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

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 $p(\tau )$ 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 {\em breaking time} $\hat{\tau }$ at which the corresponding autonomous system with a time-independent parameter $p_a=p(\hat{\tau })$ 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 $p(\tau )$, 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 ${\tau }^{\ast }-\hat{\tau }\simeq 2.338(\lambda V{)}^{-\frac{1}{3}}$ 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.

References in corpus (2)

• Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping
• Stochastic phenotype transition of a single cell in an intermediate region of gene-state switching

Cited by in corpus (3)

• Rate-induced tipping from periodic attractors: partial tipping and connecting orbits
• Delayed bifurcation in elastic snap-through instabilities
• Ergodicity Breaking and Scaling Relations for Finite-Time First-Order Phase Transition