Abstract
We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for
C^{r}
(
r=3,\dots,\infty,\omega
) dynamical systems on a manifold
\mathcal{M}
, with
\dim \mathcal{M}\geqslant 3
for diffeomorphisms and with
\dim \mathcal{M}\geqslant 4
for flows, that
C^{1}
-robust heterodimensional dynamics of coindex one appears in any generic two-parameter
C^{r}
unfolding of a homoclinic tangency to a periodic orbit such that at least one central multiplier is not real and the central dynamics is not sectionally dissipative. The heterodimensional dynamics also involves a blender exhibiting
C^{1}
-robust homoclinic tangencies. As a corollary, any system with a homoclinic tangency of the class described above belongs to the
C^{r}
-closure of the
C^{1}
-open Newhouse domain.