Existence of limit cycles for a class of quintic Kukles homogeneous system

Abstract

We investigated the existence of limit cycles for quintic Kukles polynomial differential systems depending on a parameter in this chapter. These systems are important in practical applications and theoretical advances. We first used formal series method based on Poincaré’s ideas to prove this point and determine the centre-focus problem. We then utilised the Dulac function to prove the non-existence of closed orbits. We determined the sufficient condition for the existence of the limit cycles, which bifurcate from the equilibrium point, using Hopf bifurcation theory. Lastly, we provided some numerical examples for illustration using MATLAB to plot. Note that studies on the existence and the nonexistence of limit cycles and algebraic limit cycles for Kukles systems are limited.