Tipping resonance in a chaotically forced ice age model

Abstract

Many physical systems are forced by external inputs, which can sometimes take the form of chaotic variation. A particular example is found in applications related to weather and climate, where chaotic variation is prevalent across various timescales. If the system in question has multiple attracting solutions for a given range of forcing, rate-induced tipping can be triggered by the chaotic forcing, with the difference in timescales between the forcing and the system acting as a `rate' parameter. In this paper, we explore the interplay between these two timescales in a low-order model of ice age dynamics. The model exhibits bistability between two equilibria in one region of the parameter space and between an equilibrium and a periodic orbit in another region. When chaotic variation of the parameters is allowed within these bistable regions, the solutions of the forced system undergo rate-induced tipping from one attractor to another. Simulations of the forced system show that the timescale of the chaotic forcing induces a resonance-like behaviour, with an optimal timescale at which the likelihood of rate-induced tipping is at its maximum. We combine basin instability theory, finite-time Lyapunov exponents, and linear resonance analysis under periodic forcing to explain this resonance effect.

Figures (12)

• Figure 1: Example trajectories of the transitions of interest: (a) transitioning from a state of anomalously high ice to one of anomalously low ice, and (b) transitioning from a state of anomalously low ice to large amplitude oscillations between low and high ice. The blue shading approximates the range between the forced response to the base state and an alternative response state.
• Figure 2: Two-parameter bifurcation diagram and phase portraits of System \ref{['eq:MS_unforced']}. (a) The two-parameter $(p,r)$ bifurcation diagram with two shaded regions of bistability. (b) The qualitative phase portrait of the system in region (i), showing two stable equilibria. (c) The qualitative phase portrait of the system in region (ii), showing one stable equilibrium and a large-amplitude stable periodic orbit. The other parameter $s = 0.8$.
• Figure 3: A schematic diagram of escape, rescue, and tipping events. The solution of the forced system (sky blue) escapes the basin of attraction (grey) of the moving equilibrium (dark magenta) of the unforced system at time $t_1$, is rescued at time $t_2$, and escapes again at time $t_3$. No rescue event follows the second escape; thus, the solution tips.
• Figure 4: Phase portraits of System \ref{['eq:MS_unforced']} at the boundary values of the first parameter path: (a) the parameter values $(p,r) = (p_1,r_1)=(0,1.2)$, (b) the parameter values $(p,r) = (p_2,r_2)=(0.6,1.2)$, and (c) the tipping threshold and equilibria from (b) (in black) with the tipping threshold and equilibria from (a) (in grey), showing that $e_+$ is basin unstable along the path. The other parameter $s=0.8$.
• Figure 5: Comparison of distance to the quasistatic equilibrium and Lyapunov exponents for the forced system. (a) Contours of the distance from the quasistatic solution for a selection of 100 MC simulations, with black dots denoting where the maximum FTLE is positive. (b) $\delta$-neighbourhood overlaid on the phase plane of the unforced system \ref{['eq:MS_unforced']} with $p=0.6$, $r=1.2$, and $s=0.8$. (c) The distance from the quasistatic equilibrium to one of the solutions that tipped in our Monte Carlo simulation. (d) The FTLEs $\Lambda_1$ (blue) and $\Lambda_2$ (red) along the same solution from (c). The middle red vertical shading indicates the time period from the first crossing to start of deviation from the time-dependent stable manifold of the quasi-static saddle. All model simulations use $\epsilon=0.2225$.