Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions

Abstract

<p style='text-indent:20px;'>We consider a homoclinic orbit to a saddle fixed point of an arbitrary <inline-formula><tex-math id="M1">\begin{document}$ C^\infty $\end{document}</tex-math></inline-formula> map <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> and study the phenomenon that <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory this requires <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> to have a homoclinic tangency. We show it is also necessary for <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> to satisfy a 'global resonance' condition and for the eigenvalues associated with the fixed point, <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>, to satisfy <inline-formula><tex-math id="M9">\begin{document}$ |\lambda \sigma| = 1 $\end{document}</tex-math></inline-formula>. The phenomenon is codimension-three in the case <inline-formula><tex-math id="M10">\begin{document}$ \lambda \sigma = -1 $\end{document}</tex-math></inline-formula>, but codimension-four in the case <inline-formula><tex-math id="M11">\begin{document}$ \lambda \sigma = 1 $\end{document}</tex-math></inline-formula> because here the coefficients of the leading-order resonance terms associated with <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula> at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.