STUDY OF THE BOUNDARIES OF STABILITY REGIONS OF TWO-PARAMETER SECOND ORDER DIFFERENTIAL EQUATIONS
In the article, second order nonlinear differential equations depending on two scalar parameters are studied. Construction of the boundaries of stability regions, in particular, the equilibrium point in the parameter plane is determined by the equation of the tangent to the boundary curve. The conditions under which one or more smooth boundary curves pass through the border point of stability region are specified. The scheme for determining the main scenarios of bifurcations at transition of parameters over the boundaries of stability regions is given. “Safe” and “dangerous” boundary types are determined in accordance with definition by N. Bautin. As an example, the nonlinear nonautonomous equation of van der Pol is considered. Approximate graphs of the boundaries of the stability regions in the parameter plane equation of van der Pol in the non-resonant and resonant cases are presented.