Quadratic Cremona maps transform known quadratic systems with algebraic limit cycles into a new family with a degree-5 limit cycle and recover all known higher-degree families.
Qin, On the algebraic limit cycles of second degree of the differential equation dy/dx = ∑ 0≤i+j≤2aijxiyj/ ∑ 0≤i+j≤2bijxiyj, Acta Math
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Quadratic Planar Differential Systems with Algebraic Limit Cycles via Quadratic Plane Cremona Maps
Quadratic Cremona maps transform known quadratic systems with algebraic limit cycles into a new family with a degree-5 limit cycle and recover all known higher-degree families.