On the Appearance Order of Homoclinic and Heteroclinic Intersections in a Family of Dissipative Maps of the Plane

The appearance order of homoclinic and heteroclinic intersections are analyzed in a family of dissipative maps fμ of the plane which have two saddles and an attractor. We consider a simple situation that stable and unstable manifolds of two saddles are disjoint for μ = 0, and they form a simple heteroclinic cycle for μ = 1. Thirteen routes from the initial state to the final state have been obtained. As a by-product, a new type of the first tangency which may give birth to homoclinic intersection points in conservative maps of the plane has been obtained. Finally, numerical examples are exhibited and the relation between the symmetry of the map and the occurrence of the routes is discussed.