Bifurcations in a Mathieu equation with cubic nonlinearities. II
This paper is a continuation of a previous publication dealing with solution properties of the nonautonomous Mathieu equation with a nonlinear damping of cubic type. The study, made in the neighborhood of the 2:1 resonance, is based on the analytical averaging method supposing that the forcing periodic term and the nonlinearity are sufficiently small. It was shown that a degenerate bifurcation point exists in the resulting slow flow and some of the bifurcations near this point were looked at. The introduction reminds these results with comments on a bifurcation diagram in a parameter plane, and the phase plane portraits related to the autonomous ODE (variable being amplitude and phase) obtained from the averaging method using a second-order approximation. In particular, the parameter plane represents a heteroclinic bifurcation curve, a Poincaré-Andronov bifurcation curve (here, wrongly called ``Hopf''), and a curve related to the creation of a limit cycle at infinity. With respect to the previous publication, the authors complete the former bifurcation diagram by additional bifurcations in the neighborhood of the degenerate bifurcation. Moreover, an analytic approximation to the heteroclinic bifurcation on the above parameter plane is given. This permits to refine the structure of the bifurcation plane with a new drawing of the heteroclinic and Poincaré-Andronov bifurcation curves, more the curves related to type 1 and type 2 saddle-node, and degenerated Poincaré-Andronov bifurcations. The last part is devoted to an approximation to saddle-node bifurcation of periodic orbits and degenerate Poincaré-Andronov bifurcation. This is made using a bifurcation and continuation software which generates numerically the corresponding diagrams. An analytic expression for the Poincaré-Andronov bifurcation is given and appears in agreement with the numerical results of the software.
nonautonomous ordinary differential equations
analytical methods for nonlinear ordinary differential equations
scientific article; zbMATH DE number 1901810