EXISTENCE AND COMPUTATION OF INVARIANT ALGEBRAIC CURVES FOR PLANAR QUADRATIC DIFFERENTIAL SYSTEMS

Some necessary conditions are given for the existence of invariant algebraic curves for planar quadratic differential systems in a special canonical form. An efficient algorithm is then designed for computations of invariant algebraic curves. From the algorithm, a quadratic differential system is found with two Hopf bifurcations as the parameter varies, each leading to an invariant algebraic limit cycle of degree 5. A family of degree 6 invariant algebraic limit cycles is also produced. To further demonstrate the capability of the algorithm, we provide a quadratic system with a family of degree 7 invariant algebraic curves enclosing one or two centers, and a system possessing a degree 16 irreducible invariant algebraic curve with a singular point of multiplicity 8 on the curve.