Predicting the onset of period-doubling bifurcations via dominant eigenvalue extracted from autocorrelation

Abstract

Predicting the occurrence of transitions in the qualitative dynamics of many natural systems is crucial, yet it remains a challenging task. Generic early warning signals like variance and lag-1 autocorrelation identify critical slowing down near tipping points but lack practical thresholds for predicting imminent transitions. More recent studies found that the dynamical eigenvalue is rooted in the framework of empirical dynamical modeling and then estimates the dominant eigenvalue of a system from time series, providing a threshold ($|$DEV$|$ = 1) to predict bifurcations and classify their types. However, its application requires careful calibration of the hyperparameters and focuses on reconstructing system dynamics directly from data. Here, we employ Ornstein-Uhlenbeck process to derive analytic approximations for the lag-$τ$ autocorrelation function prior to period-doubling bifurcation thereby estimating the dominant eigenvalue of dynamical systems, named dominant eigenvalue extracted from autocorrelation (DE-AC), and revealing its dynamic behaviour when approaching a period-doubling bifurcation. Theoretically, dominant eigenvalue tends to $-1$ when the system approaches a period-doubling bifurcation. In particular, we evaluated DE-AC on simulation data from cardiac alternans model and on experimental data from chick heart aggregates undergoing a period-doubling bifurcation. DE-AC reliably detected the beginning of the cardiac arrhythmia (period-doubling bifurcation) in most cases. Moreover, it demonstrated superior sensitivity and specificity as an early warning signal compared to the three widely used indicators -- variance, lag-1 autocorrelation, and dynamical eigenvalue. Our theoretical and empirical results suggest that DE-AC represents a quantitative measure for predicting the onset of potentially dangerous alternating rhythms in the heart.

Figures (6)

• Figure 1: Analytical approximation for early warning signals preceding the period-doubling bifurcation. (a) Schematic illustration of the eigenvalues in complex plane at different proximities points for the period-doubling bifurcation. The bifurcation is mathematically identified by changes in the dominant eigenvalue, $\lambda$, of the Jacobian matrix, $J$, at the tipping point. (b) Analytical approximation of the lag-$\tau$ autocorrelation function, $ACF(\tau)$, as the system approaches a period-doubling bifurcation. Analytical approximations for the $ACF(\tau)$ are drawn at different distances to period-doubling bifurcation given by $\lambda = \{-0.125, -0.5, -0.75\}$, where $\lambda$ is the real part of dominant eigenvalue of the system.
• Figure 2: Trends in indicators preceding the period-doubling bifurcations in three theoretical models. (a-c) The state trajectory (blue) and its smoothed trace (black) of a simulation undergoing a period-doubling in the Fox, Ricker, and Hénon map models. The arrow indicates the rolling window (50% of the time series) applied to calculate EWS. (d-f) Variance (y axis) as a function of simulation time (x axis). (g-i) Lag-1 autocorrelation as a function of simulation time. (j-l) Dynamical eigenvalue ($|$DEV$|$) grziwotz2023anticipating as a function of simulation time. (m-o) Dominant eigenvalue extracted from autocorrelation (DE-AC) as a function of simulation time. Measure the trend of the EWS with Kendall tau. The positive values indicate increasing trends; Negative values indicate decreasing trends. A maximal (minimal) Kendall tau value of $1$ ($-1$) indicates that every subsequent point takes a larger (smaller) value. Kendall tau values show consistent trend in dominant eigenvalue (Kendall tau $= -1.0$). The vertical dashed line indicates the tipping point (i.e., onset of period-doubling bifurcations), and the gray dashed area delineates the regime following the critical transition.
• Figure 3: The lag-$\tau$ autocorrelation function (y axis) for calculated (symbols) and fitted (lines) on the Fox model, as a function of the lag time (x axis). Each colored line represents a different time and this "Time" corresponds to those in Fig. \ref{['fig2']}. An increasing time indicates approaching the tipping point (i.e., onset of alternating cardiac rhythms). The closer coefficient of determination ($R^2$) is close to $1.0$, which means a higher fitting degree for the indicator. The solid lines match fitting curves with the form $ACF(\tau)= \lambda^{|\tau|}$, where the dominant eigenvalue $\lambda$ is the free fitting parameters during the fitting process.
• Figure 4: Trends in indicators preceding alternating cardiac rhythms for chick heart aggregates (IDs 1-12). (a--i) A period-doubling bifurcation obtained from chick heart data is shown in each panel. (Top) Inter-beat interval (IBI) trajectory (gray) and its smoothed trace (black) from the experimental data. (2nd down, blue) Variance computed using a half-length rolling window from the pre-transition data record. (3rd down, orange) Lag-1 autocorrelation computed using a half-length rolling window from the pre-transition data record. (4rd down, green) Dynamical eigenvalue ($|$DEV$|$) computed using a half-length rolling window from the pre-transition data record. (Bottom, purple) Dominant eigenvalue extracted from autocorrelation (DE-AC) computed using a half-length rolling window from the pre-transition data record. Measure the trend of the EWS with Kendall tau. The beginning of alternating heart rhythms, or onset of period-doubling bifurcation, is marked by the vertical dashed line.
• Figure 5: Trends in indicators preceding alternating cardiac rhythms for chick heart aggregates (IDs 13-23). (a--k) Each panel shows a period-doubling bifurcation obtained from chick heart data. (Top) Inter-beat interval (IBI) trajectory (gray) and its smoothed trace (black) from the experimental data. (2nd down, blue) Variance computed using a half-length rolling window from the pre-transition data record. (3rd down, orange) Lag-1 autocorrelation computed using a half-length rolling window from the pre-transition data record. (4rd down, green) Dynamical eigenvalue ($|$DEV$|$) computed using a half-length rolling window from the pre-transition data record. (Bottom, purple) Dominant eigenvalue extracted from autocorrelation (DE-AC) computed using a half-length rolling window from the pre-transition data record. Measure the trend of the EWS with Kendall tau. The beginning of alternating heart rhythms, or onset of period-doubling bifurcation, is marked by the vertical dashed line.