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Bifurcation analysis for a delayed sea-air oscillator coupling model for the ENSO

Xu Chang-Jin

Bifurcation analysis for a delayed sea-air oscillator coupling model for the ENSO

Xu Chang-Jin
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  • In this paper, a delayed sea-air oscillator coupling model for the ENSO is investigated. We obtain the sufficient condition of stability in equilibrium. By choosing delay η as a bifurcation parameter, we show that Hopf bifurcation can occur when delay η passes through a sequence of critical values. Meanwhile, based on the center manifold theory and the normal form approach, we derive the formula for determining the properties of Hopf bifurcating periodic orbit, such as the direction of Hopf bifurcation, the stability of Hopf bifurcating periodic solution and the periodic of Hopf bifurcating periodic solution. Finally, numerical simulations are carried out to illustrate the analytical results.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11261010), the Soft Science and Technology Program of Guizhou Province(Grant No. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province(Grant No. J[2012]2100), the Governor Foundation of Guizhou Province (2012), and the Doctoral Foundation of Guizhou University of Finance and Economics (2010).
    [1]

    Wang C Z 2001 J. Climate 14 98

    [2]

    Hassard B, Kazarinoff N, Wan Y H 1981 Theory and Application of Hopf Bifurcation (Combridge University Press)

    [3]

    Lin W T, Lin W T 2005 Chin. Phys. 14 875

    [4]

    Biondi F, Gershunov A, Cayan D R 2001 J. Climate 14 5

    [5]

    Kushnir Y, Robinson W A 2002 J. Climate 15 2233

    [6]

    Chao J P 1993 ENSO Dynamics (Beijing: China Meleorological Press) pp300-309) (in Chinese) [巢纪平 1993 厄尔尼诺和南方 涛动动力学(北京: 气象出版社) 第300—309页]

    [7]

    Graham N E and While W B 1990 J. Phys. Res. 96 3125

    [8]

    Lin W T, Mo J Q 2004 Chinese Science Bulletin 48 115

    [9]

    Zhu M, Liu W T, Lin Y H, Mo J Q 2011 Acta Phys. Sin. 60 339 (in Chinese) [朱敏, 林万涛, 林一华, 莫嘉琪 2011 物理学报 60 339]

    [10]

    Feng G L, Dong W J, Jia X J 2002 Acta Phys. Sin. 51 1181 (in chinese)[封国林, 董文杰, 贾小静 2002 物理学报 51 1181]

    [11]

    Mo J Q, Lin W T 2004 Acta Phys. Sin. 53 996 (in Chinese) [莫嘉琪, 林万涛 2004 物理学报 53 996]

    [12]

    Mo J Q, Wang H, Lin W T 2006 Acta Phys. Sin. 55 3229(in Chinese)[莫嘉琪, 王辉, 林万涛 2006 物理学报 55 3229]

    [13]

    Neelin J D, Battisti D S, Hirst A C 1998 J. Geophys. Res. 103 262

    [14]

    Wang C Z 2011 J. Climate 60 0205 (in Chinese) [王雯, 徐燕, 鲁世平 2001 物理学报 14 989]

    [15]

    Cooke K, Grossman Z 1982 J. Math. Anal. Appl. 86 592

    [16]

    Hale J, Lunel S V 1993 Introduction to Functional Differential Equations (New York: springer-Verlag)

  • [1]

    Wang C Z 2001 J. Climate 14 98

    [2]

    Hassard B, Kazarinoff N, Wan Y H 1981 Theory and Application of Hopf Bifurcation (Combridge University Press)

    [3]

    Lin W T, Lin W T 2005 Chin. Phys. 14 875

    [4]

    Biondi F, Gershunov A, Cayan D R 2001 J. Climate 14 5

    [5]

    Kushnir Y, Robinson W A 2002 J. Climate 15 2233

    [6]

    Chao J P 1993 ENSO Dynamics (Beijing: China Meleorological Press) pp300-309) (in Chinese) [巢纪平 1993 厄尔尼诺和南方 涛动动力学(北京: 气象出版社) 第300—309页]

    [7]

    Graham N E and While W B 1990 J. Phys. Res. 96 3125

    [8]

    Lin W T, Mo J Q 2004 Chinese Science Bulletin 48 115

    [9]

    Zhu M, Liu W T, Lin Y H, Mo J Q 2011 Acta Phys. Sin. 60 339 (in Chinese) [朱敏, 林万涛, 林一华, 莫嘉琪 2011 物理学报 60 339]

    [10]

    Feng G L, Dong W J, Jia X J 2002 Acta Phys. Sin. 51 1181 (in chinese)[封国林, 董文杰, 贾小静 2002 物理学报 51 1181]

    [11]

    Mo J Q, Lin W T 2004 Acta Phys. Sin. 53 996 (in Chinese) [莫嘉琪, 林万涛 2004 物理学报 53 996]

    [12]

    Mo J Q, Wang H, Lin W T 2006 Acta Phys. Sin. 55 3229(in Chinese)[莫嘉琪, 王辉, 林万涛 2006 物理学报 55 3229]

    [13]

    Neelin J D, Battisti D S, Hirst A C 1998 J. Geophys. Res. 103 262

    [14]

    Wang C Z 2011 J. Climate 60 0205 (in Chinese) [王雯, 徐燕, 鲁世平 2001 物理学报 14 989]

    [15]

    Cooke K, Grossman Z 1982 J. Math. Anal. Appl. 86 592

    [16]

    Hale J, Lunel S V 1993 Introduction to Functional Differential Equations (New York: springer-Verlag)

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  • Received Date:  19 May 2012
  • Accepted Date:  02 June 2012
  • Published Online:  20 November 2012

Bifurcation analysis for a delayed sea-air oscillator coupling model for the ENSO

  • 1. Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11261010), the Soft Science and Technology Program of Guizhou Province(Grant No. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province(Grant No. J[2012]2100), the Governor Foundation of Guizhou Province (2012), and the Doctoral Foundation of Guizhou University of Finance and Economics (2010).

Abstract: In this paper, a delayed sea-air oscillator coupling model for the ENSO is investigated. We obtain the sufficient condition of stability in equilibrium. By choosing delay η as a bifurcation parameter, we show that Hopf bifurcation can occur when delay η passes through a sequence of critical values. Meanwhile, based on the center manifold theory and the normal form approach, we derive the formula for determining the properties of Hopf bifurcating periodic orbit, such as the direction of Hopf bifurcation, the stability of Hopf bifurcating periodic solution and the periodic of Hopf bifurcating periodic solution. Finally, numerical simulations are carried out to illustrate the analytical results.

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