TITLE:
Co-Existence of Local Limit Cycles from Degenerate and Weak Foci in Cubic Systems
AUTHORS:
Nick Schoonover, Terence Blows
KEYWORDS:
Planar Differential Equations, Local Limit Cycles, Degenerate Foci
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.16,
October
18,
2016
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.