Algebraic Limit Cycle of Degree 2 for Linear Type Center Plus Cubic Homogeneous Polynomial Differential Systems

In this paper, we study the algebraic limit cycle of degree 2 for the system \(\dot{x}= - y + a_{30}x^3 + {a_{21}}{x^2}y + {a_{12}}x{y^2} + {a_{03}}{y^3}\) $\dot{x}=-y+a_{30}{x}^3+a_{21}{x}^{2}y+a_{12}x{y}^2+a_{03}{y}^3$ , \(\dot{y}= x + {b_{30}}{x^3} + {b_{21}}{x^2}y + {b_{12}}x{y^2} + {b_{03}}{y^3}\) $\dot{y}=x+b_{30}{x}^3+b_{21}{x}^{2}y+b_{12}x{y}^2+b_{03}{y}^3$ , which is called a linear type center plus cubic homogeneous polynomial differential system. We derive the normal form of this system with an algebraic limit cycle of degree 2. It is shown that, if an invariant algebraic curve of degree 2 is a limit cycle of the above mentioned system, then it is a unique limit cycle in the phase plane.