Anticipating bifurcations of random dynamical systems through tails of stationary densities

TL;DR

This work tackles tipping-point detection in one-dimensional random difference equations with bounded noise by exploiting the tail behavior of the stationary density near the left boundary $x_-$. The authors derive a detailed asymptotic expansion of $\phi$ near $x_-$ via a transfer-operator framework and propose two practical tail-fitting estimators (leading-order and higher-order) to recover $\lambda=f_-'(x_-)$ from time-series data; $\lambda$ approaches 1 as a topological bifurcation is approached, providing an early-warning signal. They validate the methods on linear and nonlinear maps with both uniform and truncated normal noise, showing that the tail-based indicators can succeed where variance-based warnings fail, and that boundary knowledge or estimation significantly affects accuracy. The approach delivers a data-driven, model-agnostic pathway to anticipate bifurcations in bounded-noise systems and can be extended to more general noise structures and higher-dimensional settings.

Abstract

We develop an early-warning signal for bifurcations of one-dimensional random difference equations with additive bounded noise, based on the asymptotic behaviour of the stationary density near a boundary of its support. We demonstrate the practical use in numerical examples.

Figures (18)

• Figure 1: Illustration of the one-parameter family of maps $f_a$, cf. \ref{['eq:variance']} in Appendix \ref{['appen:variance_decrease']}, together with the upper and lower extremal maps $f_a(x) \pm 0.8$. In (a), the bifurcation diagram of $f_a(x)$ for $-0.5<a<1$ is shown as a dotted line, while those of the extremal maps are plotted as solid lines. The blue lines signify stable fixed points, the red lines represent unstable fixed points, and the black vertical lines are minimal invariant intervals for selected values of $a$. In (b), the extremal maps for parameter values $a = 0, \;0.175, \;0.7, \;1$ are portrayed in dotted and solid lines, respectively. There is a topological bifurcation of the minimal invariant interval around $a\approx 0.175$, represented by a fold bifurcation of the lower extremal map, that is when $f_a(x) - 0.8$ is tangential to the diagonal, portrayed in black in (b). The bifurcation can be observed in (a), where the minimal invariant interval changes discontinuously to a larger interval. After the bifurcation, random trajectories can alternate between the two metastable states inside the enlarged minimal invariant interval, in a so-called flickering phenomenon.
• Figure 2: Results of numerical simulations of the sample variance and our proposed new early warning signal $\hat{\lambda}$ (where $\hat{\lambda}$ is an estimator of the bifurcation parameter $\lambda$, which in turn approaches $1$ as $a\rightarrow a_*$). We simulate the random map $y_{t+1} = f_a(y_t) + \xi_t$, given in \ref{['eq:variance']} in Appendix \ref{['appen:variance_decrease']}, with noise amplitude $|\xi_t|\leq 0.8$ (as in Figure \ref{['fig:modified_tanh']}) while varying $a$. This system has a topological bifurcation at $a_*\approx 0.175$, where an attractor explodes when $a$ passes through $a_*$ from below. Starting from $a=0$, a time series of length $10^6$ is generated inside the attractor, after which $a$ is increased by $0.01$, and the time series is continued for another $10^6$ iterates, after which $a$ is increased again, and so on, until $a=0.8$. Ten such time series are generated and we present the measured variances and their mean at each considered fixed value of $a$, above, in black, showing a slowly decreasing trend until the system tips (escapes from the remnant of the attractor). In the same graph, we plot our new indicator $\lambda$, in blue. Boxes represent the interquartile range (25$^{\text{th}}$ to 75$^{\text{th}}$ percentile), with whiskers extending to all data points, except for outliers marked with crosses.
• Figure 3: Illustration of the optimisation scheme (\ref{['eq:quad_method']}) following Algorithm \ref{['alg:tailfit']} in Figure (a), for the linear map $f(x) = \lambda x$ with uniform noise in (\ref{['eq:linear_noise']}), and the graphs of the extremal maps $f_{\pm}(x)=\lambda x \pm (1-\lambda)\varepsilon$ for $\varepsilon = 0.1$ is shown in Figure (b). Here we take $n = 10^5$ iterations with $\lambda = 0.684$ and construct the normalised histogram using $b = 200$ bins, shown in blue in Figure (a) on a logarithmic scale. The least squares quadratic curve for $q = 0.3$ quantile of the histogram data is shown in black according to (\ref{['eq:quad_method']}). From the figure, it becomes apparent the necessity of discarding the nearest data points to $\hat{x}_-$.
• Figure 4: Illustration of the deterministic map $f_a(x) = 3 \tanh(x/2) - a$, the extremal maps $f_a(x) \pm \varepsilon$ for $\varepsilon = 0.1$ and their bifurcation diagrams. In (a), the graphs of the map $f$ for $a\in \{-0.4, 0, 0.4\}$ are plotted alongside the diagonal identity line. In (b), the bifurcation diagram for $f_a(x)$ across parameters $-0.5\leq a\leq 1$ is shown as the dotted line while those of the extremal maps are plotted as solid lines. The blue lines signify stable fixed points while red lines represent unstable fixed points. The black vertical lines are examples of minimal invariant intervals of (\ref{['eq:non-linear_eq']}) for different $a$ values.
• Figure 5: Numerical approximations of $\lambda$ from the time series of the linear maps with uniform noise (\ref{['eq:linear_noise']}) for $0.1 \leq \lambda \leq 0.9$, using methods (\ref{['eq:quad_method']}) in blue and (\ref{['eq:quad_method_strict']}) in black, respectively, assuming the left boundary $x_-$ is known. The approximations are obtained following Algorithm \ref{['alg:tailfit']} with $n= 10^5, b=200$, and $q = 0.3$. For each parameter value $\lambda$, the estimator $\hat{\lambda}_{x_-}$ (cf. \ref{['eq:estimator']}) is computed $100$ times, using independent time series generated with different noise realisations. Figure (a) shows the box plots of the estimators $\hat{\lambda}_{x_-}$, while the corresponding errors are plotted in (b). The boxes represent the interquartile range (25$^{\text{th}}$ to 75$^{\text{th}}$ percentile), while the whiskers extend to all non-outlier data points, and the crosses mark the outliers. Sample means are indicated by solid lines.

Theorems & Definitions (9)

• Lemma 2.1
• Theorem 2.2
• Proposition B.1
• proof
• Proposition B.2
• proof
• proof : Proof of Lemma \ref{['LEMMA:main']}
• proof : Proof of Theorem \ref{['THM:main']}
• Remark D.1