  We study an $\mathcal{N}=1$ supersymmetric quantum field theory with
$O(M)\times O(N)$ symmetry. Working in $3-\epsilon$ dimensions, we calculate
the beta functions up to second loop order and analyze in detail the
Renormalization Group (RG) flow and its fixed points. We allow $N$ and $M$ to
assume general real values, which results in them functioning as bifurcation
parameters. In studying the behaviour of the model in the space of $M$ and $N$,
we demarcate the region where the RG flow is non-monotonic and determine curves
along which Hopf bifurcations take place. At a number of points in the space of
$M$ and $N$ we find that the model exhibits an interesting phenomenon: at these
points the RG flow possesses a fixed point located at real values of the
coupling constants $g_i$ but with a stability matrix $\left(\frac{\partial
\beta_i}{\partial g_j}\right)$ that is not diagonalizable and has a Jordan
block of size two with zero eigenvalue. Such points correspond to logarithmic
CFTs and represent Bogdanov-Takens bifurcations, a type of bifurcation known to
give rise to a nearby homoclinic orbit - an RG flow that originates and
terminates at the same fixed point. In the present example, we are able to
employ analytic and numeric evidence to display the existence of the homoclinic
RG flow.
