Topological bifurcation for the double cusp polynomial

Godwin, A. N.

doi:10.1017/s0305004100051112

Cited by 7 publications

(4 citation statements)

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“…The case of System (1) with bidirectional, symmetric coupling C X (y, x) = σy, C Y (x, y) = σx is a special case of the double cusp catastrophe, which is given by the potential This singularity has very rich structure [40,106,21]. R. Abraham et al [1] numerically studied a system similar to the double catastrophe, namely x = −x 3 + bx+āy, y = −y 3 + dy + cx, which is System (1) but with no constant terms (i.e., a = b = 0) and with coupling functions C X (y, x) = āy, C Y (x, y) = cx providing the only terms independent of x and independent of y, respectively.…”

Section: • Missing Data On Internet Penetration In 2010 Formentioning

confidence: 99%

“…Variants of System (1) have been studied in many contexts, including the double cusp catastrophe [40,106,21], cuspoidal nets [2,1], and coupled van der Pol oscillators [97,96,86,98,84,85,22,72] (for more information, see Appendix B). Coordination games and global games in economics are similar to System (1) in that they also permit multiple equilibria, but they lack dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…The case of System (1) with bidirectional, symmetric coupling is a special case of the double cusp catastrophe, which is given by the potential This singularity has a very rich structure [ 41 – 43 ].…”

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confidence: 99%

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Coupled catastrophes: sudden shifts cascade and hop among interdependent systems

Brummitt

Barnett

D’Souza

2015

J. R. Soc. Interface.

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An important challenge in several disciplines is to understand how sudden changes can propagate among coupled systems. Examples include the synchronization of business cycles, population collapse in patchy ecosystems, markets shifting to a new technology platform, collapses in prices and in confidence in financial markets, and protests erupting in multiple countries. A number of mathematical models of these phenomena have multiple equilibria separated by saddle-node bifurcations. We study this behaviour in its normal form as fast–slow ordinary differential equations. In our model, a system consists of multiple subsystems, such as countries in the global economy or patches of an ecosystem. Each subsystem is described by a scalar quantity, such as economic output or population, that undergoes sudden changes via saddle-node bifurcations. The subsystems are coupled via their scalar quantity (e.g. trade couples economic output; diffusion couples populations); that coupling moves the locations of their bifurcations. The model demonstrates two ways in which sudden changes can propagate: they can cascade (one causing the next), or they can hop over subsystems. The latter is absent from classic models of cascades. For an application, we study the Arab Spring protests. After connecting the model to sociological theories that have bistability, we use socioeconomic data to estimate relative proximities to tipping points and Facebook data to estimate couplings among countries. We find that although protests tend to spread locally, they also seem to ‘hop' over countries, like in the stylized model; this result highlights a new class of temporal motifs in longitudinal network datasets.

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Section: • Missing Data On Internet Penetration In 2010 Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coupled catastrophes: sudden shifts cascade and hop among interdependent systems

Brummitt

Barnett

D’Souza

2015

J. R. Soc. Interface.

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“…Recently, first conceptual steps based on Brummitt et al [45] and Abraham et al [46] have been undertaken to determine whether the network of Earth system tipping elements is capable to produce global tipping cascades [47,48]. Note that the proposed system capturing idealized interacting tipping elements is related to the double cusp catastrophe, which has been studied mathematically by, among others, Godwin [49] and Callahan [50]. More generally, coupled cell systems have been considered previously (e.g.…”

Section: Introduction 1the Concept Of Tipping Cascadesmentioning

confidence: 99%

What do we mean, ‘tipping cascade’?

Klose

Wunderling

Winkelmann

et al. 2021

Environ. Res. Lett.

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Based on suggested interactions of potential tipping elements in the Earth’s climate and in ecological systems, tipping cascades as possible dynamics are increasingly discussed and studied as their activation would impose a considerable risk for human societies and biosphere integrity. However, there are ambiguities in the description of tipping cascades within the literature so far. Here we illustrate how different patterns of multiple tipping dynamics emerge from a very simple coupling of two previously studied idealized tipping elements. In particular, we distinguish between a two phase cascade, a domino cascade and a joint cascade. While a mitigation of an unfolding two phase cascade may be possible and common early warning indicators are sensitive to upcoming critical transitions to a certain degree, the domino cascade may hardly be stopped once initiated and critical slowing down–based indicators fail to indicate tipping of the following element. These different potentials for intervention and anticipation across the distinct patterns of multiple tipping dynamics should be seen as a call to be more precise in future analyses on cascading dynamics arising from tipping element interactions in the Earth system.

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The umbilic bracelet and the double-cusp catastrophe

Zeeman

1976

Structural Stability, the Theory of Catastrophes, and Applications in the Sciences

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Topological bifurcation for the double cusp polynomial

Cited by 7 publications

References 9 publications

Coupled catastrophes: sudden shifts cascade and hop among interdependent systems

Coupled catastrophes: sudden shifts cascade and hop among interdependent systems

What do we mean, ‘tipping cascade’?

The umbilic bracelet and the double-cusp catastrophe

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