Topological bifurcations of the axis and the alternating-hyperbolic sawtooth

An outstanding problem in sawtooth physics are the observations where $q_0$ stays around $0.7$, which cannot be explained by the Kadomtsev model that predicts a reset to $q_0 = 1$. We present a sawtooth model where the crash is caused by stochastization of a core region through the transition of the magnetic axis into an alternating-hyperbolic X-point when $q_0$ reaches $2/3$, which is within measurement uncertainty of the oft-measured value of $0.7$. This transition is revealed through the identification of the structure of the magnetic field line map around the axis with elements of the Lie group $\mathrm{SL}(2, \mathbb{R})$, which shows several transitions, one of which when $q_0=2/3$. We identify a fast-growing ideal 2/3 mode localized on the axis that appears when $q_0=2/3$ which perturbs the magnetic field such as to drive the transition to the alternating-hyperbolic geometry and stochastization of the core region.