A quadratic polynomial differential systemcan be identified with a single point of ℝ 12 through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ℝ 12 having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral. © 2010 Springer-Verlag Italia.
|Journal||Rendiconti del Circolo Matematico di Palermo|
|Publication status||Published - 1 Dec 2010|
- Phase portraits
- Quadratic vector fields
- Rational first integral
Artés, J. C., Llibre, J., & Vulpe, N. (2010). Quadratic systems with a rational first integral of degree three: A complete classification in the coefficient space ℝ 12. Rendiconti del Circolo Matematico di Palermo, 59(3), 419-449. https://doi.org/10.1007/s12215-010-0032-0