Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations

We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|\lambda\gamma| = 1$. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus $|\lambda\gamma|$, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map $f$ can be $C^{r}$-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case.