On a Particular Foliation Associated with a Polynomial Family of Noninvertible Maps of the Plane

The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z0) each point having no real preimage, and (ii) (denoted by Z2) each point having two real preimages. The regions Z0, Z2 are separated by the critical curve LC, locus of points having two coincident preimages. Z2 is made up of two sheets (giving rise to a plane foliation) joining on LC, each being associated with a well-defined inverse of the map. The considered maps family is structurally unstable. For a wide choice of the parameter space it generates a singular foliation in the sense that the region Z2 is separated into two zones, Z ′ 2 and Z ′′ 2 , inside which the two preimages do not have the same qualitative behavior. Moreover, the boundary between Z ′ 2 and Z ′′ 2 is made up of points having only one real preimage at finite distance, the second one being at infinity. This situation gives rise to a nonclassical homoclinic bifurcation. The maps of the family have another important feature: their inverses present a denominator which vanish along a line of the plane. This has a great consequence on a chaotic attractor structure, when it exists. The imbedding of the map into a wider structurally stable family, generating regions Zk (k = 1, 3 being the number of real preimages), permits to understand the foliation nature when the imbedding parameter cancels leading to the structurally unstable map.