The Generalized Homoclinic Bifurcation

In this article, we study a family Xλ of vectors fields having at λ = 0 a homoclinic loop with multiplicity n. We give conditions for the displacement function to have the same zeros as all those of P(x, α(λ)) = α0(λ) + α1(λ) x + … + αn−1(λ) xn−1 + xn in a positive neighborhood of x = 0, where α(λ) is continuous. These conditions determine the versality of the family Xλ in a neighborhood of the loop. To obtain the result, we use the concept of Chebychev systems.