have been obtained in the terms of original equations and do not require to pass to normal forms and using theorems on a central manifold.