Invariant curves and integrability of planar \(\mathcal C^r\) differential systems
This paper considers \({\mathcal C}^r\) real planar differential systems of the form \[ \dot{x} = P(x,y),\;\;\dot{y}=Q(x,y), \] where \((x,y) \in D \subseteq \mathbb{R}^2\), \(D\) is the domain of definition of the system and \(P,Q\in {\mathcal C}^r\) where \(r\) is a positive integer, or \(r = \infty\), or \(r = \omega\) (meaning that the system is analytic). This paper improves the known expression of the \({\mathcal C}^r\) differential systems in the plane having a given \({\mathcal C}^{r+1}\) invariant curve, or a given \({\mathcal C}^{r+1}\) first integral. Application of the expressions to polynomial systems also improve known results on polynomial differential systems having either an invariant algebraic curve, or a first integral. The main theorem shows that if a polynomial differential system has an invariant algebraic curve \(g = g(x,y) = 0\), and \(g_x \not\equiv 0\), then the system has the normal form \[ \dot{x} = A g - B g_y, \;\;\;\dot{y} = B g_x, \] where \(A\) and \(B\) are suitable rational functions.
scientific article; zbMATH DE number 6958791
scientific article