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There is a growing awareness that catastrophic phenomena in biology andmedicine can be mathematically represented in terms of saddle-nodebifurcations. In particular, the term `tipping', or critical transition has inrecent years entered the discourse of the general public in relation toecology, medicine, and public health. The saddle-node bifurcation and itsassociated theory of catastrophe as put forth by Thom and Zeeman has seenapplications in a wide range of fields including molecular biophysics,mesoscopic physics, and climate science. In this paper, we investigate a simplemodel of a non-autonomous system with a time-dependent parameter $p(\tau)$ andits corresponding `dynamic' (time-dependent) saddle-node bifurcation by themodern theory of non-autonomous dynamical systems. We show that the actualpoint of no return for a system undergoing tipping can be significantly delayedin comparison to the {\em breaking time} $\hat{\tau}$ at which thecorresponding autonomous system with a time-independent parameter $p_{a}=p(\hat{\tau})$ undergoes a bifurcation. A dimensionless parameter$\alpha=\lambda p_0^3V^{-2}$ is introduced, in which $\lambda$ is the curvatureof the autonomous saddle-node bifurcation according to parameter $p(\tau)$,which has an initial value of $p_{0}$ and a constant rate of change $V$. Wefind that the breaking time $\hat{\tau}$ is always less than the actual pointof no return $\tau^*$ after which the critical transition is irreversible;specifically, the relation $\tau^*-\hat{\tau}\simeq 2.338(\lambdaV)^{-\frac{1}{3}}$ is analytically obtained. For a system with a small $\lambdaV$, there exists a significant window of opportunity $(\hat{\tau},\tau^*)$during which rapid reversal of the environment can save the system fromcatastrophe.

Time Dependent Saddle Node Bifurcation: Breaking Time and the Point of No Return in a Non-Autonomous Model of Critical Transitions