Non-algebraic invariant curves for polynomial planar vector fields

Abstract

In this paper we give, as far as we know, the first method to detect non-algebraic invariant curves for polynomial planar vector fields. This approach is based on the existence of a generalized cofactor for such curves. As an application of this algorithmic method we give some Lotka-Volterra systems with non-algebraic invariant curves.

A note on Liouvillian first integrals and invariant algebraic curves

“…In order to find a first integral for system (1) we can also use non-algebraic invariant curves with polynomial cofactor, see [5,6]. Some generalizations of the Liouvillian integrability theory are given in [8,9,12] where a new kind of first integrals, not only the Liouvillian ones, appears.…”

“…where, without loss of generality, we have privileged in (6) the variable y with respect to variable x writing the polynomials P (x, y) and Q(x, y) of ( 1) as polynomials in y with coefficients polynomials in x. The particular case studied in [11] is system (6) with P (x, y) = 1.…”

A Survey on the Inverse Integrating Factor

“…for all (x, y) ∈ U. The notion of invariant curve was first introduced in [53]. The identity (8) can be rewritten by X f = kf .…”

“…In case f (x, y) = 0 defines a curve in the real plane, this definition implies that the function X f is equal to zero on the points such that f (x, y) = 0. In the article [53] an invariant curve is defined as a C 1 function f (x, y) defined in the open set U ⊆ R 2 , such that, the function X f is zero in all the points {(x, y) ∈ U | f (x, y) = 0}. We notice that our definition of invariant curve is a particular case of the previous one but, for the sake of our results, the cofactor is very important and that's why we always assume its existence.…”

Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in ℂ2

“…Later on the Liouvillian theory of integrability has been extended to n-dimensional autonomous or non-autonomous differential systems in [18]. A method for detecting non-algebraic invariant curves for some polynomial differential systems was given in [12].…”