Effects of Periodic Forcing on a Paleoclimate Delay Model

Quinn, Courtney; Sieber, Jan; Heydt, Anna S. von der

doi:10.1137/18m1203079

Cited by 7 publications

(5 citation statements)

References 37 publications

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“…If this process were analysed in isolation, this might seem to be a potentially reversible transition between the TH and SA states,whereas in actuality the crisis has already occurred and the system is destined to eventually converge to the SA attractor. Similar behaviour has previously been seen in quasi-periodically forced delay models [21,22] and in systems with delayed Hopf bifurcations (see e.g. [13] and references therein).…”

Section: Chaotic Saddle Collisionssupporting

confidence: 82%

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

Axelsen¹,

Quinn²,

Bassom³

2023

Preprint

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We consider a coupling of the Stommel Box and the Lorenz models, with the goal of investigating the so-called "crises" that are known to occur given sufficient forcing. In this context, a crisis is characterised as the bifurcation of a system that is produced by the application of forcing to a bistable system, thereby reducing it to a monostable system. We document the variety of chaotic attractors and crises possible in our model and demonstrate the possibility of a merging between the stable chaotic attractor that persists after a crisis with either a chaotic transient or a ghost attractor. An investigation of the finite time Lyapunov exponents around crisis levels of forcing reveals a strong alignment between the first Stommel and neutral Lorenz exponents, particularly at points of a trajectory most sensitive to divergence around these levels. We discuss possible predictors that may identity those chaotic attractors liable to collapse as a consequence of a crisis and show the chaotic saddle collisions that occur in a boundary crisis. Finally, we comment on the generality of our findings by coupling the Stommel Box model with other strange attractors and thereby show that many of the behaviours are quite generic and robust.

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Section: Chaotic Saddle Collisionssupporting

confidence: 82%

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

Axelsen¹,

Quinn²,

Bassom³

2023

Preprint

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“…Of course the computation of global manifolds, especially of two-dimensional stable and unstable manifolds constitutes a much more difficult task. Towards this direction, Quinn et al [34] have exploited the equation-free approach to compute the one-dimensional stable manifold of an one-dimensional delay differential equation. Other points that require further investigation are the analysis of the convergence properties of the algorithm, the numerical convergence properties of the scheme with respect to the amplitude of the noise to stochasticity, the sensitivity of the approximation with respect to the discretization of the domain around the saddle as well as the issue of finding confidence intervals for the coefficients of the polynomial expansion.…”

Section: Discussionmentioning

confidence: 99%

“…Frewen et al [12] traced within the equation-free framework two-dimensional slow manifolds to get out of potential wells. Finally, Quinn et al [34] have exploited the concept of equation-free approach to compute the one-dimensional stable manifold of a one-dimensional delay differential equation.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

Siettos

Russo

2021

Numer Algor

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We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

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“…Convergence analysis with general a-priori error estimates has been performed mostly for the idealized case where the D-dimensional microscopic problem has a d-dimensional attracting invariant slow manifold C, which is rarely encountered in the practical applications listed above. Exceptions are, for example, a study of bursting neurons [8] and the application of implicit equation-free computations to generalize an algorithm for growing stable manifolds of fixed points of two-dimensional maps a delay-differential equation with an unknown two-dimensional slow manifold [38]. Even for this idealized case one faces the geometric difficulty illustrated in Figure 2.1.…”

Section: Current State Of Analysismentioning

confidence: 99%

Convergence of Equation-Free Methods in the Case of Finite Time Scale Separation with Application to Deterministic and Stochastic Systems

Sieber

Marschler

Starke

2018

SIAM J. Appl. Dyn. Syst.

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A common approach to studying high-dimensional systems with emergent lowdimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): first, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a user-defined restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for finite time-scale separation with respect to a method parameter, the so-called healing time. Our convergence result justifies equation-free methods as a tool for performing high-level tasks such as bifurcation analysis on high-dimensional systems.More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then an implicit formulation of the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. A demonstration with Michaelis-Menten kinetics shows that the error estimates of our theorem are sharp.The ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic systems, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically finite. We investigate a low-dimensional stochastic differential equation where the ratio between the decay rates of fast and slow variables is 2.

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Effects of Periodic Forcing on a Paleoclimate Delay Model

Cited by 7 publications

References 37 publications

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

Convergence of Equation-Free Methods in the Case of Finite Time Scale Separation with Application to Deterministic and Stochastic Systems

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