# gst: the evasion of tipping; pattern formation near a Turing-fold bifurcation

<< ️Model studies indicate that many climate subsystems, especially ecosystems, may be vulnerable to 'tipping': a 'catastrophic process' in which a system, driven by gradually changing external factors, abruptly transitions (or 'collapses') from a preferred state to a less desirable one. In ecosystems, the emergence of spatial patterns has traditionally been interpreted as a possible 'early warning signal' for tipping. More recently, however, pattern formation has been proposed to serve a fundamentally different role: as a mechanism through which an (eco)system may 'evade tipping' by forming stable patterns that persist beyond the tipping point. >>

<< ️Mathematically, tipping is typically associated with a saddle-node bifurcation, while pattern formation is normally driven by a Turing bifurcation. Therefore, (AA) study the co-dimension 2 Turing-fold bifurcation and investigate the question: 'When can patterns initiated by the Turing bifurcation enable a system to evade tipping?' >>

AA << develop (their) approach for a class of phase-field models and subsequently apply it to -component reaction-diffusion systems -- a class of PDEs often used in ecosystem modeling. (AA) demonstrate that a two-component system of modulation equations governs pattern formation near a Turing-fold bifurcation, and that tipping will be evaded when a critical parameter, β, is positive. (They) derive explicit expressions for β, allowing one to determine whether a given system may evade tipping. Moreover, (They) show numerically that this system exhibits rich behavior, ranging from stable, stationary, spatially quasi-periodic patterns to irregular, spatio-temporal, chaos-like dynamics. >>

Dock Staal, Arjen Doelman. The evasion of tipping: pattern formation near a Turing-fold bifurcation. arXiv: 2506.22251v1 [math.DS]. Jun 27, 2025.

https://arxiv.org/abs/2506.22251

Also: disorder & fluctuations, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, disorder & fluctuations, pattern formation, Turing-fold bifurcation, transitions, tipping, collapse

# gst: tipping in a low-dimensional model of a (tropical) cyclone.

 <<Tipping is the rapid, and often irreversible, change in the state of a system [Ashwin P. et al (2017)] >> 

AA << present a case study of both rate and noise-induced tipping between the stable states, relating to the destabilization or formation of a tropical cyclone. While the stochastic system exhibits transitions both to and from the non-storm state, noise-induced tipping is more likely to form a storm, whereas rate-induced tipping is more likely to be the way a storm is destabilized, and in fact, rate-induced tipping can never lead to the formation of a storm when acting alone. For rate-induced tipping acting as a destabilizer of the storm, a striking result is that both wind shear and maximal potential velocity have to increase, at a substantial rate, in order to effect tipping away from the active hurricane state. For storm formation through noise-induced tipping, (AA) identify a specific direction along which the non-storm state is most likely to get activated. >> 

Katherine Slyman, John A. Gemmer, et al. Tipping in a Low-Dimensional Model of a Tropical Cyclone. arXiv: 2307.15583v1 [math.DS]. Jul 28, 2023.

https://arxiv.org/abs/2307.15583

 

Also: vortex, instability, noise, in https://www.inkgmr.net/kwrds.html

Keywords: gst, climate, storm, cyclone, vortex, noise, tipping, noise-induced tipping