The Generalized Homoclinic Bifurcation

In this article, we study a family Xλ of vector 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.