1996/01/01 by Jeremiah H. Li, Jeremiah Li, Felix X.-F. Ye +4 · 1 citation
Biochemistry, Genetics and Molecular Biology · Earth and Planetary Sciences · Environmental Science · Mathematics · Physics and Astronomy · Social Sciences · #Aesthetics #Bifurcation #Bifurcation theory #Classical mechanics #Dimensionless quantity #Dynamical systems theory #Earth Systems and Cosmic Evolution #Ecosystem dynamics and resilience #Environmental ethics #Epistemology #Geometry #Infinite-period bifurcation #Law and economics #Mathematical optimization #Mathematics #Mechanics #Node (physics) #Nonlinear system #Philosophy #Physics #Political Philosophy and Ethics #Political science #Quantum mechanics #Saddle #Saddle point #Saddle-node bifurcation #Sociology #Statistical physics #math.DS #q-bio.QM #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/j.physd.2019.02.005
openalex publication_date 1996/01/01 · openalex created_date 2016/06/24 · arxiv created 2019/01/03 · arxiv updated 2019/10/29 · openalex updated_date 2026/08/01
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(τ) 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 τ at which the corresponding autonomous system with a time-independent parameter pa= p(τ) undergoes a bifurcation. A dimensionless parameter α=λp03V-2 is introduced, in which λ is the curvature of the autonomous saddle-node bifurcation according to parameter p(τ), which has an initial value of p0 and a constant rate of change V. We find that the breaking time τ is always less than the actual point of no return τ^* after which the critical transition is irreversible; specifically, the relation τ^*-τ≃ 2.338(λV)-(1)/(3) is analytically obtained. For a system with a small λV, there exists a significant window of opportunity (τ,τ^*) during which rapid reversal of the environment can save the system from catastrophe.