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

a b s t r a c t 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''structureanddependsstronglyonthesaddlevalue,i.e. onthearea-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 homoclinictangencytoasaddle-focusisalwaysaccompaniedbystabilitywindows;howevertheviolation of the area-contraction property can make the stability windows invisible in one-parameter families.

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