Indicating the synchronization bifurcation points using the early warning signals in two case studies: Continuous and explosive synchronization

Many dynamical systems, including lakes, organisms, ocean circulation patterns, or financial markets, are now thought to have tipping points where critical transitions to a contrasting state can happen. Because critical transitions can occur unexpectedly and are difficult to manage, there is a need for methods that can be used to identify when a critical transition is approaching. Recent theory shows that we can identify the proximity of a system to a critical transition using a variety of so-called ‘early warning signals’, and successful empirical examples suggest a potential for practical applicability. However, while the range of proposed methods for predicting critical transitions is rapidly expanding, opinions on their practical use differ widely, and there is no comparative study that tests the limitations of the different methods to identify approaching critical transitions using time-series data. Here, we summarize a range of currently available early warning methods and apply them to two simulated time series that are typical of systems undergoing a critical transition. In addition to a methodological guide, our work offers a practical toolbox that may be used in a wide range of fields to help detect early warning signals of critical transitions in time series data.

There is growing interest in the use of critical slowing down and critical fluctuations as early warning signals for critical transitions in different complex systems. However, while some studies found them effective, others found the opposite. In this paper, we investigated why this might be so, by testing three commonly used indicators: lag-1 autocorrelation, variance, and low-frequency power spectrum at anticipating critical transitions in the very-high-frequency time series data of the Australian Dollar-Japanese Yen and Swiss Franc-Japanese Yen exchange rates. Besides testing rising trends in these indicators at a strict level of confidence using the Kendall-tau test, we also required statistically significant early warning signals to be concurrent in the three indicators, which must rise to appreciable values. We then found for our data set the optimum parameters for discovering critical transitions, and showed that the set of critical transitions found is generally insensitive to variations in the parameters. Suspecting that negative results in the literature are the results of low data frequencies, we created time series with time intervals over three orders of magnitude from the raw data, and tested them for early warning signals. Early warning signals can be reliably found only if the time interval of the data is shorter than the time scale of critical transitions in our complex system of interest. Finally, we compared the set of time windows with statistically significant early warning signals with the set of time windows followed by large movements, to conclude that the early warning signals indeed provide reliable information on impending critical transitions. This reliability becomes more compelling statistically the more events we test.

Loss of resilience in population numbers in response to environmental perturbations may be predicted with statistical metrics called early warning signals (EWS) that are derived from abundance time series. These signals, however, have been shown to have limited success, leading to the development of trait‐based EWS that are based on information collected from phenotypic traits such as body size. Experimental work assessing the efficacy of EWS under varying ecological and environmental factors are rare. In addition, disentangling how such warning signals are affected under varying ecological and environmental factors is key to their application in biological conservation. Here, we experimentally test how different rates of environmental forcing (i.e. warming) and varying ecological factors (i.e. habitat quality and phenotypic diversity) affected population stability and predictive power of early warning signals of population collapse. We analyzed population density and body size time series data from three phenotypically different populations of a protozoan ciliate Askenasia volvox in two levels of habitat quality subjected to three different treatments of warming (i.e. no warming, fast warming and slow warming). We then evaluated how well abundance‐ and trait‐based EWS predicted population collapses under different levels of phenotypic diversity, habitat quality and warming treatments. Our results suggest that habitat quality and warming treatments had more profound effects than phenotypic diversity had on both population stability and on the performance of abundance‐based signals of population collapse. In addition, trait‐based EWS generally performed well, were reliable and more robust in forecasting population collapse than abundance‐based EWS, regardless of variation in environmental and ecological factors. Our study points towards the development of a predictive framework that includes information from phenotypic traits such as body size as an indicator of loss of resilience of ecological systems in response to environmental perturbations.

Towards improving detection of early warning signals within organizations : an approach to the identification and utilization of underlying factors from an organizational perspective

Critical transitions occur when a system undergoes a sudden shift from one state to another. Early warning signals (EWS) are indicators that can be used to potentially anticipate critical transitions in such systems, which may be temporal or spatio-temporal. Temporal systems are those whose state varies over time, whereas spatio-temporal systems also vary over a spatial domain. While temporal EWS can be applied to spatio-temporal systems by averaging over the spatial domain, spatially-informed EWS should, in principle, be able to outperform their temporal counterparts by making use of the additional spatial information. We seek to understand how EWS for spatial systems compare to those used for temporal systems. To facilitate comparison, we explore how EWS performance is measured. We use the strength of EWS trends, quantified using Kendall’s τ, as a proxy for performance. Other factors, such as robustness to choices of parameters used for detrending, statistical significance, and agreement with expected EWS behaviour, are considered. This assessment of EWS based on these factors enables an informed comparison and decision regarding which signals to apply to different systems for potential indications of critical transitions. We find that while spatially-informed EWS generally offer improved performance over temporal EWS for the example systems studied, we find the choice is system specific.

Early warning signals (EWSs) have the potential to predict tipping points where catastrophic changes occur in ecological systems. However, EWSs are plagued by false negatives, leading to undetected catastrophes. One reason may be because EWSs do not occur equally for all species in a system, so whether and how strongly EWSs are detected depends on which species is being observed. Here, we illustrate how the strength of EWSs is determined by each species' relationship to properties of the noise, the system's response to that noise, and the occurrence of critical slowing down (the dynamical phenomenon that gives rise to EWSs). Using these relationships, we present general rules for maximizing EWS detection in ecological communities. We find that for two-species competitive and mutualistic systems, one should generally monitor the species experiencing smaller intraspecific effects to maximize EWS performance, while in consumer-resource systems, one should monitor the species imposing the smaller interspecific effects. These guidelines appear to hold for at least some larger communities as well. We close by extending the theoretical basis for our rules to systems with any number of species and more complex forms of noise. Our findings provide important guidance on how to monitor systems for EWSs to maximize detection of tipping points.

Many animals disperse to find their own territory, mates to reproduce or suitable environments to live. Dispersal can be described as a three-phase process consisting of two stationary phases (S1 and S2) at the beginning and the end of a dispersal event. These stationary phases are temporally separated by a transient phase (T), where the animal moves from S1 to a new area S2 in space. The net squared displacement (NSD) is a frequently used metric to identify these phases from animal tracking data. We tested whether early warning signals (EWSs) on time series of the NSD, can be used to predict dispersal events. To identify EWSs we conducted a rolling window approach and evaluated the dispersal events by performing a spatial cluster analysis with the mechanistic range shift analysis (MRSA). We used data from 22 GPS-collared red foxes (Vulpes vulpes) as an example of a mammal species in which the juvenile (sub-) adult transition usually involves dispersal. Applying EWSs resulted in the identification of both transitions from S1 to T and from T to S2. For 10 individuals we detected EWSs. For 8 out of these 10 individuals (80%) we identified a spatial shift between S1 and S2 via a MRSA. Accordingly, for 8 out of 22 individuals (36%) we observed a transient phase (T) which led to a major and persistent transformation of red fox locations. Even though the identification of dispersal events based on movement data is challenging using well known techniques such as state space models or the MRSA, our results suggested that EWS in combination with MRSA is appropriate to detect and identify dispersal events in radio-collared mammals. Thus, in the context of identifying dispersal events using EWSs we recommend to evaluate the existence of stationary and transient phases using the MSRA. The benefit of using EWSs is the calculation of the NSD and simple statistics (standard deviation, autocorrelation) and no requirement of high resolution tracking data. Additionally, transitions to the stationary or transient phase might be detected where home range calculations are not possible.

Early Warning Signals (EWS) are indicators that can be used to anticipate tipping points i.e. abrupt changes in dynamical systems. Detecting EWS is a crucial part of climate science, especially in the context of climate change. Several methods are used to identify tipping points using time series of climate state variables (e.g. temperature, precipitation, etc), but few consider spatial correlations [1]. Spatial detection could identify the starting location of a transition process from a state to another and be directly applied to satellite observations. We consider different state variables on a numerical grid as a complex network, where grid points displaying correlation are connected and the temporal evolution of this network is studied. We seek for network properties that can be used as EWS when approaching the state transition. The network is generated and analysed using the pyUnicorn package [2], and compared to classical statistical methods. The networks are constructed using two methods: Pearson correlation coefficient and mutual information, allowing us to compare a linear and a causal approach. Multiple network indicators such as the degree of correlation, the average path length, and the area weighted connectivity are compared. To test the method robustness, we look at the network dependencies in terms of the time window, the interval over which the forcing is changed, and the effect of reducing the extent of the network (limited, for example, over polar or equatorial regions). These indicators show tipping points at the global scale, as simulated in a coupled-aquaplanet configuration with the MIT general circulation model, using as forcing parameter the atmospheric CO2 content or the input of solar energy [3] . The application of such indicators as EWS is discussed. [1] van der Mheen et al. Geophysical Research Letters 40, 11  (2013)[2] Donges et al. Chaos 25, 113101 (2015)[3] Brunetti \& Ragon, Physical Review E 107, 054214 (2023)

Objective To analyze the effectiveness and response status of China Infectious Disease Automated Alert and Response System (CIDARS) in Hunan province from 2012 to 2016 for improving the system. Methods To collect the early warning signals, the number of suspected events, the result of on-site investigation, the signal response time and the result of public health emergencies, and the χ2 test, correlation analysis and non-parametric test were used to analyze the information on CIDARS in Hunan Province during the period from 2012 to 2016. Results A total of 108 188 signals were generated by the CIDARS in Hunan Province; The warning involved 30 kinds of infectious diseases and 138 counties (districts), and each county (district) received 3.00 weekly warning messages on average; 100% early warning signal was responded, 2 h response rate was 92.43%; The median response time (P25-P75) was 0.28 (0.11-0.77) h in the single case warning, and the five-year timely response rate showed an upward trend year by year (trend χ2=58.89, P 0.05); the time series warning response was more timely than the single case warning, and the difference between the two timely response rates was statistically significant(χ2=5156.60, P 0.05); the system had the best effect in rubella, and the positive likelihood ratio of rubella is 4.36. Conclusions CIDARS in Hunan province runs well overall with high sensitivity and timely response to the early warning signals, but the specificity of the early warning system needs to be improved. The quality of the epidemic report of infectious diseases needs to be further improved, and the early warning parameters and warning method should be adjusted according to the characteristics of different diseases and the differences among different regions. Key words: Early-warning system; Infectious disease; Early-warning response

Synthesis The quickly expanding literature on early warning signals for critical transitions in ecosystems suggests that critical slowing down is a key phenomenon to measure the distance to a tipping point in ecosystems. Such work is broadly misinterpreted as showing that slowing down is specific to tipping points. In this contribution, we show why this is not the case. Early warning signals based on critical slowing down indicate a broader class of situations where a system becomes increasingly sensitive to perturbations. Ecosystem responses to external changes can surprise us by their abruptness and irreversibility. Models have helped identifying indicators of impending catastrophic shifts, referred to as ‘generic early warning signals’. These indicators are linked to a phenomenon known as ‘critical slowing down’ which describes the fact that the recovery rate of a system after a perturbation decreases when the system approaches a bifurcation – such as the classical fold bifurcation associated to catastrophic shifts. However, contrary to what has sometimes been suggested in the literature, a decrease in recovery rate cannot be considered as specific to approaching catastrophic shifts. Here, we analyze the behavior of early warning signals based on critical slowing down in systems approaching a range of catastrophic and non‐catastrophic situations. Our results show that slowing down generally happens in situations where a system is becoming increasingly sensitive to external perturbations, independently of whether the impeding change is catastrophic or not. These results highlight that indicators specific to catastrophic shifts are still lacking. More importantly, they also imply that in systems where we have no reason to expect catastrophic transitions, slowing down may still be used in a more general sense as a warning signal for a potential decrease in stability.

Successfully anticipating sudden major changes in complex systems is a practical concern. Such complex systems often form a heterogeneous network, which may show multi-stage transitions in which some nodes experience a regime shift earlier than others as an environment gradually changes. Here we investigate early warning signals for networked systems undergoing a multi-stage transition. We found that knowledge of both the ongoing multi-stage transition and network structure enables us to calculate effective early warning signals for multi-stage transitions. Furthermore, we found that small subsets of nodes could anticipate transitions as well as or even better than using all the nodes. Even if we fix the network and dynamical system, no single best subset of nodes provides good early warning signals, and a good choice of sentinel nodes depends on the tipping direction and the current stage of the dynamics within a multi-stage transition, which we systematically characterize.

In recent years, various early warning signals of critical transition have been presented, such as autocorrelation at lag 1 [AR(1)], variance, the propagator based on detrended fluctuation analysis (DFA-propagator), and so on. Many studies have shown that the climate system has the characteristics of long-term memory (LTM). Will the LTM characteristics of the climate system change as it approaches possible critical transition points? In view of this, the present paper first studies whether the LTM of several folding (folded bifurcation) models changes consistently as they approach their critical points slowly by the rescaled range (R/S) analysis. The results of numerical experiments show that when the control parameters of the folding model are close to its critical threshold, the Hurst exponent H exhibits an almost monotonic increase (significance level α = 0.05). We compare the performance of R/S with the existing indicators, including AR(1), variance, and DFA-propagator, and find that R/S is a perfectly valid alternative. When there is no extra false noise, AR(1) and variance have good early warning effects. After the addition of extra Gaussian white noise of different intensities, the values of AR(1) and variance change significantly. As a result, the DFA-propagator based on AR(1) calibration also changed significantly. Compared with the other three indicators, the early warning effect of H has stronger ability to resist the interference of external false signals. To further verify the validity of increasing H, paleoclimate reconstruction of Cariaco Basin sediment core grayscale record with long trends filtered out is studied by R/S analysis. The other three early warning signals are calculated in the same way. The data contain a well-known abrupt climate change: the transition between the Younger Dryas (YD) and the Holocene. We find that approximately 300 years before this abrupt climate change occurred, before 11.7 kyr BP, the LTM exponents for Cariaco Basin deglacial grayscale data present an obvious increasing trend at a significant level of α = 0.05. Meanwhile, the variation trend of H and DFA-propagator is basically similar. This shows that increasing H by R/S analysis is an effective early warning signal, which indicates that a dynamic system is approaching its possible critical transition points; H is a completely valid alternative signal for AR(1) and DFA-propagator. The main conclusion of this paper is based on numerical experiments. The precise relationship between H and the stability of the underlying state approaching the transition needs to be further studied. Significance Statement Dynamic systems have critical transition points, and these systems will suddenly change from a stable state to another alternative one beyond these points. Using several simple theoretical models and paleoclimate data, we study whether the characteristics of long-term memory, which are ubiquitous in complex systems in nature and society, change as a system approaches its critical transition point. The results show that the long-term memory of a dynamic system increases significantly with the approach of the critical point, whether in theoretical models or in paleoclimate data.

Early warning of critical transitions in crude oil price

Early-warning signals (EWS) have been successfully employed to predict transitions in research fields such as biology, ecology, and psychiatry. The predictive properties of EWS might aid in foreseeing transitions in mood episodes (i.e. recurrent episodes of mania and depression) in bipolar disorder (BD) patients. We analyzed actigraphy data assessed during normal daily life to investigate the feasibility of using EWS to predict mood transitions in bipolar patients. Actigraphy data of 15 patients diagnosed with BD Type I collected continuously for 180 days were used. Our final sample included eight patients that experienced a mood episode, three manic episodes and five depressed episodes. Actigraphy data derived generic EWS (variance and kurtosis) and context-driven EWS (autocorrelation at lag-720) were used to determine if these were associated to upcoming bipolar episodes. Spectral analysis was used to predict changes in the periodicity of the sleep/wake cycle. The study procedures were pre-registered. Results indicated that in seven out of eight patients at least one of the EWS did show a significant change-up till four weeks before episode onset. For the generic EWS the direction of change was always in the expected direction, whereas for the context-driven EWS the observed effect was often in the direction opposite of what was expected. The actigraphy data derived EWS and spectral analysis showed promise for the prediction of upcoming transitions in mood episodes in bipolar patients. Further studies into false positive rates are suggested to improve effectiveness for EWS to identify upcoming bipolar episode onsets.

Using natural archives to detect climate and environmental tipping points in the Earth System