The effect of timescale separation on the tipping window for chaotically forced systems

TL;DR

This work extends the chaotic tipping window framework to continuous-time bistable systems, showing how the relative forcing–response timescale $\gamma$ determines whether tipping is governed by forcing extrema (slow forcing) or forcing means (fast forcing). By analyzing two coupled ODEs forced with chaotic Lorenz dynamics and leveraging unstable periodic orbits (UPOs) on the forcing attractor, the authors derive limiting-endpoint formulas and demonstrate rich intermediate-regime behavior where UPOs rearrange to shape tipping boundaries. The study introduces dynamic tipping windows under parameter drift, linking autonomous tipping to nonautonomous tipping and highlighting the delay/advance of tipping caused by ramp rates. Overall, the results provide a rigorous, UPO-driven, ergodic-optimization perspective on how timescale separation governs chaotically forced tipping and its evolution under drift, with potential implications for climate and other multiscale systems.

Abstract

Tipping behavior can occur when an equilibrium of a dynamical system loses stability in response to a slowly varying parameter crossing a bifurcation threshold, or where noise drives a system from one attractor to another, or some combination of these effects. Similar behavior can be expected when a multistable system is forced by a chaotic deterministic system rather than by noise. In this context, the chaotic tipping window was recently introduced and investigated for discrete-time dynamics. In this paper, we find tipping windows for continuous-time nonlinear systems forced by chaos. We characterize the tipping window in terms of forcing by unstable periodic orbits of the chaos, and we show how the location and structure of this window depend on the relative timescales between the forcing and the responding system. We illustrate this by finding tipping windows for two examples of coupled bistable ODEs forced with chaos. Additionally, we describe the dynamic tipping window in the setting of a changing system parameter.

Figures (11)

• Figure 1: Typical trajectories of the Lorenz-double-well system \ref{['eq:LorDoubleWell']} with $a=0.03$, $\gamma=1$, randomly chosen initial conditions of the Lorenz-$y$-dynamics (i.e. typical with respect to the physical ergodic measure $m$ on the Lorenz-63 attractor), and three different values of $\eta$. The $x$-dynamics are initialized at $x=-1.5$ in the basin of attraction of the double-well attractor $x_-(\eta)$. The blue curves ($\eta=1.7$) remain in the vicinity of the attractor of the $x$-dynamics they started in. The orange curves ($\eta=1.85$) all leave the vicinity where they were initialized in ("they tip"), but the time of leaving this vicinity strongly depends on the specific $y$-trajectory that forces the $x$-dynamics. The green curves ($\eta=2.01$) always immediately leave towards the attractor $x_+(\eta)$, independently of the specific Lorenz trajectory. The $y$-trajectories are generated by taking the final state of the previous ensemble member as the initial condition for the next.
• Figure 2: The coloured lines show for each $a$ the lowest value of $\eta$ for which the $x$ dynamics (initialized in $x=-1.5$) fulfilled $x>0$ at some point during the integration time (i.e. the $x$ dynamics leave the vicinity of $x_-(\eta)$ and go to $x_+(\eta)$ - they "tip"), when forcing the double-well dynamics given by Equation (\ref{['eq:doubleWell']}) with the mean (left plot) or the maximum (right plot) of the colour-coded UPO. The law of motion is of the form of Equation (\ref{['eq:addforcedSyst']}) with $\phi(y(\gamma t))=\bar{y}_{1,UPO_k}$ given by the mean (left plot) or $\phi(y(\gamma t))=\max(y_{1,UPO_k})$ the maximum (right plot) of the $k$-th UPO. The values $\bar{y}_{1,UPO_k}$ and $\max(y_{1,UPO_k})$ are given in Tab. \ref{['tab:DWgrayShadingTimes']}. For each UPO, we fix discrete values of $a\in[0,0.04]$ with spacing of $0.01$ and then do bisections in $\eta\in(0.4,1.6)$ to approximate the lowest $\eta$ up to an accuracy of $5\times10^{-3}$ for which the system, initialized in $x=-1.5$, tips, i.e. it fulfills $x>0$ during the integration time. Note that in the left plot, the lines are ordered from left to right by decreasing mean $y_1$-value of the associated UPOs, and each line is given by $\eta_k(a)=\eta^*-a\bar{y}_{1,UPO_k}$ as derived in section \ref{['sec:inf_ts_sep']}. In the right plot, the lines are ordered from left to right by decreasing maximum $y_1$-value of the associated UPOs, and each line is given by $\eta_k(a)=\eta^*-a\max(y_{1,UPO_k})$.
• Figure 3: The colored lines show for each $a$ the lowest value of $\eta$ for which tipping is observed when forcing the double-well dynamics given by Equation (\ref{['eq:doubleWell']}) with the color-coded UPO. The system is of the form of Equation (\ref{['eq:addforcedSyst']}) with $y_1(t)=y_{1,UPO_k}$ given by the $k$th UPO and $\gamma\in(0.01, \, 0.1, \, 1, \, 10)$. For each UPO, we fix discrete values of $a\in[0,0.04]$ with spacing $0.01$ and then do bisections in $\eta\in(0.4,1.6)$ to approximate the lowest $\eta$ up to an accuracy of $5\times10^{-3}$ for which the system initialized in $x=-1.5$ tips to the positive attractor i.e. $x>0$ within the integration time. The black, dark gray, light gray, and white shading show in which time interval tipping of the double-well system \ref{['eq:LorDoubleWell']} initialized in the basin of the attractor $x_-(\eta)$ (at $x=-1.5$), was observed as a result of forcing with a randomly chosen Lorenz-trajectory. The Lorenz initial condition of one run is given by letting the final condition of the previous run evolve for $5$ Lorenz-timesteps. The first initial condition of the Lorenz system is given by starting the system at $(y_1,y_2,y_3)=(0.1,0.1,25.1)$ and letting it relax to the attractor for $5$ Lorenz-timesteps. In the white areas, tipping to the other attractor was observed at a very small time, and in the black areas, tipping was not observed at all during the simulation time. The gray shadings show tipping for intermediate times. The time intervals corresponding to the different shading colors are given in Table \ref{['tab:DWgrayShadingTimes']} for each value of $\gamma$.
• Figure 4: For fixed $a=0.04$, we plot the critical values of $\eta$ from Figure \ref{['fig:DWgamma_intermediate']} vs. $\gamma$. The dots at the top and bottom show the $\eta$ value at tipping of the tipping of the $x$ dynamics for the limiting cases of timescale separation.
• Figure 5: For a set of fixed $\gamma\in\{0.01,0.02,\ldots,0.1\}$, we approximate the value of $a$ for which tipping of the Lorenz-double-well system's $x$ dynamics in response to UPO4 forcing occurs at the same value of $\eta$ as tipping in response to UPO1 forcing using a bisection method. We do this by fixing a $\gamma$ value, setting $\eta(a)=\eta^*-\frac{a}{2}(\max(y_{1,UPO_1}) + \bar{y}_{1,UPO1} )$ (which is an approximation of the $\eta$ value for which the system tips in response to UPO1 forcing), and then doing a bisection in $a$ to find the smallest value of $a$ for which the $x$ dynamics tip in response to UPO4 forcing. This gives $a$ as a function of $\gamma$ , shown here in blue in the log-log plot. The orange line corresponds to a least-squares fit of the function $a=c\gamma^2$ to the blue dots with best fit $c= 21.15$ and standard error $\pm 0.00129$.

Theorems & Definitions (2)

• Theorem 2.1
• proof