A Criterion for Safe Overshoot in Coupled Tipping Systems

Abstract

Abrupt transitions are a central concern in climate and ecological research, and may arise when critical thresholds known as tipping points are crossed. However, previous work has shown that finite-time overshoots of tipping points can be safe, and that such behavior is captured by an inverse-square-law criterion when overshoots are sufficiently small and slow. So far studied in isolated systems with external drivers, (un)safe overshoots may also emerge from interactions between subsystems. Here, we investigate safe-overshoot phenomena in unidirectionally coupled slow-fast systems featuring both nonlinear interactions and coupling through time-derivatives. Specifically, we derive a criterion for the occurrence of safe overshoots analogous to the inverse-square law for isolated systems, but adapted to interactive settings, and expressed explicitly in terms of the timescale separation and coupling strength between subsystems. We illustrate these results using two conceptual models in which the Atlantic Meridional Overturning Circulation interacts with either the Amazon rainforest or the Greenland Ice Sheet.

Figures (7)

• Figure 1: Trajectories and bifurcation structure of the two example systems defined by \ref{['eq:systexx']}-\ref{['eq:systexy']} under the slowly varying forcing \ref{['eq:muillustr']}. Blue, green, and orange curves indicate trajectories for increasing coupling strengths $\gamma$, resulting in no overshoot, safe overshoot, and unsafe overshoot, respectively. (a) From top to bottom: trajectory of the slow subsystem $x$; trajectory of the coupling functions $c_1(x)$ (dark blue) and $c_2(x)$ (dark orange); trajectories of the fast subsystem $y_1$ coupled via $c_1(x)$; and trajectories of the fast subsystem $y_2$ coupled via $c_2(x)$. (b) Bifurcation diagram of the slow subsystem showing stable (solid black) and unstable (dashed black) equilibria. The slow-subsystem trajectory is overlaid in grey. (c) Bifurcation diagrams of the fast subsystems $y_1$ (left) and $y_2$ (right), showing stable (solid black) and unstable (dashed black) equilibria. Fast-subsystem trajectories are overlaid in blue, green, and orange.
• Figure 2: Overshoot regimes of the two example systems defined by \ref{['eq:systexx']}-\ref{['eq:systexy']} under the slowly varying forcing \ref{['eq:muillustr']}. No-overshoot (blue), safe-overshoot (green), and unsafe-overshoot (orange) regions are shown as a function of timescale separation $\epsilon$ and coupling strength $\gamma$. (a) Nonlinear coupling via $c_1(x)$. (b) Derivative coupling via $c_2(x)$. The solid black line shows the approximation of the boundary between safe and unsafe overshoot provided by Result \ref{['res:1']}. Stars indicate the parameter combinations used in Fig. \ref{['fig:Fig1']}.
• Figure 3: Overshoot regimes of the example system defined by \ref{['eq:systexx']}-\ref{['eq:systexy']} in the case of derivative coupling via $c_2(x)$, for different values of the forcing rate parameter $r$. No-overshoot (blue), safe-overshoot (green), and unsafe-overshoot (orange) regions are shown as a function of timescale separation $\epsilon$ and coupling strength $\gamma$. The solid and dashed black lines show the approximation of the boundary between safe and unsafe overshoot provided by Result \ref{['res:1']} and Corollary \ref{['corr:1']}, respectively.
• Figure 4: Coupled AMOC-Amazon overshoot dynamics under the slowly varying forcing \ref{['eq:Fvar']}. (a) Trajectories of AMOC strength $Q$ (top) and Amazon tree cover $T$ (bottom) for three values of the coupling strength $\gamma = 0.50$, $0.57$, and $0.62$ (blue, green, and orange, respectively). (b) The same trajectories as in (a), shown on the bifurcation diagrams of the AMOC (left) and Amazon (right), with stable (solid black) and unstable (dashed black) equilibria. (c) Overshoot regimes of the coupled AMOC-Amazon system. No-overshoot (blue), safe-overshoot (green), and unsafe-overshoot (orange) regions are shown as a function of timescale separation $\epsilon$ and coupling strength $\gamma$. The solid black line shows the approximation of the boundary between safe and unsafe overshoot provided by Result \ref{['res:1']}. Stars indicate the parameter combinations shown in (a-b). The vertical dotted grey line denotes the reference value of $\epsilon$.
• Figure 5: Coupled GIS-AMOC overshoot dynamics under the slowly varying forcing \ref{['eq:vardT']}. (a) Trajectories of GIS ice-volume fraction $V$ (top) and AMOC strength $Q$ (bottom) for three values of the timescale separation $\epsilon = 1.0$, $1.6$, and $2.4$ (blue, green, and orange, respectively). (b) The same trajectories as in (a), shown on the bifurcation diagrams of the GIS (left) and AMOC (right), with stable (solid black) and unstable (dashed black) equilibria. (c) Overshoot regimes of xthe coupled GIS-AMOC system. No-overshoot (blue), safe-overshoot (green), and unsafe-overshoot (orange) regions are shown as a function of timescale separation $\epsilon$ and coupling strength $\gamma$. The dashed black line shows the approximation of the boundary between safe and unsafe overshoot provided by Corollary \ref{['corr:1']}. Stars indicate the parameter combinations shown in (a-b). The vertical and horizontal dotted grey lines denote the reference values of $\epsilon$ and $\gamma$, respectively.