On the effect of invisibility of stable periodic orbits at homoclinic bifurcations

We study bifurcations of a homoclinic tangency to a saddle-focus periodic point. We show that the stability domain for single-round periodic orbits which bifurcate near the homoclinic tangency has a characteristic “comb-like” structure and depends strongly on the saddle value, i.e. on the area-contracting properties of the map at the saddle-focus. In particular, when the map contracts two-dimensional areas, we have a cascade of periodic sinks in any one-parameter family transverse to the bifurcation surface that corresponds to the homoclinic tangency. However, when the area-contraction property is broken (while three-dimensional volumes are still contracted), the cascade of single-round sinks appears with “probability zero” only. Thus, if three-dimensional volumes are contracted, chaos associated with a homoclinic tangency to a saddle-focus is always accompanied by stability windows; however the violation of the area-contraction property can make the stability windows invisible in one-parameter families.