117,639 research outputs found

    Tipping points near a delayed saddle node bifurcation with periodic forcing

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    We consider the effect on tipping from an additive periodic forcing in a canonical model with a saddle node bifurcation and a slowly varying bifurcation parameter. Here tipping refers to the dramatic change in dynamical behavior characterized by a rapid transition away from a previously attracting state. In the absence of the periodic forcing, it is well-known that a slowly varying bifurcation parameter produces a delay in this transition, beyond the bifurcation point for the static case. Using a multiple scales analysis, we consider the effect of amplitude and frequency of the periodic forcing relative to the drifting rate of the slowly varying bifurcation parameter. We show that a high frequency oscillation drives an earlier tipping when the bifurcation parameter varies more slowly, with the advance of the tipping point proportional to the square of the ratio of amplitude to frequency. In the low frequency case the position of the tipping point is affected by the frequency, amplitude and phase of the oscillation. The results are based on an analysis of the local concavity of the trajectory, used for low frequencies both of the same order as the drifting rate of the bifurcation parameter and for low frequencies larger than the drifting rate. The tipping point location is advanced with increased amplitude of the periodic forcing, with critical amplitudes where there are jumps in the location, yielding significant advances in the tipping point. We demonstrate the analysis for two applications with saddle node-type bifurcations

    Extreme sensitivity and climate tipping points

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    A climate state close to a tipping point will have a degenerate linear response to perturbations, which can be associated with extreme values of the equilibrium climate sensitivity (ECS). In this paper we contrast linearized (`instantaneous') with fully nonlinear geometric (`two-point') notions of ECS, in both presence and absence of tipping points. For a stochastic energy balance model of the global mean surface temperature with two stable regimes, we confirm that tipping events cause the appearance of extremes in both notions of ECS. Moreover, multiple regimes with different mean sensitivities are visible in the two-point ECS. We confirm some of our findings in a physics-based multi-box model of the climate system.Comment: 11 figure

    Tipping and the Dynamics of Segregation

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    In a classic paper, Schelling (1971) showed that extreme segregation can arise from social interactions in white preferences: once the minority share in a neighborhood exceeds a critical "tipping point," all the whites leave. We use regression discontinuity methods and Census tract data from 1970 through 2000 to test for discontinuities in the dynamics of neighborhood racial composition. White population flows exhibit tipping-like behavior in most cities, with a distribution of tipping points ranging from 5% to 20% minority share. The estimated discontinuities are robust to controls for a wide variety of neighborhood characteristics, and are as strong in the suburbs as in tracts close to high-minority neighborhoods, ruling out the main alternative explanations for apparent tipping behavior. In contrast to white population flows, there is no systematic evidence that rents or housing prices exhibit non-linearities around the tipping point. Finally, we relate the location of the estimated tipping points in different cities to measures of the racial attitudes of whites, and find that cities with more tolerant whites have higher tipping points.

    Inverse-square law between time and amplitude for crossing tipping thresholds

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    A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses} near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher dimensional system, a model for the Indian Summer Monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.Comment: 20 pages, 6 figures, 1 table, Supplementary Material found at https://figshare.com/articles/Monsoon_supplementary_material_final_pdf/760582

    The tipping point: a mathematical model for the profit-driven abandonment of restaurant tipping

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    The custom of voluntarily tipping for services rendered has gone in and out of fashion in America since its introduction in the 19th century. Restaurant owners that ban tipping in their establishments often claim that social justice drives their decisions, but we show that rational profit-maximization may also justify the decisions. Here, we propose a conceptual model of restaurant competition for staff and customers, and we show that there exists a critical conventional tip rate at which restaurant owners should eliminate tipping to maximize profit. Because the conventional tip rate has been increasing steadily for the last several decades, our model suggests that restaurant owners may abandon tipping en masse when that critical tip rate is reached.Comment: 14 pages, 5 figures, supplementary material include

    Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping

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    We discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We express tipping in terms of pullback attraction and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a smooth stable branch of equilibria over the range of values of the parameter shift, the pullback attractor will remain close to (track) this branch for small enough rates, though larger rates may lead to rate-induced tipping. More generally, we show that one can track certain stable paths that go along several stable branches by pseudo-orbits of the system, for small enough rates. For these local pullback point attractors, we define notions of bifurcation-induced and irreversible rate-induced tipping of the non-autonomous system. In one-dimension, we give a number of sufficient conditions for the presence or absence of rate-induced tipping, and we discuss some applications of our results to give criteria for irreversible rate-induced tipping in a conceptual climate model example

    Book Review: The Tipping Point

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    Book review of The Tipping Point by Malcom Gladwell, written for the Spring 2016 Semester MGMT 641: Organizational Leadership and Project Team Management class
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