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Pullback Attractor Crisis in a Delay Differential ENSO Model

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Advances in Nonlinear Geosciences

Abstract

We study the pullback attractor (PBA) of a seasonally forced delay differential model for the El Niño–Southern Oscillation (ENSO); the model has two delays, associated with a positive and a negative feedback. The control parameter is the intensity of the positive feedback and the PBA undergoes a crisis that consists of a chaos-to-chaos transition. Since the PBA is dominated by chaotic behavior, we refer to it as a strange PBA. Both chaotic regimes correspond to an overlapping of resonances but the two differ by the properties of this overlapping. The crisis manifests itself by a brutal change not only in the size but also in the shape of the PBA. The change is associated with the sudden disappearance of the most extreme warm (El Niño) and cold (La Niña) events, as one crosses the critical parameter value from below. The analysis reveals that regions of the strange PBA that survive the crisis are those populated by the most probable states of the system. These regions are those that exhibit robust foldings with respect to perturbations. The effect of noise on this phase-and-parameter space behavior is then discussed. It is shown that the chaos-to-chaos crisis may or may not survive the addition of small noise to the evolution equation, depending on how the noise enters the latter.

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Notes

  1. 1.

    Relying, for instance, on the Takens embedding theorem (Takens 1981).

  2. 2.

    Still, a segment [s′, t′] of the forcing may drive the system in a way that is similar to that over the segment [s, t], even when g(t) is a white noise, provided the system’s solutions exhibit recurrent patterns as time evolves; see Chekroun et al. (2011a), Kondrashov et al. (2013).

  3. 3.

    Here compact set is understood in the sense of point set topology (Kelley 1975).

  4. 4.

    This set is equivalently defined as the set of elements ψ in X obtained as the pullback limit \(\psi =\mathop{\lim }\limits_{ k \rightarrow \infty }U(t,s_{k})\phi _{k}\), with s k → − and ϕ k B.

  5. 5.

    Heavy curves have been used for a better visualization of the overall evolution in the three-dimensional representation used in Fig. 2.

  6. 6.

    Allowing, for instance, for a weighted combination over the possible accumulation points in X of the trajectory sU(t, s)x.

  7. 7.

    In this case, U(t, s) = S(ts) becomes a (semi-)flow and μ t is time independent.

  8. 8.

    The other aspect of the problem that renders the analysis difficult is tied to the lack of smoothing of the flow in probability space by the Liouville equation (Chekroun et al., 2014)— in the present, deterministic setting—as compared to the Fokker–Planck equation, which is its counterpart in the presence of noise; see Chekroun et al. (submitted).

  9. 9.

    Here ϕ(−1) corresponds to the value of the initial histories at − 1 years.

  10. 10.

    The “true” embedding dimension d given by the Takens embedding theorem may be greater than 2; see Robinson (2008) for a version of this theorem in the context of PBAs.

  11. 11.

    With respect to the Lebesgue measure in \(\mathbb{R}^{d}\).

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Acknowledgements

This work has been partially supported by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) grant N00014-12-1-0911 and N00014-16-1-2073 (MDC & MG), by the National Science Foundation grants OCE-1243175 (MDC & MG), DMS-1616981(MDC), and AGS-1540518 grant (JDN).

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Chekroun, M.D., Ghil, M., Neelin, J.D. (2018). Pullback Attractor Crisis in a Delay Differential ENSO Model. In: Tsonis, A. (eds) Advances in Nonlinear Geosciences. Springer, Cham. https://doi.org/10.1007/978-3-319-58895-7_1

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