  It is conjectured that the generalization of the Constantin-Lax-Majda model
(gCLM) $\omega_t + a u\omega_x = u_x \omega$ due to Okamoto, Sakajo and Wunsch
can develop a finite time singularity from smooth initial data for $a < 1$. For
the endpoint case where $a$ is close to and less than $1$, we prove finite time
asymptotically self-similar blowup of gCLM on a circle from a class of smooth
initial data. For the gCLM on a circle with the same initial data, if the
strength of advection $a$ is slightly larger than $1$, we prove that the
solution exists globally with $|| \omega(t)||_{H^1}$ decaying in a rate of
$O(t^{-1})$ for large time. The transition threshold between two different
behaviors is $a=1$, which corresponds to the De Gregorio model.
