Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space \(\mathbb R^{12}\). (English) Zbl 1213.34004
Authors’ abstract: A quadratic polynomial differential system can be identified with a single point of \(\mathbb R^{12}\) 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 algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in \(\mathbb R^{12}\) 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 a first integral for it.
Reviewer: Alexey Remizov (Trieste)
MSC:
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
Keywords:
quadratic vector fields; integrability; rational first integral; phase portraits; invariantsReferences:
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