Time-dependent saddle–node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions

Highlights: • Dynamical saddle–node bifurcation and its applications are studied. • Pullback attractors of non-autonomous dynamical systems are presented analytically and numerically. • The time window to avoid the catastrophic shift is investigated. -- Abstract: There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle–node bifurcations. In particular, the term "tipping", or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle–node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter $p\left(\tau \right)$ and its corresponding "dynamic" (time-dependent) saddle–node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the breaking time $\hat{\tau }$ at which the corresponding autonomous system with a time-independent parameter $p_a=p\left(\hat{\tau }\right)$ undergoes a bifurcation. A dimensionless parameter $\alpha =\lambda p_0^3{V}^{-2}$ is introduced, in which $\lambda$ is the curvature of the autonomous saddle–node bifurcation according to parameter $p\left(\tau \right)$, which has an initial value of $p_0$ and a constant rate of change $V$. We find that the breaking time $\hat{\tau }$ is always less than the actual point of no return ${\tau }^{\ast }$ after which the critical transition is irreversible; specifically, the relation ${\tau }^{\ast }-\hat{\tau }\simeq 2.338{\left(\lambda V\right)}^{-\frac{1}{3}}$ is analytically obtained. For a system with a small $\lambda V$, there exists a significant window of opportunity $(\hat{\tau },{\tau }^{\ast })$ during which rapid reversal of the environment can save the system from catastrophe.