Critical saddle-node bifurcations and Morse-Smale maps
The authors study the saddle-node bifurcation in diffeomorphisms with a critical homoclinic orbit to a saddle point. More precisely, one considers a typical smooth family \(F_{\gamma}\) of diffeomorphisms that undergoes a saddle-node bifurcation at \(\gamma=0\). For \(\gamma<0\) the diffeomorphisms exhibit two periodic points which coalesce at \(\gamma=0\) and disappear for \(\gamma>0\). In order to characterize the behaviour of the family \(F_{\gamma}\), one recalls that a subset \(\Gamma\) of the parameter space has a positive density at \(0^{+}\) if \[ \liminf_{\gamma\searrow 0}{{m(\Gamma\cap[0,\gamma))}\over{\gamma}}>0, \] where \(m\) is the Lebesgue measure. In [\textit{L. J. Díaz, J. Rocha} and \textit{M. Viana}, Invent. Math. 125, 37--74 (1996; Zbl 0865.58034)] it is proved that for this type of bifurcation, there exist subsets \(\Gamma_{A}\) and \(\Gamma_{H}\) of the parameter space which have positive density at \(0^{+}\), and such that for \(\gamma\in \Gamma_{A}\), \(F_{\gamma}\) has an absolutely continuous invariant measure, while for \(\gamma\in \Gamma_{H}\), \(F_{\gamma}\) has a hyperbolic attracting periodic orbit. In the paper under review, one proves that if the saddle-node points have a critical homoclinic orbit there exists a set \(\Gamma_{M}\subset \Gamma_{H}\) with positive density at \(0^{+}\) such that \(f_{\gamma}\) is a Morse-Smale diffeomorphism for each \(\gamma\in\Gamma_{M}\). It is also proved that the boundary of the set of Morse-Smale diffeomorphisms possesses comb-like structures. On the other hand, one proves that if the criticalities are sufficiently formed, then \(F_{\gamma}\) cannot be Morse-Smale for any \(\gamma>0\). In the process of characterization of the set \(\Gamma_{M}\), the authors show that such unfoldings of bifurcations in diffeomorphisms are related to families of circle endomorphisms.
scientific article; zbMATH DE number 2117607
scientific article