Detecting and Predicting Tipping Points

Abstract

Tipping points are sudden, and sometimes irreversible and catastrophic, changes in a system’s dynamical regime. Complex networks are now widely used in the analysis of time series from a complex system. In this paper, we investigate the scope of network methods to indicate tipping points. In particular, we verify that the permutation entropy of transition networks constructed from time series observations of the logistic map can distinguish periodic and chaotic regimes and indicate bifurcations. The permutation entropy of transition networks, the mean edge betweenness of visibility graphs and the number of code words in compression networks, are each shown to indicate the onset of transition of a pitchfork bifurcation system. Our study shows that network methods are effective in detecting transitions. Network-based forecasts can be applied to models of real systems, as we illustrate by considering a lake eutrophication model.

Permutation Entropy as a Universal Disorder Criterion: How Disorders at Different Scale Levels Are Manifestations of the Same Underlying Principle

“…When stress levels increase further, a tipping point is reached at which central coherence by hub structures is suddenly lost, causing a steep rise in permutation entropy scores. These conclusions are in line with experimental data that show how changes in network topology may contribute to the formation of tipping points [ 10 ]. Our model seems to explain several generic changes in internal message passing of living systems under rising levels of stress.…”

“…There may be several factors that contribute to CSD, but a generic mechanism that underlies CSD at multiple scale levels so far remains elusive. Critical slowing down may be due to a gradual increase in the number and strength of recurrent connections between system components (e.g., computers, genes, neurons, or people) [ 10 ]. Such components continuously enforce each other’s activity, for which reason it will take longer for the system to quiet down after initial perturbation (‘hysteresis’ or slowing down: this would explain the increase in autocorrelations).…”

“…Examples of such states in nonliving systems are melting or boiling points, where, e.g., ice represents a highly ordered state with strong connections between water molecules and only a small increase in temperature is sufficient to cause a cascading failure of hydrogen bonds (i.e., melting), allowing all water molecules to move around more freely as water. The exact origins of tipping points are still unknown, but network topology appears to be an important factor [ 10 , 130 ]. In systems of coupled oscillators, the flow of energy may arrange system components in such a way that it will arrive at a point where only a few central nodes are responsible for connecting all of the system’s nodes into one giant ‘connected component’ [ 11 ].…”

Pattern Breaking: A Complex Systems Approach to Psychedelic Medicine

“…In non-resilient complex systems, tipping in one subsystem can result in tipping cascades and turbulence in the whole system (Carreras et al 2002;Krönke et al 2020;Klose et al 2021;Sharpe and Lenton, 2021). Interestingly, tipping and cascading may not necessarily have detrimental effects, e.g., if these phenomena can be planned for and harnessed for functional purposes, such as large scale information transfer or functionally useful global system transformations (Brouwer and Carhart-Harris, 2021) The prediction of tipping can be useful for preventing, preparing for, or controlling their occurrence or consequences (Peng et al 2019). Tipping can be induced in complex systems by intentional perturbations such as 'temperature' increases, causing bifurcations, fluctuations or noise near critical states and rate-dependent variations of drift in control parameters of the dynamics (Ambika and Kurths, 2021).…”

System identification near a hopf bifurcation via the noise-induced dynamics in the fixed-point regime

“…Close to a bifurcation, changes in the dynamics of a system can be detected in its network properties. For example, Peng et al (2019) showed that the network propertiessuch as the entropy of transition networks and the mean edge betweenness of visibility graphs-decrease as the system approaches a pitchfork bifurcation. By analyzing a mathematical model of lake eutrophication, these researchers showed that these network-based precursors can forecast real-world bifurcations.…”