Integral invariants and limit sets of planar vector fields

Abstract

We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact -or -limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then V vanishes at at least one point of K (and on all of K if V is real analytic or Morse).

The inverse integrating factor and the Poincaré map

“…More precisely, in [3], Berrone and Giacomini proved that, if p 0 is a hyperbolic saddle point of system (1), then any inverse integrating factor V defined in a neighborhood of p 0 vanishes on all four separatrices of p 0 , provided V (p 0 ) = 0. We emphasize here that this result does not hold, in general, for nonhyperbolic singularities (see [11] for example).…”

On the remarkable values of the rational first integrals of polynomial vector fields

“…Some results on the vanishing set of the inverse integrating factor can be found in [1,9,10,13]. Next we prove Theorem 4, which is a new result related to the vanishing set of V .…”

“…A good way to study integrable vector fields is through the inverse integrating factor V , for more details see [2]. If X is a real vector field and V : U → R is an inverse integrating factor of X on the open subset U of R 2 , then V becomes very important because V −1 (0) contains a lot of information about the skeleton or separatrices of the phase portrait of X in U , see [1,9,10,13].…”

“…for the pairs (u i , α i ) given by (h 1 , 2), (h 2 , 3), (y, 10) and (x, 2). We get the solutions shown in Table 2.…”

Nilpotent Systems Admitting an Algebraic Inverse Integrating Factor over $${{\mathbb{C}}((x,y))}$$

“…There are other reasons to study the existence of inverse integrating factors, among them: this concept plays an important role in the study of the existence of limit cycles of a vector field, because the zero-set {V = 0}, formed by orbits of the system (1), contains the limit cycles of the system (1) which are in U, whenever they exist, see [10,14,15]. The zero-set {V = 0} also contains the homoclinic and heteroclinic connections between hyperbolic saddle equilibria, see [13].…”