Quadratic systems with a symmetrical solution

Abstract

In this paper we study the existence and uniqueness of limit cycles for so-called quadratic systems with a symmetrical solution: \begin{equation*} \begin{split} \frac{dx(t)}{dt}& = P_2(x,y) \equiv a_{00}+a_{10}x+a_{01}y+a_{20}x^2+a_{11}xy+a_{02}y^2\\ \frac{dy(t)}{dt}& = Q_2(x,y) \equiv b_{00}+b_{10}x+b_{01}y+b_{20}x^2+b_{11}xy+b_{02}y^2 \end{split} \end{equation*} where $(x,y) \in \mathbb{R}^2, t \in \mathbb{R}, a_{ij}, b_{ij} \in \mathbb{R}$, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic.

Keywords

• ordinary differential equations
• limit cycle
• algebraic curve