Statistical warning indicators for abrupt transitions in dynamical systems with slow periodic forcing

Abstract

There is growing interest in anticipating critical transitions in natural systems, often pursued through statistical detection of early warning signals associated with dynamical bifurcations. In stochastic dynamical systems, such signals commonly rely on manifestations of critical slowing down. However, we still need additional development for the underlying theory for critical transitions in non-autonomous systems. This extension is relevant for natural systems, whose behaviour often emerges from seasonal periodic forcing. In this study, we systematically investigate the feasibility of anticipating the termination of oscillatory behavior in a bistable system with slow periodic forcing. In this setting, existing approaches of estimating linear characteristics of the return map fail in practical scenarios due to the unfavourable time-scale separation. Instead, we propose two statistical indicators for the anticipation of critical transitions in the periodic behaviour: (i) conventional early warning indicators, such as increasing variance and autocorrelation, evaluated across system cycles, and (ii) indicators derived from the phase of the seasonal forcing. By statistically comparing their predictive performance, we find that phase-based indicators provide the strongest early warning capability. Our results offer guidance for the detection of critical transitions in periodically forced systems and, more broadly, systematically extend early-warning signs towards non-autonomous dynamical systems.

Figures (6)

• Figure 1: Example realization of the slowly forced Duffing oscillator showing a breakdown of the relaxation oscillation due to a decreasing forcing amplitude $D_a$. The trajectory alternates between wells once per half forcing cycle until a late-cycle transition fails; afterward, the dynamics remain confined to a single-well response with small seasonal modulation. The vertical dashed line marks the breakdown onset (first missed transition).
• Figure 2: Critical-manifold geometry on the phase cylinder for the fast--slow formulation $\omega\,dx/ds=f(x,s)$ with $f(x,s)=x-\tfrac{1}{3}x^3+D_a\cos s$. (a) For $D_a>2/3$ the critical manifold has folds and exhibits relaxation oscillations via slow drift along attracting sheets connected by fast jumps; markers indicate representative jump and drop locations. (b) For $D_a<2/3$ the folds vanish and only small-amplitude, single-well seasonal responses persist. Light arrows indicate the reduced flow direction.
• Figure 3: Illustration of cycle-averaged fluctuation indicators under piecewise-constant forcing amplitudes. (a) Example trajectory $x(t)$ for successive amplitude levels $D_a\in\{1.00,0.90,0.80,0.72\}$. (b) Per-cycle variance $\mathrm{Var}_n$ computed from the concatenated detrended residuals of the two between-jump segments in each cycle. (c) Corresponding per-cycle lag-1 autocorrelation $\mathrm{AC1}_n$ computed from the same concatenated residual series. Vertical dotted lines mark amplitude changes; thick black bars denote the mean of the estimated cycle-wise quantities across the time stretch of constant amplitude.
• Figure 4: Illustration of phase-based indicators under the same piecewise-constant amplitude protocol as in Figure \ref{['fig:var_acf_multilevel']}. (a) Example trajectory $x(t)$. (b) Per-jump phase $\Delta_j$ (signed phase difference to the nearest forcing extremum) together with within-level circular mean (black) and dispersion (shaded band). As $D_a$ decreases toward the fold threshold, the jump phase drifts toward the associated extremum and the jump-phase spread increases.
• Figure 5: Indicator slopes grouped by whether the respective trajectory exhibited a breakdown, shown for a subsample of 150 runs. The indicators are (a) $\text{slope}_\mathrm{Var}$, (b) $\text{slope}_{\mathrm{AC1}}$, (c) $\text{slope}_{\mathrm{mean\ phase}}$, (d) $\text{slope}_{\mathrm{phase\ std}}$. The mean-phase slope provides the clearest class separation, followed by $\mathrm{AC1}$; variance and phase dispersion exhibit stronger overlap.