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本文研究了一类具有时滞的厄尔尼诺-南方波涛动模型. 得到了该模型在平衡点稳定的充分条件.通过选择时滞 η为分岔参数,得到了当时滞 η 通过一系列的临界值时, Hopf分岔产生,然后,应用中心流形和规范型理论, 得到了确定Hopf分岔特性 (例如Hopf分岔方向和分岔周期解的稳定性以及Hopf分岔周期解的周期等) 的计算公式. 最后进行数值模拟验证了所得结果的正确性.
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关键词:
- 时滞 /
- 厄尔尼诺-南方波涛动模型 /
- Hopf分岔 /
- 稳定性
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.-
Keywords:
- delay /
- sea-air oscillator coupling model /
- Hopf bifurcation /
- stability
[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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