Isochronous oscillations for cubic systems
Planar vector fieldsẋ = -y + P₃(x,y)ẏ = x + q₃(x, y).with a center non necessarily at the origin are considered, where p₃ and q₃ are homogeneous polynomials of degree 3. We are concerned with the behavior of the periods of the periodic solutions near the center, and in determining when the center is isochronous, i.e., when all periodic solutions have the same period. It is proved that, modulus a linear change of coordinates, there are only four systems which have an isochronous center. Each of them has an isochronous center at the origin, and its other centers are also isochronous but not necessarily with the same period. Also the maximal weakness of the non isochronous centers, is obtained.