Co-Existence of Local Limit Cycles from Degenerate and Weak Foci in Cubic Systems

Abstract

In this paper, we investigate the existence of local limit cycles obtained by perturbing degenerate and weak foci of two-dimensional cubic systems of differential equations. In particular, we consider a specific class of such systems where the origin is a degenerate focus. By utilizing a Liapunov function method and the stability results that follow, we first determine constraints on the system to maximize the number of local limit cycles that can be obtained by perturbing the degenerate focus at the origin. Once this is established, we add on the additional assumption that the system has a weak focus at , where , and determine conditions to maximize the number of additional local limit cycles that can be obtained near this fixed point. We will ultimately achieve an example of a cubic system with three local limit cycles about the degenerate focus and one local limit cycle about the weak focus.