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In this paper, we analyze the stability of solution of the nonlinear function of physics, the Logistic function. It is found that the solution has a special character that it can change abruptly from one stable state to another when the initial value and parameters of function are selected. Abrupt change level and abrupt change rate are related to parameters of function, which can be described by defining the abrupt change intensity index. By using the character of solution, we build an ideal time series to imitate climate abrupt change in mean of climate system, investigate what behaviors the recovery rate and recovery force can have when the system approaches to a critical threshold, and to ascertain how it warns the abrupt change of the system early. Besides, we also find that even the system is disturbed by some noise signals, the recovery rate and recovery force also make an early response to the arrival of the abrupt change of system. Finally, the result of testing the Pacific Decadal Oscillation (PDO) index showes that the early warning of the abrupt change appeared in 1973, much more early than the abrupt change of PDO index happening in 1976/1977, which means that the recovery rate and recovery force can be used as the early warning signals of the abrupt change in mean.
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Keywords:
- nonlinear /
- changes in mean /
- recovery rate /
- recovery force
[1] Thom R 1972 Stability Structural and Morphogenesis (Sichuan: Sichuan Education Press) 73 105 (in Chinese) [雷内· 托姆 1972结构稳定性与形态发生学 (四川: 四川教育出版社) 73 105 ］
[2] Fu C B, Wang Q 1992 Scientia Atmospherica Sinica 4 482 (in Chinese) [符淙斌, 王强 1992 大气科学 4 482]
[3] Wei F Y 1999 Modern Climatic Statistical Diagnosis and Forecasting Technology (Beijing: China MeteorologicalPress) pp62-76 (in Chinese) [魏凤英 1999 现代气候统计诊断与预测技术 北京: 气象出版社 第62-76页]
[4] Feng G L, Gong Z Q, Dong W J 2005 Acta Phys. Sin. 54 5494 (in Chinese) [封国林, 龚志强, 董文杰 2005 54 5494]
[5] Wan S Q, Feng G L, Dong W J 2005 Acta Phys. Sin. 54 5487 (in Chinese) [万仕全, 封国林, 董文杰 2005 54 5487]
[6] Gong Z Q, Feng G L, Dong W J, Li J P 2006 Acta Phys. Sin. 55 3180 (in Chinese) [龚志强, 封国林, 董文杰, 李建平 2006 55 3180]
[7] Gong Z Q, Feng G L 2007 Acta Phys. Sin. 56 3619 (in Chinese) [龚志强, 封国林 2007 56 3619]
[8] Feng G L, Gong Z Q, Zhi R 2008 Acta Meteorol. Sin. 66 892 (in Chinese) [封国林, 龚志强, 支容 2008 气象学报 66 892]
[9] He W P, Wu Q, Zhang W, Wang Q G, Zhang Y 2009 Acta Phys. Sin. 58 2862 (in Chinese) [何文平, 吴琼, 张文, 王启光, 张勇 2009 58 2862]
[10] He W P, Deng B S, Wu Q, Zhang W, Cheng H Y 2010 Acta Phys. Sin. 59 8264 (in Chinese) [何文平, 邓北胜, 吴琼, 张文, 成海英 2010 59 8264]
[11] He W P, Wu Q, Cheng H Y, Zhang W 2011 Acta Phys. Sin. 60 029203 (in Chinese) [何文平, 吴琼, 程海英, 张文 2011 60 029203]
[12] He W P, Feng G L, Wu Q, He T, Wan S Q, Chou J F 2011 Int. J. Climatology DOI: 10.1002/joc.2367
[13] He W P, Feng G L, Wu Q, Wan S Q, Chou J F 2008 Nonlinear Processes in Geophysics 15 601
[14] Alley R B, Marotzke J, Nordhaus W D 2005 Science 299 5615
[15] Scheffer M, Bascompte J, William A 2009 Nature 461 53
[16] Carpenter S R, Brook W A 2006 Ecology Letters 9 311
[17] Guttal V, Jayaprakash C 2008 Ecology Letters 11 450
[18] May R 1976 Nature 261 459
[19] Mamtua N J, Mantua S R, Zhang H Y 1997 Bull. Amet. Meteor. Soc. 78 1069-1080
[20] Gu D J, Wang D X, Li C H 2003 Journal of Tropical Meteorology 19 136 (in Chinese) [谷德军, 王东晓, 李春辉 2003 热带气象学报 19 136]
[21] Gu W, Li C Y 2010 Transactions of Atmospheric Sciences 33 401 (in Chinese) [顾薇, 李崇银 2010 大气科学学报 33 401]
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[1] Thom R 1972 Stability Structural and Morphogenesis (Sichuan: Sichuan Education Press) 73 105 (in Chinese) [雷内· 托姆 1972结构稳定性与形态发生学 (四川: 四川教育出版社) 73 105 ］
[2] Fu C B, Wang Q 1992 Scientia Atmospherica Sinica 4 482 (in Chinese) [符淙斌, 王强 1992 大气科学 4 482]
[3] Wei F Y 1999 Modern Climatic Statistical Diagnosis and Forecasting Technology (Beijing: China MeteorologicalPress) pp62-76 (in Chinese) [魏凤英 1999 现代气候统计诊断与预测技术 北京: 气象出版社 第62-76页]
[4] Feng G L, Gong Z Q, Dong W J 2005 Acta Phys. Sin. 54 5494 (in Chinese) [封国林, 龚志强, 董文杰 2005 54 5494]
[5] Wan S Q, Feng G L, Dong W J 2005 Acta Phys. Sin. 54 5487 (in Chinese) [万仕全, 封国林, 董文杰 2005 54 5487]
[6] Gong Z Q, Feng G L, Dong W J, Li J P 2006 Acta Phys. Sin. 55 3180 (in Chinese) [龚志强, 封国林, 董文杰, 李建平 2006 55 3180]
[7] Gong Z Q, Feng G L 2007 Acta Phys. Sin. 56 3619 (in Chinese) [龚志强, 封国林 2007 56 3619]
[8] Feng G L, Gong Z Q, Zhi R 2008 Acta Meteorol. Sin. 66 892 (in Chinese) [封国林, 龚志强, 支容 2008 气象学报 66 892]
[9] He W P, Wu Q, Zhang W, Wang Q G, Zhang Y 2009 Acta Phys. Sin. 58 2862 (in Chinese) [何文平, 吴琼, 张文, 王启光, 张勇 2009 58 2862]
[10] He W P, Deng B S, Wu Q, Zhang W, Cheng H Y 2010 Acta Phys. Sin. 59 8264 (in Chinese) [何文平, 邓北胜, 吴琼, 张文, 成海英 2010 59 8264]
[11] He W P, Wu Q, Cheng H Y, Zhang W 2011 Acta Phys. Sin. 60 029203 (in Chinese) [何文平, 吴琼, 程海英, 张文 2011 60 029203]
[12] He W P, Feng G L, Wu Q, He T, Wan S Q, Chou J F 2011 Int. J. Climatology DOI: 10.1002/joc.2367
[13] He W P, Feng G L, Wu Q, Wan S Q, Chou J F 2008 Nonlinear Processes in Geophysics 15 601
[14] Alley R B, Marotzke J, Nordhaus W D 2005 Science 299 5615
[15] Scheffer M, Bascompte J, William A 2009 Nature 461 53
[16] Carpenter S R, Brook W A 2006 Ecology Letters 9 311
[17] Guttal V, Jayaprakash C 2008 Ecology Letters 11 450
[18] May R 1976 Nature 261 459
[19] Mamtua N J, Mantua S R, Zhang H Y 1997 Bull. Amet. Meteor. Soc. 78 1069-1080
[20] Gu D J, Wang D X, Li C H 2003 Journal of Tropical Meteorology 19 136 (in Chinese) [谷德军, 王东晓, 李春辉 2003 热带气象学报 19 136]
[21] Gu W, Li C Y 2010 Transactions of Atmospheric Sciences 33 401 (in Chinese) [顾薇, 李崇银 2010 大气科学学报 33 401]
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