Investigating irregular behavior in a model for the El Niño southern oscillation with positive and negative delayed feedback. (English) Zbl 1351.37270


37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37M20 Computational methods for bifurcation problems in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
86A10 Meteorology and atmospheric physics
Full Text: DOI


[1] A. G. Barnston and C. F. Ropelewski, Prediction of ENSO episodes using canonical correlation analysis, J. Climate, 5 (1992), pp. 1316–1345.
[2] D. S. Battisti and A. C. Hirst, Interannual variability in a tropical atmosphere-ocean model: Influence of the basic state, ocean geometry and nonlinearity, J. Atmos. Sci., 46 (1989), pp. 1687–1712.
[3] J. Bjerknes, Atmospheric teleconnections from the equatorial Pacific 1, Monthly Weather Rev., 97 (1969), pp. 163–172.
[4] J.-P. Boulanger and C. Menkes, Propagation and reflection of long equatorial waves in the Pacific Ocean during the 1992–1993 El Nin͂o, J. Geophys. Res. Oceans, 100 (1995), pp. 25041–25059.
[5] R. C. Calleja, A. R. Humphries, and B. Krauskopf, Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays, preprint, arXiv:1607.02683, 2016. · Zbl 1371.34115
[6] M. A. Cane, M. Münnich, and S. F. Zebiak, A study of self-excited oscillations of the tropical ocean-atmosphere system. Part I: Linear analysis, J. Atmos. Sci., 47 (1990), pp. 1562–1577.
[7] M. A. Cane, S. E. Zebiak, and S. C. Dolan, Experimental forecasts of El Nin͂o, Nature, 321 (1986), pp. 827–832.
[8] P. Chang, L. Ji, B. Wang, and T. Li, Interactions between the seasonal cycle and El Nin͂o-Southern Oscillation in an intermediate coupled ocean-atmosphere model, J. Atmos. Sci., 52 (1995), pp. 2353–2372.
[9] S. A. Changnon, Impacts of 1997–98 El Nin͂o generated weather in the United States, Bull. Amer. Meteor. Soc., 80 (1999), pp. 1819–1827.
[10] D. Chen, M. A. Cane, A. Kaplan, S. E. Zebiak, and D. Huang, Predictability of El Nin͂o over the past 148 years, Nature, 428 (2004), pp. 733–736.
[11] A. J. Clarke, An Introduction to the Dynamics of El Nin͂o & the Southern Oscillation, Academic Press, New York, 2008.
[12] A. J. Clarke and S. Van Gorder, Improving El Nin͂o prediction using a space-time integration of Indo-Pacific winds and equatorial Pacific upper ocean heat content, Geophys. Res. Lett., 30 (2003), 1399.
[13] A. J Clarke, J. Wang, and S. Van Gorder, A simple warm-pool displacement ENSO model, J. Phys. Ocean., 30 (2000), pp. 1679–1691.
[14] M. Eichmann, A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays, Doctoral dissertation, Justus-Liebig-University of Giessen, Giessen, Germany, 2006.
[15] K. Engelborghs, T. Luzyanina, and G. Samaey, DDE-BIFTOOL: A MATLAB Package for Bifurcation Analysis of Delay Differential Equations, TW report, 305 (2000).
[16] J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Phys. D, 4 (1982), pp. 366–393. · Zbl 1194.37052
[17] M. Ghil, M. R. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. W. Robertson, A. Saunders, Y. Tian, F. Varadi, and P. Yiou, Advanced spectral methods for climatic time series, Rev. Geophys., 40 (2002).
[18] M. Ghil, I. Zaliapin, and S. Thompson, A delay differential model of ENSO variability: Parametric instability and the distribution of extremes, Nonlinear Process. Geophys., 15 (2008), pp. 417–433.
[19] I. Gyori and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst., 18 (2007), p. 773. · Zbl 1144.34051
[20] S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll, and W. Kinzel, Strong and weak chaos in nonlinear networks with time-delayed couplings, Phys. Rev. Lett., 107 (2011), 234102.
[21] Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), pp. 2801–2840. · Zbl 1214.34057
[22] A. R. Humphries, O. DeMasi, F. M. G. Magpantay, and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete Contin. Dyn. Syst. A, 32 (2012), pp. 2701–2727. · Zbl 1246.34073
[23] N. Jiang, J. D. Neelin, and M. Ghil, Quasi-quadrennial and quasi-biennial variability in the equatorial Pacific, Climate Dynam., 12 (1995), pp. 101–112.
[24] F.-F. Jin, An equatorial ocean recharge paradigm for ENSO. Part I: Conceptual model, J. Atmos. Sci., 54 (1997), pp. 811–829.
[25] F.-F. Jin, J. D. Neelin, and M. Ghil, El Nin͂o on the devil’s staircase: Annual subharmonic steps to chaos, Science, 264 (1994), pp. 70–72.
[26] I.-S. Kang and J.-S. Kug, An El-Nin͂o prediction system using an intermediate ocean and a statistical atmosphere, Geophys. Res. Lett., 27 (2000), pp. 1167–1170.
[27] H. Kaper and H. Engler, Mathematics and Climate, SIAM, Philadelphia, 2013. · Zbl 1285.86001
[28] A. Keane, A Dynamical Systems Approach to Understanding the Interplay between Delayed Feedback and Seasonal Forcing in the El Nin͂o Southern Oscillation, Ph.D. thesis, University of Auckland, Auckland, New Zealand, 2016.
[29] A. Keane, B. Krauskopf, and C. M. Postlethwaite, Delayed feedback versus seasonal forcing: Resonance phenomena in an El Nin͂o Southern Oscillation model, SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 1229–1257. · Zbl 1320.37038
[30] B. Krauskopf and K. Green, Computing unstable manifolds of periodic orbits in delay differential equations, J. Comput. Phys., 186 (2003), pp. 230–249. · Zbl 1017.65102
[31] B. Krauskopf and J. Sieber, Bifurcation analysis of delay-induced resonances of the El-Nin͂o Southern Oscillation, Philos. Trans. Roy. Soc. A, 470 (2014). · Zbl 1320.86004
[32] J.-S. Kug, I.-S. Kang, and S. E. Zebiak, The impacts of the model assimilated wind stress data in the initialization of an intermediate ocean and the ENSO predictability, Geophys. Res. Lett., 28 (2001), pp. 3713–3716.
[33] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1995. · Zbl 0829.58029
[34] M. Münnich, M. A. Cane, and S. E. Zebiak, A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: Nonlinear cases, J. Atmos. Sci., 48 (1991), pp. 1238–1248.
[35] M. Nakagawa, K. Tanaka, T. Nakashizuka, T. Ohkubo, T. Kato, T. Maeda, K. Sato, H. Miguchi, H. Nagamasu, K. Ogino, S. Teo, A. A. Hamid, and L. H. Seng, Impact of severe drought associated with the 1997–1998 El Nin͂o in a tropical forest in Sarawak, J. Tropical Ecology, 16 (2000), pp. 355–367.
[36] J. D. Neelin, D. S. Battisti, A. C. Hirst, F.-F. Jin, Y. Wakata, T. Yamagata, and S. E. Zebiak, ENSO theory, J. Geophys. Res. Oceans, 103 (1998), pp. 14261–14290.
[37] J. D. Neelin and M. Latif, El Nin͂o dynamics, Physics Today, 51 (1998), p. 32.
[38] J. Picaut, F. Masia, and Y. du Penhoat, An advective-reflective conceptual model for the oscillatory nature of the ENSO, Science, 277 (1997), pp. 663–666.
[39] R. A. Pielke Jr. and C. N. Landsea, La Nina, El Nin͂o and Atlantic hurricane damages in the United States, Bull. Amer. Meteor. Soc., 80 (1999), pp. 2027–2033.
[40] Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), pp. 189–197.
[41] D. A. Randall, General Circulation Model Development: Past, Present, and Future, Vol. 70, Academic Press, New York, 2000.
[42] D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in Numerical Continuation Methods for Dynamical Systems, B. Krauskopf, H. M. Osinga, and J. Galán-Vioque, eds., Springer, New York, 2007, pp. 359–399. · Zbl 1132.34001
[43] S. Saha, S. Nadiga, C. Thiaw, J. Wang, W. Wang, Q. Zhang, H. M. Van den Dool, H.-L. Pan, S. Moorthi, D. Behringer, D. Stokes, M. Pen͂a, S. Lord, G. White, W. Ebisuzaki, P. Peng, and P. Xie, The NCEP climate forecast system, J. Climate, 19 (2006), pp. 3483–3517.
[44] E. S. Sarachik and M. A. Cane, The El Nin͂o-Southern Oscillation Phenomenon, Cambridge University Press, Cambridge, 2010.
[45] J. Sieber, Finding Periodic Orbits in State-Dependent Delay Differential Equations as Roots of Algebraic Equations, preprint, arXiv:1010.2391, 2010.
[46] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey, and D. Roose, DDE-BIFTOOL Manual—Bifurcation Analysis of Delay Differential Equations, preprint, arXiv:1406.7144, 2014.
[47] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, Harlow, UK, 1989.
[48] L. Stone, P. I. Saparin, A. Huppert, and C. Price, El Nin͂o chaos: The role of noise and stochastic resonance on the ENSO cycle, Geophys. Res. Lett., 25 (1998), pp. 175–178.
[49] M. J. Suarez and P. S. Schopf, A delayed action oscillator for ENSO, J. Atmos. Sci., 45 (1988), pp. 3283–3287.
[50] E. Tziperman, M. A. Cane, and S. E. Zebiak, Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos, J. Atmos. Sci., 52 (1995), pp. 293–306.
[51] E. Tziperman, M. A. Cane, S. E. Zebiak, Y. Xue, and B. Blumenthal, Locking of El Nin͂o’s peak time to the end of the calendar year in the delayed oscillator picture of ENSO, J. Climate, 11 (1998).
[52] E. Tziperman, L. Stone, M. A. Cane, and H. Jarosh, El Nin͂o chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science, 264 (1994), pp. 72–73.
[53] H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, J. Differential Equations, 244 (2008), pp. 1910–1945. · Zbl 1146.34048
[54] C. Wang and J. Picaut, Understanding ENSO physics—A review, Earth’s Climate: The Ocean-Atmosphere Interaction, 147 (2004), pp. 21–48.
[55] C. Wang, R. H. Weisberg, and H. Yang, Effects of the wind speed-evaporation-SST feedback on the El Nin͂o-Southern Oscillation, J. Atmos. Sci., 56 (1999), pp. 1391–1403.
[56] S. Wieczorek, B. Krauskopf, and D. Lenstra, Unnested islands of period doublings in an injected semiconductor laser, Phys. Rev. E, 64 (2001), 056204.
[57] A. A. Williams and D. J. Karoly, Extreme fire weather in Australia and the impact of the El Nin͂o-Southern Oscillation, Aust. Meteor. Mag., 48 (1999), pp. 15–22.
[58] E. Winston, Uniqueness of solutions of state dependent delay differential equations, J. Math. Anal. Appl., 47 (1974), pp. 620–625. · Zbl 0286.34112
[59] S. Yanchuk and P. Perlikowski, Delay and periodicity, Phys. Rev. E, 79 (2009), 046221.
[60] I. Zaliapin and M. Ghil, A delay differential model of ENSO variability—Part 2: Phase locking, multiple solutions and dynamics of extrema, Nonlinear Process. Geophys., 17 (2010), pp. 123–135.
[61] S. E. Zebiak and M. A. Cane, A model El Nin͂o-Southern Oscillation, Monthly Weather Rev., 115 (1987), pp. 2262–2278.
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