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2018 | OriginalPaper | Chapter

Pullback Attractor Crisis in a Delay Differential ENSO Model

Authors : Mickaël D. Chekroun, Michael Ghil, J. David Neelin

Published in: Advances in Nonlinear Geosciences

Publisher: Springer International Publishing

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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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Footnotes
1
Relying, for instance, on the Takens embedding theorem (Takens 1981).
 
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
Here compact set is understood in the sense of point set topology (Kelley 1975).
 
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
Heavy curves have been used for a better visualization of the overall evolution in the three-dimensional representation used in Fig. 2.
 
6
Allowing, for instance, for a weighted combination over the possible accumulation points in X of the trajectory sU(t, s)x.
 
7
In this case, U(t, s) = S(ts) becomes a (semi-)flow and μ t is time independent.
 
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
Here ϕ(−1) corresponds to the value of the initial histories at − 1 years.
 
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
With respect to the Lebesgue measure in \(\mathbb{R}^{d}\).
 
Literature
go back to reference Arnold, V.I. 1988. Geometrical methods in the theory of ordinary differential equations, 2nd ed., Grundlehren der mathematischen Wissenschaften, vol. 250. Berlin: Springer. Arnold, V.I. 1988. Geometrical methods in the theory of ordinary differential equations, 2nd ed., Grundlehren der mathematischen Wissenschaften, vol. 250. Berlin: Springer.
go back to reference Arnold, L. 2013. Random dynamical systems. Berlin: Springer Science & Business Media. Arnold, L. 2013. Random dynamical systems. Berlin: Springer Science & Business Media.
go back to reference Battisti, D.S., and A.C. Hirst. 1989. Interannual variability in a tropical atmosphere–ocean model: Influence of the basic state, ocean geometry and nonlinearity. Journal of the Atmospheric Sciences 46(12): 1687–1712.CrossRef Battisti, D.S., and A.C. Hirst. 1989. Interannual variability in a tropical atmosphere–ocean model: Influence of the basic state, ocean geometry and nonlinearity. Journal of the Atmospheric Sciences 46(12): 1687–1712.CrossRef
go back to reference Bjerknes, J. 1969. Atmospheric teleconnections from the equatorial Pacific. Monthly Weather Review 97(3): 163–172.CrossRef Bjerknes, J. 1969. Atmospheric teleconnections from the equatorial Pacific. Monthly Weather Review 97(3): 163–172.CrossRef
go back to reference Blanke, B., J.D. Neelin, and D. Gutzler. 1997. Estimating the effects of stochastic wind stress forcing on ENSO irregularity. Journal of Climate 10: 1473–1486.CrossRef Blanke, B., J.D. Neelin, and D. Gutzler. 1997. Estimating the effects of stochastic wind stress forcing on ENSO irregularity. Journal of Climate 10: 1473–1486.CrossRef
go back to reference Bódai, T., and T. Tél. 2012. Annual variability in a conceptual climate model: snapshot attractors, hysteresis in extreme events, and climate sensitivity. Chaos: An Interdisciplinary Journal of Nonlinear Science 22(2): 023110.CrossRef Bódai, T., and T. Tél. 2012. Annual variability in a conceptual climate model: snapshot attractors, hysteresis in extreme events, and climate sensitivity. Chaos: An Interdisciplinary Journal of Nonlinear Science 22(2): 023110.CrossRef
go back to reference Bódai, T., G. Károlyi, and T. Tél. 2011. A chaotically driven model climate: extreme events and snapshot attractors. Nonlinear Processes in Geophysics 18(5): 573–580.CrossRef Bódai, T., G. Károlyi, and T. Tél. 2011. A chaotically driven model climate: extreme events and snapshot attractors. Nonlinear Processes in Geophysics 18(5): 573–580.CrossRef
go back to reference Bódai, T., G. Károlyi, and T. Tél. 2013. Driving a conceptual model climate by different processes: snapshot attractors and extreme events. Physical Review E 87(2): 022822.CrossRef Bódai, T., G. Károlyi, and T. Tél. 2013. Driving a conceptual model climate by different processes: snapshot attractors and extreme events. Physical Review E 87(2): 022822.CrossRef
go back to reference Botev, Z.I., J.F. Grotowski, and D.P. Kroese. 2010. Kernel density estimation via diffusion. The Annals of Statistics 38(5): 2916–2957.CrossRef Botev, Z.I., J.F. Grotowski, and D.P. Kroese. 2010. Kernel density estimation via diffusion. The Annals of Statistics 38(5): 2916–2957.CrossRef
go back to reference Cane, M.A. 1986. Experimental forecasts of el Niño. Nature 321: 827–832.CrossRef Cane, M.A. 1986. Experimental forecasts of el Niño. Nature 321: 827–832.CrossRef
go back to reference Caraballo, T., J.A. Langa, and J.C. Robinson. 2001. Attractors for differential equations with variable delays. Journal of Mathematical Analysis and Applications 260(2): 421–438.CrossRef Caraballo, T., J.A. Langa, and J.C. Robinson. 2001. Attractors for differential equations with variable delays. Journal of Mathematical Analysis and Applications 260(2): 421–438.CrossRef
go back to reference Caraballo, T., P. Marın-Rubio, and J. Valero. 2005. Autonomous and non-autonomous attractors for differential equations with delays. Journal of Differential Equations 208(1): 9–41.CrossRef Caraballo, T., P. Marın-Rubio, and J. Valero. 2005. Autonomous and non-autonomous attractors for differential equations with delays. Journal of Differential Equations 208(1): 9–41.CrossRef
go back to reference Carvalho, A., J.A. Langa, and J. Robinson. 2013. Attractors for infinite-dimensional non-autonomous dynamical systems In Applied mathematical sciences, vol. 182. Berlin: Springer. Carvalho, A., J.A. Langa, and J. Robinson. 2013. Attractors for infinite-dimensional non-autonomous dynamical systems In Applied mathematical sciences, vol. 182. Berlin: Springer.
go back to reference Chekroun, M.D., and N.E. Glatt-Holtz. 2012. Invariant measures for dissipative dynamical systems: Abstract results and applications. Communications in Mathematical Physics 316(3): 723–761.CrossRef Chekroun, M.D., and N.E. Glatt-Holtz. 2012. Invariant measures for dissipative dynamical systems: Abstract results and applications. Communications in Mathematical Physics 316(3): 723–761.CrossRef
go back to reference Chekroun, M.D., D. Kondrashov, and M. Ghil. 2011a. Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation. Proceedings of the National Academy of Sciences of the United States of America 108: 11766–11771. Chekroun, M.D., D. Kondrashov, and M. Ghil. 2011a. Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation. Proceedings of the National Academy of Sciences of the United States of America 108: 11766–11771.
go back to reference Chekroun, M.D., E. Simonnet, and M. Ghil. 2011b. Stochastic climate dynamics: random attractors and time-dependent invariant measures. Physica D 240(21): 1685–1700. Chekroun, M.D., E. Simonnet, and M. Ghil. 2011b. Stochastic climate dynamics: random attractors and time-dependent invariant measures. Physica D 240(21): 1685–1700.
go back to reference Chekroun, M.D., J.D. Neelin, D. Kondrashov, J.C. McWilliams, and M. Ghil. 2014. Rough parameter dependence in climate models: the role of Ruelle-Pollicott resonances. Proceedings of the National Academy of Sciences of the United States of America 111(5): 1684–1690.CrossRef Chekroun, M.D., J.D. Neelin, D. Kondrashov, J.C. McWilliams, and M. Ghil. 2014. Rough parameter dependence in climate models: the role of Ruelle-Pollicott resonances. Proceedings of the National Academy of Sciences of the United States of America 111(5): 1684–1690.CrossRef
go back to reference Chekroun, M.D., H. Liu, and S. Wang. 2015a. Approximation of stochastic invariant manifolds: stochastic manifolds for nonlinear SPDEs I. Springer briefs in mathematics. New York: Springer. Chekroun, M.D., H. Liu, and S. Wang. 2015a. Approximation of stochastic invariant manifolds: stochastic manifolds for nonlinear SPDEs I. Springer briefs in mathematics. New York: Springer.
go back to reference Chekroun, M.D., H. Liu, and S. Wang. 2015b. Parameterizing manifolds and non-Markovian reduced equations: stochastic manifolds for nonlinear SPDEs II. Springer briefs in mathematics. New York: Springer. Chekroun, M.D., H. Liu, and S. Wang. 2015b. Parameterizing manifolds and non-Markovian reduced equations: stochastic manifolds for nonlinear SPDEs II. Springer briefs in mathematics. New York: Springer.
go back to reference Chekroun, M.D., E. Park, and R. Temam. 2016a. The Stampacchia maximum principle for stochastic partial differential equations and applications. Journal of Differential Equations 260(3): 2926–2972. Chekroun, M.D., E. Park, and R. Temam. 2016a. The Stampacchia maximum principle for stochastic partial differential equations and applications. Journal of Differential Equations 260(3): 2926–2972.
go back to reference Chekroun, M.D., M. Ghil, H. Liu, and S. Wang. 2016b. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete and Continuous Dynamical System A 36(8): 4133–4177. Chekroun, M.D., M. Ghil, H. Liu, and S. Wang. 2016b. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete and Continuous Dynamical System A 36(8): 4133–4177.
go back to reference Chekroun, M.D., A. Tantet, H.A Dijkstra, and J.D. Neelin, Mixing Spectrum in reduced phase spaces of stochastic differential equations. Part I: theory (submitted) Chekroun, M.D., A. Tantet, H.A Dijkstra, and J.D. Neelin, Mixing Spectrum in reduced phase spaces of stochastic differential equations. Part I: theory (submitted)
go back to reference Chepyzhov, V.V., and M.I. Vishik. 2002. Attractors for equations of mathematical physics, vol. 49 of Colloquium publications. Providence, RI: American Mathematical Society. Chepyzhov, V.V., and M.I. Vishik. 2002. Attractors for equations of mathematical physics, vol. 49 of Colloquium publications. Providence, RI: American Mathematical Society.
go back to reference Cong, N.D. 1996. Topological classification of linear hyperbolic cocycles. Journal of Dynamics and Differential Equations 8(3): 427–467.CrossRef Cong, N.D. 1996. Topological classification of linear hyperbolic cocycles. Journal of Dynamics and Differential Equations 8(3): 427–467.CrossRef
go back to reference Cong, N.D. 1997. Topological dynamics of random dynamical systems. Oxford: Oxford University Press. Cong, N.D. 1997. Topological dynamics of random dynamical systems. Oxford: Oxford University Press.
go back to reference Crauel, H., and F. Flandoli. 1994. Attractors for random dynamical systems. Probability Theory and Related Fields 100(3): 365–393.CrossRef Crauel, H., and F. Flandoli. 1994. Attractors for random dynamical systems. Probability Theory and Related Fields 100(3): 365–393.CrossRef
go back to reference Crauel, H., A. Debussche, and F. Flandoli. 1997. Random attractors. Journal of Dynamics and Differential Equations 9(2): 307–341.CrossRef Crauel, H., A. Debussche, and F. Flandoli. 1997. Random attractors. Journal of Dynamics and Differential Equations 9(2): 307–341.CrossRef
go back to reference Diekmann, O., S.A. van Gils, S.M. Verduyn Lunel, and H.-O. Walther. 1995. Delay equations: functional, complex, and nonlinear analysis. Applied mathematical sciences, vol. 110. New York: Springer. Diekmann, O., S.A. van Gils, S.M. Verduyn Lunel, and H.-O. Walther. 1995. Delay equations: functional, complex, and nonlinear analysis. Applied mathematical sciences, vol. 110. New York: Springer.
go back to reference Drótos, G., T. Bódai, and T. Tél. 2015. Probabilistic concepts in a changing climate: a snapshot attractor picture. Journal of Climate 28(8): 3275–3288.CrossRef Drótos, G., T. Bódai, and T. Tél. 2015. Probabilistic concepts in a changing climate: a snapshot attractor picture. Journal of Climate 28(8): 3275–3288.CrossRef
go back to reference Eckert, C., and M. Latif. 1997. Predictability of a stochastically forced hybrid coupled model of El Niño. Journal of Climate 10(7): 1488–1504.CrossRef Eckert, C., and M. Latif. 1997. Predictability of a stochastically forced hybrid coupled model of El Niño. Journal of Climate 10(7): 1488–1504.CrossRef
go back to reference Eckmann, J.-P., and D. Ruelle. 1985. Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57: 617–656.CrossRef Eckmann, J.-P., and D. Ruelle. 1985. Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57: 617–656.CrossRef
go back to reference Foias, C., O. Manley, R. Rosa, and R. Temam. 2001. Navier-Stokes equations and turbulence. Encyclopedia of mathematics and its applications, vol. 83. Cambridge: Cambridge University Press. Foias, C., O. Manley, R. Rosa, and R. Temam. 2001. Navier-Stokes equations and turbulence. Encyclopedia of mathematics and its applications, vol. 83. Cambridge: Cambridge University Press.
go back to reference Galanti, E., and E. Tziperman. 2000. ENSO’s phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes. Journal of the Atmospheric Sciences 57(17): 2936–2950.CrossRef Galanti, E., and E. Tziperman. 2000. ENSO’s phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes. Journal of the Atmospheric Sciences 57(17): 2936–2950.CrossRef
go back to reference Ghil, M. 2017. The wind-driven ocean circulation: applying dynamical systems theory to a climate problem. Discrete and Continuous Dynamical Systems-A 37(1): 189–228.CrossRef Ghil, M. 2017. The wind-driven ocean circulation: applying dynamical systems theory to a climate problem. Discrete and Continuous Dynamical Systems-A 37(1): 189–228.CrossRef
go back to reference Ghil, M., and N. Jiang. 1998. Recent forecast skill for the El Niño/Southern Oscillation. Geophysical Research Letters 25(2): 171–174.CrossRef Ghil, M., and N. Jiang. 1998. Recent forecast skill for the El Niño/Southern Oscillation. Geophysical Research Letters 25(2): 171–174.CrossRef
go back to reference Ghil, M., and I. Zaliapin. 2015. Understanding ENSO variability and its extrema: a delay differential equation approach. In Observations, modeling and economics of extreme events, ed. M. Ghil, M. Chavez, and J. Urrutia-Fucugauchi. Geophysical monographs, vol. 214, 63–78. Washington, DC: American Geophysical Union & Wiley. Ghil, M., and I. Zaliapin. 2015. Understanding ENSO variability and its extrema: a delay differential equation approach. In Observations, modeling and economics of extreme events, ed. M. Ghil, M. Chavez, and J. Urrutia-Fucugauchi. Geophysical monographs, vol. 214, 63–78. Washington, DC: American Geophysical Union & Wiley.
go back to reference Ghil, M., I. Zaliapin, and S. Thompson. 2008a. A delay differential model of ENSO variability: parametric instability and the distribution of extremes. Nonlinear Processes in Geophysics 15(3): 417–433. Ghil, M., I. Zaliapin, and S. Thompson. 2008a. A delay differential model of ENSO variability: parametric instability and the distribution of extremes. Nonlinear Processes in Geophysics 15(3): 417–433.
go back to reference Ghil, M., M.D. Chekroun, and E. Simonnet. 2008b. Climate dynamics and fluid mechanics: natural variability and related uncertainties. Physica D 237: 2111–2126. Ghil, M., M.D. Chekroun, and E. Simonnet. 2008b. Climate dynamics and fluid mechanics: natural variability and related uncertainties. Physica D 237: 2111–2126.
go back to reference Grassberger, P., and I. Procaccia. 1983. Characterization of strange attractors. Physical Review Letters 505: 346.CrossRef Grassberger, P., and I. Procaccia. 1983. Characterization of strange attractors. Physical Review Letters 505: 346.CrossRef
go back to reference Grassberger, P., and I. Procaccia. 2004. Measuring the strangeness of strange attractors. In The theory of chaotic attractors, ed. B.R. Hunt, T.-Y. Tien-Yien Li, J.A. Kennedy, and H.E. Nusse, 170–189. Berlin: Springer.CrossRef Grassberger, P., and I. Procaccia. 2004. Measuring the strangeness of strange attractors. In The theory of chaotic attractors, ed. B.R. Hunt, T.-Y. Tien-Yien Li, J.A. Kennedy, and H.E. Nusse, 170–189. Berlin: Springer.CrossRef
go back to reference Grebogi, C., E. Ott, F. Romeiras, and J.A. Yorke. 1987. Critical exponents for crisis-induced intermittency. Physical Review A 36(11): 5365.CrossRef Grebogi, C., E. Ott, F. Romeiras, and J.A. Yorke. 1987. Critical exponents for crisis-induced intermittency. Physical Review A 36(11): 5365.CrossRef
go back to reference Hale, J.K., and S.M. Verduyn-Lunel. 1993. Introduction to functional-differential equations. Applied mathematical sciences. vol. 99. New York: Springer. Hale, J.K., and S.M. Verduyn-Lunel. 1993. Introduction to functional-differential equations. Applied mathematical sciences. vol. 99. New York: Springer.
go back to reference Haraux, A. 1991. Systèmes Dynamiques Dissipatifs et applications, vol. 17. Paris: Masson. Haraux, A. 1991. Systèmes Dynamiques Dissipatifs et applications, vol. 17. Paris: Masson.
go back to reference Horita, T., H. Hata, H. Mori, T. Morita, K. Tomita, K. Shoichi, and H. Okamoto. 1988. Local structures of chaotic attractors and q-phase transitions at attractor-merging crises in the sine-circle maps. Progress of Theoretical Physics 80(5): 793–808.CrossRef Horita, T., H. Hata, H. Mori, T. Morita, K. Tomita, K. Shoichi, and H. Okamoto. 1988. Local structures of chaotic attractors and q-phase transitions at attractor-merging crises in the sine-circle maps. Progress of Theoretical Physics 80(5): 793–808.CrossRef
go back to reference Hunt, B.R., and V.Y. Kaloshin. 1997. How projections affect the dimension spectrum of fractal measures. Nonlinearity 10(5): 1031–1046.CrossRef Hunt, B.R., and V.Y. Kaloshin. 1997. How projections affect the dimension spectrum of fractal measures. Nonlinearity 10(5): 1031–1046.CrossRef
go back to reference Jensen, M.H., P. Bak, and T. Bohr. 1984. Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps. Physical Review A 30(4): 1960. Jensen, M.H., P. Bak, and T. Bohr. 1984. Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps. Physical Review A 30(4): 1960.
go back to reference Jiang, N., M. Ghil, and D. Neelin. 1995. Forecasts of equatorial Pacific SST anomalies by an autoregressive process using singular spectrum analysis. Experimental Long-Lead Forecast Bulletin 4(1): 24–27. Jiang, N., M. Ghil, and D. Neelin. 1995. Forecasts of equatorial Pacific SST anomalies by an autoregressive process using singular spectrum analysis. Experimental Long-Lead Forecast Bulletin 4(1): 24–27.
go back to reference Jin, F.-F., and J.D. Neelin. 1993. Modes of interannual tropical ocean-atmosphere interaction-A unified view. Part III: analytical results in fully coupled cases. Journal of the Atmospheric Sciences 50(21): 3523–3540.CrossRef Jin, F.-F., and J.D. Neelin. 1993. Modes of interannual tropical ocean-atmosphere interaction-A unified view. Part III: analytical results in fully coupled cases. Journal of the Atmospheric Sciences 50(21): 3523–3540.CrossRef
go back to reference Jin, F.-F., J.D. Neelin, and M. Ghil. 1994. El Niño on the Devil’s Staircase: annual subharmonic steps to chaos. Science 274: 70–72.CrossRef Jin, F.-F., J.D. Neelin, and M. Ghil. 1994. El Niño on the Devil’s Staircase: annual subharmonic steps to chaos. Science 274: 70–72.CrossRef
go back to reference Jin, F.-F., J.D. Neelin, and M. Ghil. 1996. El Niño/Southern oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity. Physica D 98: 442–465.CrossRef Jin, F.-F., J.D. Neelin, and M. Ghil. 1996. El Niño/Southern oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity. Physica D 98: 442–465.CrossRef
go back to reference Keane, A., Krauskopf, B., and C. Postlethwaite. 2015. Delayed feedback versus seasonal forcing: resonance phenomena in an El Niño Southern Oscillation model. SIAM Journal on Applied Dynamical Systems 14(3): 1229–1257.CrossRef Keane, A., Krauskopf, B., and C. Postlethwaite. 2015. Delayed feedback versus seasonal forcing: resonance phenomena in an El Niño Southern Oscillation model. SIAM Journal on Applied Dynamical Systems 14(3): 1229–1257.CrossRef
go back to reference Keane, A., B. Krauskopf, and C. Postlethwaite. 2016. Investigating irregular behavior in a model for the El Niño Southern Oscillation with positive and negative delayed feedback. SIAM Journal on Applied Dynamical Systems 15(3): 1656–1689.CrossRef Keane, A., B. Krauskopf, and C. Postlethwaite. 2016. Investigating irregular behavior in a model for the El Niño Southern Oscillation with positive and negative delayed feedback. SIAM Journal on Applied Dynamical Systems 15(3): 1656–1689.CrossRef
go back to reference Kelley, J.L. 1975. General topology. Berlin: Springer Science & Business Media. Kelley, J.L. 1975. General topology. Berlin: Springer Science & Business Media.
go back to reference Kleeman, R., and S.B. Power. 1994. Limits to predictability in a coupled ocean-atmosphere model due to atmospheric noise. Tellus A 46(4): 529–540.CrossRef Kleeman, R., and S.B. Power. 1994. Limits to predictability in a coupled ocean-atmosphere model due to atmospheric noise. Tellus A 46(4): 529–540.CrossRef
go back to reference Kleeman, R., and A.M. Moore. 1997. A theory for the limitation of ENSO predictability due to stochastic atmospheric transients Journal of the Atmospheric Sciences 54(6): 753–767. Kleeman, R., and A.M. Moore. 1997. A theory for the limitation of ENSO predictability due to stochastic atmospheric transients Journal of the Atmospheric Sciences 54(6): 753–767.
go back to reference Kondrashov, D., M.D. Chekroun, A.W. Robertson, and M. Ghil. 2013. Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation. Geophysical Research Letters 40: 5305–5310. doi:10.1002/grl.50991.CrossRef Kondrashov, D., M.D. Chekroun, A.W. Robertson, and M. Ghil. 2013. Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation. Geophysical Research Letters 40: 5305–5310. doi:10.1002/grl.50991.CrossRef
go back to reference Kondrashov, D., M.D. Chekroun, and M. Ghil. 2015. Data-driven non-Markovian closure models. Physica D: Nonlinear Phenomena 297: 33–55.CrossRef Kondrashov, D., M.D. Chekroun, and M. Ghil. 2015. Data-driven non-Markovian closure models. Physica D: Nonlinear Phenomena 297: 33–55.CrossRef
go back to reference Kostelich, E.J., and H.L. Swinney. 1989. Practical considerations in estimating dimension from time series data. Physica Scripta 40(3): 436CrossRef Kostelich, E.J., and H.L. Swinney. 1989. Practical considerations in estimating dimension from time series data. Physica Scripta 40(3): 436CrossRef
go back to reference Krauskopf, B., and J. Sieber. 2014. Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation. Proc R Soc A 470(2169). The Royal Society. Krauskopf, B., and J. Sieber. 2014. Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation. Proc R Soc A 470(2169). The Royal Society.
go back to reference Lu, K., Q. Wang, and L.-S. Young. 2013. Strange attractors for periodically forced parabolic equations, vol. 224. Memoirs of the American mathematical society, vol. 1054. Providence, RI: American Mathematical Society. Lu, K., Q. Wang, and L.-S. Young. 2013. Strange attractors for periodically forced parabolic equations, vol. 224. Memoirs of the American mathematical society, vol. 1054. Providence, RI: American Mathematical Society.
go back to reference Lukaszewicz, G., and J.C. Robinson. 2014. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous and Dynamical Systems A 34(10): 4211–4222.CrossRef Lukaszewicz, G., and J.C. Robinson. 2014. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous and Dynamical Systems A 34(10): 4211–4222.CrossRef
go back to reference Mechoso, C.R., J.D. Neelin, and J.-Y. Yu. 2003. Testing simple models of ENSO. Journal of the Atmospheric Sciences 60: 305–318.CrossRef Mechoso, C.R., J.D. Neelin, and J.-Y. Yu. 2003. Testing simple models of ENSO. Journal of the Atmospheric Sciences 60: 305–318.CrossRef
go back to reference Mori, H., and Y. Kuramoto. 2013. Dissipative structures and chaos. New York: Springer. Mori, H., and Y. Kuramoto. 2013. Dissipative structures and chaos. New York: Springer.
go back to reference Münnich, M., M.A. Cane, and S.E. Zebiak. 1991. A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: nonlinear cases. Journal of the Atmospheric Sciences 48(10): 1238–1248.CrossRef Münnich, M., M.A. Cane, and S.E. Zebiak. 1991. A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: nonlinear cases. Journal of the Atmospheric Sciences 48(10): 1238–1248.CrossRef
go back to reference Neelin, J.D., D.S. Battisti, A.C. Hirst, F.-F. Jin, Y. Wakata, T. Yamagata, and S.E. Zebiak. 1998. ENSO theory. Journal of Geophysical Research: Oceans (1978–2012) 103(C7): 14261–14290.CrossRef Neelin, J.D., D.S. Battisti, A.C. Hirst, F.-F. Jin, Y. Wakata, T. Yamagata, and S.E. Zebiak. 1998. ENSO theory. Journal of Geophysical Research: Oceans (1978–2012) 103(C7): 14261–14290.CrossRef
go back to reference Neelin, J.D., F.-F. Jin, and H.-H. Syu. 2000. Variations in ENSO phase-locking. Journal of Climate 13: 2570–2590.CrossRef Neelin, J.D., F.-F. Jin, and H.-H. Syu. 2000. Variations in ENSO phase-locking. Journal of Climate 13: 2570–2590.CrossRef
go back to reference Philander, S.G.H. 1992. El Niño, La Niña, and the Southern oscillation. San Diego: Academic. Philander, S.G.H. 1992. El Niño, La Niña, and the Southern oscillation. San Diego: Academic.
go back to reference Pierini, S., M. Ghil, and M.D. Chekroun. 2016. Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case. Journal of Climate 29(11): 4185–4202.CrossRef Pierini, S., M. Ghil, and M.D. Chekroun. 2016. Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case. Journal of Climate 29(11): 4185–4202.CrossRef
go back to reference Robinson, J.C. 2008. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete and Continuous Dynamical Systems. Series B 9(3–4): 731–741.CrossRef Robinson, J.C. 2008. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete and Continuous Dynamical Systems. Series B 9(3–4): 731–741.CrossRef
go back to reference Romeiras, F.J., C. Grebogi, and E. Ott. 1990. Multifractal properties of snapshot attractors of random maps. Physical Review A 41(2): 784.CrossRef Romeiras, F.J., C. Grebogi, and E. Ott. 1990. Multifractal properties of snapshot attractors of random maps. Physical Review A 41(2): 784.CrossRef
go back to reference Roulston, M.S., and J.D. Neelin. 2000. The response of an ENSO model to climate noise, weather noise and intraseasonal forcing. Geophysical Research Letters 27: 3723–3726.CrossRef Roulston, M.S., and J.D. Neelin. 2000. The response of an ENSO model to climate noise, weather noise and intraseasonal forcing. Geophysical Research Letters 27: 3723–3726.CrossRef
go back to reference Ruelle, D. 1999. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. Journal of Statistical Physics 95(1): 393–468.CrossRef Ruelle, D. 1999. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. Journal of Statistical Physics 95(1): 393–468.CrossRef
go back to reference Suarez, M.J., and P.S. Schopf. 1988. A delayed action oscillator for ENSO. Journal of the Atmospheric Sciences 45(21): 3283–3287.CrossRef Suarez, M.J., and P.S. Schopf. 1988. A delayed action oscillator for ENSO. Journal of the Atmospheric Sciences 45(21): 3283–3287.CrossRef
go back to reference Takens, F. 1981. Detecting strange attractors in turbulence. In Dynamical systems and turbulence, Warwick 1980, 366–381. New York: Springer.CrossRef Takens, F. 1981. Detecting strange attractors in turbulence. In Dynamical systems and turbulence, Warwick 1980, 366–381. New York: Springer.CrossRef
go back to reference Temam, R. 1997. Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68. New York: Springer. Temam, R. 1997. Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68. New York: Springer.
go back to reference Tziperman, E., L. Stone, M.A. Cane, and H. Jarosh. 1994. El Niño chaos: overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science 264(5155): 72–74.CrossRef Tziperman, E., L. Stone, M.A. Cane, and H. Jarosh. 1994. El Niño chaos: overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science 264(5155): 72–74.CrossRef
go back to reference Tziperman, E., M.A. Cane, and S.E. Zebiak. 1995. Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos. Journal of the Atmospheric Sciences 52(3): 293–306.CrossRef Tziperman, E., M.A. Cane, and S.E. Zebiak. 1995. Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos. Journal of the Atmospheric Sciences 52(3): 293–306.CrossRef
go back to reference Wang, Q., and L.-S. Young. 2001. Strange attractors with one direction of instability. Communications in Mathematical Physics 218(1): 1–97.CrossRef Wang, Q., and L.-S. Young. 2001. Strange attractors with one direction of instability. Communications in Mathematical Physics 218(1): 1–97.CrossRef
go back to reference Wang, Q., and L.-S. Young. 2003. Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Communications in Mathematical Physics 240(3): 509–529.CrossRef Wang, Q., and L.-S. Young. 2003. Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Communications in Mathematical Physics 240(3): 509–529.CrossRef
go back to reference Wang, Q., and L.-S. Young. 2008. Toward a theory of rank one attractors. Annals of Mathematics 167: 349–480.CrossRef Wang, Q., and L.-S. Young. 2008. Toward a theory of rank one attractors. Annals of Mathematics 167: 349–480.CrossRef
go back to reference Young, L.-S. 2016. Generalizations of SRB measures to nonautonomous, random, and infinite dimensional systems. Journal of Statistical Physics 166: 494–515.CrossRef Young, L.-S. 2016. Generalizations of SRB measures to nonautonomous, random, and infinite dimensional systems. Journal of Statistical Physics 166: 494–515.CrossRef
Metadata
Title
Pullback Attractor Crisis in a Delay Differential ENSO Model
Authors
Mickaël D. Chekroun
Michael Ghil
J. David Neelin
Copyright Year
2018
DOI
https://doi.org/10.1007/978-3-319-58895-7_1