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The saturation property of mean growth of initial error and the relation between saturation value and predictability limit of chaos system are studied in a frame of the nonlinear error growth dynamics. Firstly, the saturation property of mean relative growth of initial error (RGIE) of Lorenz96 system is investigated. It is found that there exists a simple linear relationship between the logarithm of saturation value of mean RGIE and initial error. The sum of logarithms of the two is constant that is independent of the magnitude of the initial error. It is proven by experiment that this conclusion is suitable for other chaotic systems too. With this conclusion, once the constant sum has been determined, the saturation values of mean RGIE at any magnitude of initial error can be calculated easily. Furthermore, to make the study of the relation between error growth saturation and the predictability limits more convenient, just as the definition of the mean RGIE, a definition of the mean absolute growth of initial error (AGIE) is introduced and theoretical analysis reveals that the AGIE has a similar saturation property as RGIE. The saturation value of mean AGIE is constant, which means for a given chaos system, once the control parameters of the system has been determined, the saturation of AGIE is determined. Finally a model for calculating predictability limit quantitatively is given as follows: Tp=1/∧ln(Es/δ0)+c, where Es is the saturation value of mean AGIE. It is shown that this model can work with complicated and high dimension chaos system very well.
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Keywords:
- relative growth of initial error /
- nonlinear error growth dynamics /
- chaos /
- predictability limit
[1] Lorenz E N 1969 Tellus 21 289
[2] Eckmann J P, Ruelle D 1985 Rev. Mod. Phys. 57 617
[3] Ding R Q, Li J P 2007 Phys. Lett. A 364 396
[4] Li J P, Ding R Q, Chen B H 2006 Frontier and Prospect of Atmospheric Sciences at the Beginning of the 21th Century (Beijing: China Meteorology Press) p96 (in Chinese) [李建平, 丁瑞强, 陈宝花 2006 21世纪大气科学发展的回顾与展望 (北京:气象出版社) 第96页]
[5] Chen B H, Li J P, Ding R Q 2006 Sci. China D 49 1111
[6] Ding R Q, Li J P 2007 Chin. J. Atmos. Sci. 31 571 (in Chinese) [丁瑞强, 李建平 2007 大气科学 31 571]
[7] Ding R Q, Li J P 2008 Acta Phys. Sin. 57 7494 (in Chinese) [丁瑞强, 李建平 2008 57 7494]
[8] Wolf A, Swift J B, Swinney H L 1985 Physica D 16 285
[9] Sano M, Sawada Y 1985 Phys. Rev. Lett. 55 1082
[10] Li J P, Ding R Q 2011 Relationship between the Predictability Limit and Initial Error in Chaotic System Esteban Tlelo-Cuautle (Ed.) 39-50
[11] Lorenz E N 1995 Proceedings of a Seminar Held at ECMWF on Predictability (Reading: ECMWF) p1
[12] Diego Pazo, Ivan G Szendro 2008 Phys. Rev. E 78 16209
[13] Lorenz E N 1963 J. Atmos. Sci. 20 130
[14] Henon M 1976 Comm. Math. Phys. 50 69
[15] Rossler O E 1976 Phys. Lett. A 57 397
[16] Hu Y H 2009 Sci. Tech. Engng. 11 2856 (in Chinese) [胡杨慧 2009 科学技术与工程 11 2856]
[17] Orrell D, Smith L A 2001 Nonlin. Proc. Geo. 8 357
[18] Orrell D 2003 J. Atmos. Sci. 60 2219
[19] Lv J H, Lu J A, Chen S H 2005 (in Chinese) [吕金虎, 陆君安, 陈士华 2005 混沌时间序列分析及其应用(第二版) (武昌:武汉大学出版社) 第27页]
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[1] Lorenz E N 1969 Tellus 21 289
[2] Eckmann J P, Ruelle D 1985 Rev. Mod. Phys. 57 617
[3] Ding R Q, Li J P 2007 Phys. Lett. A 364 396
[4] Li J P, Ding R Q, Chen B H 2006 Frontier and Prospect of Atmospheric Sciences at the Beginning of the 21th Century (Beijing: China Meteorology Press) p96 (in Chinese) [李建平, 丁瑞强, 陈宝花 2006 21世纪大气科学发展的回顾与展望 (北京:气象出版社) 第96页]
[5] Chen B H, Li J P, Ding R Q 2006 Sci. China D 49 1111
[6] Ding R Q, Li J P 2007 Chin. J. Atmos. Sci. 31 571 (in Chinese) [丁瑞强, 李建平 2007 大气科学 31 571]
[7] Ding R Q, Li J P 2008 Acta Phys. Sin. 57 7494 (in Chinese) [丁瑞强, 李建平 2008 57 7494]
[8] Wolf A, Swift J B, Swinney H L 1985 Physica D 16 285
[9] Sano M, Sawada Y 1985 Phys. Rev. Lett. 55 1082
[10] Li J P, Ding R Q 2011 Relationship between the Predictability Limit and Initial Error in Chaotic System Esteban Tlelo-Cuautle (Ed.) 39-50
[11] Lorenz E N 1995 Proceedings of a Seminar Held at ECMWF on Predictability (Reading: ECMWF) p1
[12] Diego Pazo, Ivan G Szendro 2008 Phys. Rev. E 78 16209
[13] Lorenz E N 1963 J. Atmos. Sci. 20 130
[14] Henon M 1976 Comm. Math. Phys. 50 69
[15] Rossler O E 1976 Phys. Lett. A 57 397
[16] Hu Y H 2009 Sci. Tech. Engng. 11 2856 (in Chinese) [胡杨慧 2009 科学技术与工程 11 2856]
[17] Orrell D, Smith L A 2001 Nonlin. Proc. Geo. 8 357
[18] Orrell D 2003 J. Atmos. Sci. 60 2219
[19] Lv J H, Lu J A, Chen S H 2005 (in Chinese) [吕金虎, 陆君安, 陈士华 2005 混沌时间序列分析及其应用(第二版) (武昌:武汉大学出版社) 第27页]
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