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

Abstract

The remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by Poincaré. He was mainly interested in their algebraic aspects. Here we are interested in their dynamic aspects; i.e. how they contribute to the phase portrait of the system, to its separatrices, to its singular points, etc. The relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor.

A New Algorithm for Finding Rational First Integrals of Polynomial Vector Fields

“…Here if c = ∞ then f + cg denotes g. The curves in the factorization of f + cg, when c is a remarkable value, are called remarkable curves. It is proved in [3] that there are finitely many remarkable values for a given rational first integral H. These curves appear to be very important in the phase portrait of the vector field, as it is shown in [8,9].…”

“…In the process of searching the rational first integral our algorithm allows to compute all the remarkable curves of the system, which is in general a difficult task (see [6,9]). The remarkable curves are very important in the phase portrait of the polynomial systems having a rational first integral, because they provide many separatrices of the phase portrait.…”

Quantitative analysis of competition models

“…The numerator of all the first integrals in Table 1 has the form x p y q , for some p, q ∈ N. See [5] for details about this fact.…”

Detection of Special Curves Via the Double Resultant

“…The remarkable values and remarkable curves of rational first integrals of planar differential systems were first introduced by Poincaré in [23], and afterwards studied by several authors, see [8,10,11]. It has been shown in the literature that the remarkable curves play an important role in the phase portrait as they are strongly related to its separatrices.…”

“…It is proved in [8] that there are a finite number of them. In [8,11] they are related with the inverse integrating factor.…”