Invariant algebraic curves of large degree for quadratic system
This paper presents several interesting examples of quadratic systems (QS) having invariant algebraic curves of large degree. These examples are: (i) QS having algebraic limit cycles of degree 5 and 6; (ii) QS having algebraic saddle loops of degree 3 and 5 surrounding a strong focus; (iii) the QS, \(\dot x=1+x^2+xy,\,\) \(\dot y=57/2-81/2x^2+3y^2,\) which is not Liouvillian integrable and has an invariant algebraic curve of degree 12. It is worth to mention that until this paper appeared, the highest degree known for algebraic limit cycles of QS was 4. The new examples having limit cycles of degrees 5 and 6 have been constructed following the next clever idea: By applying to a QS of the form \[ dx/dt=\alpha x+\beta y+2ex^2+bxy+cy^2,\quad dy/dt=\gamma x+\delta y+ exy+fx^2, \] the transformation \(X=x/y^2, Y=1/y\) and the change of time \(dt/dT=Y,\) the new system still remains in the class of QS. Then, starting from the known QS having algebraic limit cycles of degree 4, new systems having also algebraic limit cycles of degrees 5 and 6 are constructed. It is unknown whether there are examples of QS having algebraic limit cycles of degree higher than~6.
scientific article; zbMATH DE number 2153754