Constraining safe and unsafe overshoots in saddle-node bifurcations

Abstract

We consider a dynamical system undergoing a saddle-node bifurcation with an explicitly time dependent parameter~$p(t)$. The combined dynamics can be considered as a dynamical systems where $p$ is a slowly evolving parameter. Here, we investigate settings where the parameter features an overshoot. It crosses the bifurcation threshold for some finite duration $t_e$ and up to an amplitude $R$, before it returns to its initial value. We denote the overshoot as safe when the dynamical system returns to its initial state. Otherwise, one encounters runaway trajectories (tipping), and the overshoot is unsafe. For shallow overshoots (small $R$) safe and unsafe overshoots are discriminated by an inverse square-root border, $t_e \propto R^{-1/2}$, as reported in earlier literature. However, for larger overshoots we here establish a crossover to another power law with an exponent that depends on the asymptotics of $p(t)$. For overshoots with a finite support we find that $t_e \propto R^{-1}$, and we provide examples for overshoots with exponents in the range $[-1, -1/2]$. All results are substantiated by numerical simulations, and it is discussed how the analytic and numeric results pave the way towards improved risks assessments separating safe from unsafe overshoots in climate, ecology and nonlinear dynamics.

Figures (7)

• Figure 1: The border dividing the parameter space spanned by the dimensionless overshoot amplitude $H$ and overshoot duration $T$ (both introduced in Eq. (\ref{['eq:HT']})) into a safe region (no tipping) and an unsafe region (tipping). The different line refer to different shapes of the overshoot with the same amplitude and duration. The uppermost, dotted black line corresponds to the results from literatureRitchie and follows a power law. Numerical results from the present article for an overshoot following a Gaussian shape are represented by the red solid line, and those for distributions decaying to their asymptotic values with power laws with exponents $2$ (Cauchy distribution), $3$ and $4$ by blue, orange and green solid lines respectively. The solid gray line provides the boundary for a discontinuous, piecewise constant dependence of $r(t)$.
• Figure 2: The different curves represent solutions for $x(t)$ during the overshoot ($|t|< t_e/2$) and afterwards ($t>t_e/2$), as outlined in Eqs. (\ref{['eq:solutiontan']}) and \ref{['eq:threecases']}, respectively. The stable fixed point $x=-x_c$ serves as a common initial condition before the overshoot starts at $t=-t_e/2$. The blue curves represent the divergent solutions with $x(t_e/2) > x_c$. The green curves represent the solutions with $x(t_e/2) < x_c$ that converge back to the stable fixed point at $-x_c$. The thick red line corresponds to $x(t_e/2) = +x_c$, the constant solution at the unstable fixed point. (a) Trajectories for $H=0.2$ with green lines for $T = 2, \; 4, \; 5, \; 5.14, \; 5.144$, blue lines for $T = 5.14415, \; 5.15, \; 5.5, \; 8$, and the red line $T_c = (2 /\sqrt{0.2}) \; \text{atan}( 1/\sqrt{0.2} ) \simeq 5.144128$. (b) Trajectories for $H=5$ with green lines for $T = 0.1, \; 0.2, \; 0.3, \; 0.37, \; 0.376$, blue lines for $T = 0.3762, \; 0.377, \; 0.38, \; 0.4$, and the red line $T_c = (2 /\sqrt{5}) \; \text{atan}( 1/\sqrt{5} ) \simeq 0.3761373$. The thickness of the lines decreases with increasing distance of $T$ from the $T_c$.
• Figure 3: Panels (a) and (b) show different triangular-shaped, i.e. piecewise-linear profiles $r(t)$, where $\alpha$ denotes the position of their respective maxima. They are all bounded from above (panel a) by a profile that takes the value $R$ on an interval of duration $(1+d^2)\,t_e$ and $-R_d$ outside the interval. They are all bounded from below (panel b) by a profile that takes the value $R_U$ on an interval of duration $(1-R_U/R) \, t_e$ and $-R_d$ outside the interval. The profiles in panels (a) and (b) are shifted by $t_L(\alpha)$ and $t_U(\alpha)$, respectively, in order to emphasize that $\alpha$ only induces a trivial time shift of the piecewise-constant bounds. According to Eqs. (\ref{['eq:bound-triangle']}) these bounds provide the boundaries indicated by the green and the blue line in panel (c). The numerical values for the position of the stability boundary are marked by lines with colors matching those in the other two panels. The dotted green line indicates that for sufficiently large $d$ one can improve the lower bound by considering overshoots that do not reach down to $-R_d$, as in panel (a), but rather take a larger value $-R_d < -R_L < 0$. Full details are provided in the main text.
• Figure 4: Notations adopted to derive optimal bounds for general unimodal overshoots.
• Figure 5: The stability border for Gaussian overshoots, as defined in Eq. (\ref{['eq:Gaussian-overshoots']}). The thick red line shows numerical data for the boundary. The other lines show upper and lower bounds for large and small values of $d$, respectively, as indicated in the legend.