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Early warning signs for saddle-escape transitions in complex networks.

Kuehn C, Zschaler G, Gross T - Sci Rep (2015)

Bottom Line: We illustrate our results in two network models and epidemiological data.This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition.In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

View Article: PubMed Central - PubMed

Affiliation: Vienna University of Technology, 1040 Vienna, Austria.

ABSTRACT
Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

No MeSH data available.


Dynamics near a planar saddle with small noise.(a) Phase space (x1, x2) with a trajectory (black) passing near the saddle point. The stationary state (dark green dot) and a circle (gray) of radius r = 0.5 indicate a neighborhood of the stationary state outside of which the trajectory is shown as a dashed curve. (b) Time series for x1 (red) and x2 (blue). The gray vertical lines indicate entry and exit to the ball . The black squares are predicted values from the warning signals obtained inside B. (c) Plot of the logarithmic distance reduction d(T) as crosses; (see Supplementary Information, Section 2). The red/blue linear interpolants yield two approximations for the stable eigenvalue λs ≈ −1.10, −0.99 and the black lines for the important unstable eigenvalue λu ≈ 0.51, 0.57; the true values are (λs, λu) = (−1, 0.5). The black squares in (b) can be obtained from . Note that the choice of B is a choice of sliding window length (or lead time) for prediction as in the case for bifurcation-induced tipping.
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f1: Dynamics near a planar saddle with small noise.(a) Phase space (x1, x2) with a trajectory (black) passing near the saddle point. The stationary state (dark green dot) and a circle (gray) of radius r = 0.5 indicate a neighborhood of the stationary state outside of which the trajectory is shown as a dashed curve. (b) Time series for x1 (red) and x2 (blue). The gray vertical lines indicate entry and exit to the ball . The black squares are predicted values from the warning signals obtained inside B. (c) Plot of the logarithmic distance reduction d(T) as crosses; (see Supplementary Information, Section 2). The red/blue linear interpolants yield two approximations for the stable eigenvalue λs ≈ −1.10, −0.99 and the black lines for the important unstable eigenvalue λu ≈ 0.51, 0.57; the true values are (λs, λu) = (−1, 0.5). The black squares in (b) can be obtained from . Note that the choice of B is a choice of sliding window length (or lead time) for prediction as in the case for bifurcation-induced tipping.

Mentions: The example above differs fundamentally from bifurcation-induced transitions, because the qualitative change is not induced by a change of environmental parameters, but rather by a specific ‘rare’ perturbation. A system is susceptible to such perturbation-induced transitions if it resides in a saddle point, a state that is stable with respect to some perturbations, but unstable with respect to others. In the following we refer to critical transitions caused by the departure from saddle points far from bifurcations as saddle-escape transitions; see also (Fig. 1) for a mathematical normal form example for passage near a saddle; we remark that this example is generic in the sense that mathematical theory guarantees that other systems with nondegenerate saddles show the same dynamics up to coordinate changes and by using a suitable notion of equivalence for the dynamics4.


Early warning signs for saddle-escape transitions in complex networks.

Kuehn C, Zschaler G, Gross T - Sci Rep (2015)

Dynamics near a planar saddle with small noise.(a) Phase space (x1, x2) with a trajectory (black) passing near the saddle point. The stationary state (dark green dot) and a circle (gray) of radius r = 0.5 indicate a neighborhood of the stationary state outside of which the trajectory is shown as a dashed curve. (b) Time series for x1 (red) and x2 (blue). The gray vertical lines indicate entry and exit to the ball . The black squares are predicted values from the warning signals obtained inside B. (c) Plot of the logarithmic distance reduction d(T) as crosses; (see Supplementary Information, Section 2). The red/blue linear interpolants yield two approximations for the stable eigenvalue λs ≈ −1.10, −0.99 and the black lines for the important unstable eigenvalue λu ≈ 0.51, 0.57; the true values are (λs, λu) = (−1, 0.5). The black squares in (b) can be obtained from . Note that the choice of B is a choice of sliding window length (or lead time) for prediction as in the case for bifurcation-induced tipping.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4544003&req=5

f1: Dynamics near a planar saddle with small noise.(a) Phase space (x1, x2) with a trajectory (black) passing near the saddle point. The stationary state (dark green dot) and a circle (gray) of radius r = 0.5 indicate a neighborhood of the stationary state outside of which the trajectory is shown as a dashed curve. (b) Time series for x1 (red) and x2 (blue). The gray vertical lines indicate entry and exit to the ball . The black squares are predicted values from the warning signals obtained inside B. (c) Plot of the logarithmic distance reduction d(T) as crosses; (see Supplementary Information, Section 2). The red/blue linear interpolants yield two approximations for the stable eigenvalue λs ≈ −1.10, −0.99 and the black lines for the important unstable eigenvalue λu ≈ 0.51, 0.57; the true values are (λs, λu) = (−1, 0.5). The black squares in (b) can be obtained from . Note that the choice of B is a choice of sliding window length (or lead time) for prediction as in the case for bifurcation-induced tipping.
Mentions: The example above differs fundamentally from bifurcation-induced transitions, because the qualitative change is not induced by a change of environmental parameters, but rather by a specific ‘rare’ perturbation. A system is susceptible to such perturbation-induced transitions if it resides in a saddle point, a state that is stable with respect to some perturbations, but unstable with respect to others. In the following we refer to critical transitions caused by the departure from saddle points far from bifurcations as saddle-escape transitions; see also (Fig. 1) for a mathematical normal form example for passage near a saddle; we remark that this example is generic in the sense that mathematical theory guarantees that other systems with nondegenerate saddles show the same dynamics up to coordinate changes and by using a suitable notion of equivalence for the dynamics4.

Bottom Line: We illustrate our results in two network models and epidemiological data.This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition.In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

View Article: PubMed Central - PubMed

Affiliation: Vienna University of Technology, 1040 Vienna, Austria.

ABSTRACT
Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.

No MeSH data available.