Simultaneous Bifurcation of Limit Cycles and Critical Periods

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

The Existence of n Periodic Solutions on One Element n-Degree Polynomial Differential Equation

This paper deals with a class of one element n-degree polynomial differential equations. By the fixed point theory, we obtain n periodic solutions of the equation. This paper generalizes some related conclusions of some papers.