Fractals in rate-induced tipping

TL;DR

The paper addresses rate-induced tipping (R-tipping) in non-autonomous systems and shows that non-attracting fractal saddles imprint fractal structures on parameter and rate spaces, producing a continuum of tipping thresholds. Using a general discrete-time formalism, the authors demonstrate fractal track/tip boundaries in three paradigmatic systems: a piecewise-linear 1D map, the Hénon map, and a forced pendulum. They establish a unifying mechanism with three conditions, proving that the fractal co-dimension $\alpha$ of the basin boundary at the final parameter $\lambda_+$ matches the co-dimensions of the flip in initial-condition space and in parameter/rate spaces ($\alpha_1=\alpha_2=\alpha_3$). The results quantify extreme sensitivity to parameter changes via $\alpha$, with practical implications for forecasting and controlling tipping in climate and ecological models that undergo non-autonomous parameter protocols. This framework highlights how fractal edge states govern tipping behavior and may extend to broader non-autonomous dynamical systems.

Abstract

When parameters of a dynamical system change sufficiently fast, critical transitions can take place even in the absence of bifurcations. This phenomenon is known as rate-induced tipping and has been reported in a variety of systems, from simple ordinary differential equations and maps to mathematical models in climate sciences and ecology. In most examples, the transition happens at a critical rate of parameter change, a rate-induced tipping point, and is associated with a simple unstable orbit (edge state). In this work, we show how this simple picture changes when non-attracting fractal sets exist in the autonomous system, a ubiquitous situation in non-linear dynamics. We show that these fractals in phase space induce fractals in parameter space, which control the rates and parameter changes that result in tipping. We explain how such rate-induced fractals appear and how the fractal dimensions of the different sets are related to each other. We illustrate our general theory in three paradigmatic systems: a piecewise linear one-dimensional map, the two-dimensional Hénon map, and a forced pendulum.

Figures (8)

• Figure 1: Graphical representation of the modified tent map $T(x)$ in Eq. (\ref{['eq.example1']}). There are four fixed points at the intersection of $T(x)$ (solid line) and the diagonal $y=x$ (dashed blue line): two unstable as in the open tent map ($x^{u,1}=0$ and $x^{u,2}=3/4$), a third unstable ($x^{u,3}=4/3$) in the intermediate interval ($37/33 < x < 3/2$), and one stable ($x^*=55/18$). We investigate R-tipping out of $x^*$ by shifting this map along the diagonal -- see Eqs. \ref{['eq.lambdaTent']}-\ref{['eq:tanh_drift']}.
• Figure 2: The three fractals in the piecewise-linear map (\ref{['eq.lambdaTent']}) have the same co-dimension $\alpha$. (a) Plot of the function $\phi(r,x_0;\Lambda(s))$ in Eq. (\ref{['eq.phi']}) -- with blue corresponding to $\phi=$track and yellow to $\phi=$tip -- as a function of different parameters (top to bottom): initial conditions $x=x_0$ (with $r=\lambda_+=0$, Fractal 1); range of parameter variation $\Delta \lambda = 2\lambda_+$ used in $\Lambda(s)$ (with $r\rightarrow \infty$, Fractal 2); and rate of change $r$ (with $\Delta \lambda = 4$, Fractal 3). (b) Estimation of the fractal co-dimension $\alpha$ of the track/tip boundary for the three fractals shown in panel (a). We use the uncertainty algorithm McDondald1985lai_transient_2011 to estimate $\alpha$ as the slope of the scaling between the fraction of $\varepsilon$-uncertain points $f(\varepsilon)$ and $\varepsilon$; see Appendix \ref{['app.fractal']} for details. Data for Fractal 1 is multiplied by 2 for clarity.
• Figure 3: Tracking and tipping trajectories in the non-autonomous piecewise-linear map defined by Eqs. (\ref{['eq.example1']})-\ref{['eq:tanh_drift']}. The three panels show the frozen paths of the stable -- $(x^*(\Lambda(s)),s)$ as solid line -- and unstable fixed points -- $(x^{u,1}(\Lambda(s)),s)$ and $(x^{u,3}(\Lambda(s)),s)$ as dashed and dotted black lines, respectively --, and the trajectory $x_n$ obtained starting at $x^*$ for $n\rightarrow -\infty$ and evolving using Eq. (\ref{['eq.map']}) with $F=F_1$ -- in Eq. (\ref{['eq.lambdaTent']}) -- and $\Lambda(s)$ -- in Eq. (\ref{['eq:tanh_drift']}) -- with parameters $\lambda_+=2$ and different values of the rate $r$ of parameter change: (a) small rate $r= 0.77 < r_{c1} \approx 0.7708$, leading to a tracking trajectory; (b) two similar rates $r_1=0.96243528753$ and $r_2=r_1+10^{-9}$ close to a critical rate $r_{c1}<r_c<r_{c2}$ (boundary point), leading to both tracking and a tipping trajectories; and (c) large rate $r=1.08 > r_{c2} \approx 1.0727$, leading to a tipping trajectory.
• Figure 4: The basin of attraction of the fixed point of the Hénon map can have smooth or fractal boundaries. (a) Parameter space $\lambda=(a,b)$ of the Hénon map \ref{['eq:henon']}. The gray shaded region corresponds to the region $\Gamma$ for which the fixed point (\ref{['eq.henonfixedpoint']}) is stable, defined in Eq. (\ref{['eq.Gamma']}). The two highlighted points indicate the values $\lambda_0=(a_0,b_0)$ and $\lambda_+=(a_+,b_+)$ used in our investigation (see legend). (b) The basin of attraction (in blue) of the stable fixed point (red symbol) for $\lambda=\lambda_0$ (left, smooth boundary) and $\lambda=\lambda_+$ (right, fractal boundary).
• Figure 5: Fractal boundary between tracking and tipping regions in the parameter space $\Gamma$ of the Hénon map. Initial conditions at the fixed point $(x_0, y_0)= (x^*_0,y^*_0)$ of $\lambda_0=(a_0,b_0)$ are iterated using Eq. (\ref{['eq:henon']}) with $\lambda=\lambda_+=(a_+,b_+)$. (a) Dependence of $\phi$ on $(a_+,b_+)$ for a fixed $(a_0,b_0)=(0.20,0.25)$. (b) Dependence of $\phi$ on $(a_0,b_0)$ for a fixed $(a_+,b_+)=(1.38,-0.54)$. The fixed values of $\lambda_0$ and $\lambda_+$ (red symbols, see legend) are the same used in Fig. \ref{['fig.henon1']}. Parameters $\lambda_0$ and $\lambda_+$ showing $\phi=$ tip will display R-tipping with at least one critical tipping rate $r_c$.