Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems. (English) Zbl 1091.34017
Consider a planar polynomial differential equation \(\dot x=P(x,y), \dot y=Q(x,y).\) This interesting paper gives necessary conditions for the equation to have an invariant algebraic curve. Recall that a real polynomial in two variables \(f(x,y)\) defines an invariant algebraic curve \(f(x,y)=0\) for the differential equation if there exists another polynomial \(k(x,y)\) such that
\[
P(x,y)f_x(x,y)+Q(x,y)f_y(x,y)=k(x,y)f(x,y).
\]
This polynomial \(k(x,y)\) is called the cofactor of \(f(x,y).\) This set of conditions is given on the values of the cofactors of the possible invariant curves at the singular points of the differential equation. In order to enlarge the set of conditions on the cofactors, the authors also consider infinite singular points after extending the differential equation to the complex projective plane. It is clear that if \(p\) is a real or complex singular point of the extended differential equation and \(f(p)\neq0\), then \(k(p)=0.\) When \(f(p)=0\), the authors obtain for several types of singularities the value of \(k(p)\) in terms of the eigenvalues of the differential equation at \(p.\) One of the main advantages of including infinite critical points is that the value of the cofactor at these singular points can be used to get information about the degree of the polynomial invariant curve. The results obtained are applied to several examples of planar differential equations. In particular, they prove that for all known quadratic systems having an invariant algebraic curve, it is the only invariant algebraic curve of the differential equation. As a consequence, they show that none of these systems has a Liouvillian first integral.
This polynomial \(k(x,y)\) is called the cofactor of \(f(x,y).\) This set of conditions is given on the values of the cofactors of the possible invariant curves at the singular points of the differential equation. In order to enlarge the set of conditions on the cofactors, the authors also consider infinite singular points after extending the differential equation to the complex projective plane. It is clear that if \(p\) is a real or complex singular point of the extended differential equation and \(f(p)\neq0\), then \(k(p)=0.\) When \(f(p)=0\), the authors obtain for several types of singularities the value of \(k(p)\) in terms of the eigenvalues of the differential equation at \(p.\) One of the main advantages of including infinite critical points is that the value of the cofactor at these singular points can be used to get information about the degree of the polynomial invariant curve. The results obtained are applied to several examples of planar differential equations. In particular, they prove that for all known quadratic systems having an invariant algebraic curve, it is the only invariant algebraic curve of the differential equation. As a consequence, they show that none of these systems has a Liouvillian first integral.
Reviewer: Armengol Gasull (Barcelona)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C10 | Dynamics induced by flows and semiflows |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
Keywords:
planar ordinary differential equation; critical point; algebraic solution; algebraic limit cycle; quadratic systemReferences:
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