On the appearance order of homoclinic and heteroclinic intersections in a family of dissipative maps of the plane
The paper considers a one-parameter family \(f_{\mu}\), \(\mu \in [0,1]\) of dissipative orientation-preserving diffeomorphisms having three fixed points, two of which being the hyperbolic saddles and the third one stays in a neighborhood of the two saddles and may be a sink or a saddle. It is assumed that for \(\mu=0\), the stable and unstable manifolds of the two saddles have no common points and for \(\mu=1\) they form a simple heteroclinic cycle. The structure change of the system is triggered by the creation of the horseshoes, which often starts at a microscopic level and is not observable physically. Therefore it is important to catch the critical values of the parameter at which homoclinic intersections of lower order appear. The authors obtain all possible routes from the initial state to the final one when the first homoclinic tangencies appears and discover a new type of the first tangency which may give birth to homoclinic intersections in conservative maps of the plane. Finally they give numerical examples and discuss the relation between the symmetry of the map and the occurrence of the routes.
scientific article; zbMATH DE number 1370472
scientific article