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Estimating parameters of chaotic system with variational method

Cao Xiao-Qun Song Jun-Qiang Zhang Wei-Min Zhao Jun Zhang Li-Lun

Estimating parameters of chaotic system with variational method

Cao Xiao-Qun, Song Jun-Qiang, Zhang Wei-Min, Zhao Jun, Zhang Li-Lun
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  • In this paper a method is presented to estimate the unknown parameters of chaotic system based on the variational principle, which can be applied to all chaotic systems governed by the following equation:x= F(x,θ). Firstly,the equation of the chaotic system is included into the objective functional. Secondly, the universal formulas of the adjoint equation for chaotic systems and the functional gradient for unknown parameters are derived using the variational principle. Thirdly, the algorithm to estimate unknown parameters of chaotic system is designed according to above formulas. Finally, all unknown parameters of the typical Lorenz chaotic system and the hyperchaotic Chen system are estimated separately. Numerical simulations show that the effectiveness and the feasibility of the proposed method to estimate unknown parameters of chaotic systems.
    • Funds:
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    Liu F C, Liang XM 2005 Acta Phys. Sin. 54 4584 (in Chinese) [刘福才、梁晓明 2005 物理学报 54 4584]

    [2]

    Li L X, Peng H P, Lu H B, Guan X P 2001 Acta Phys. Sin. 50 629 (in Chinese) [李丽香、彭海朋、卢辉斌、关新平 2001 物理学报 50 629]

    [3]

    Wang X Y, Wu X J 2006 Acta Phys. Sin. 55 605 (in Chinese) [王兴元、武相军 2006 物理学报55 605]

    [4]

    Wu Z Q, Tan F X, Wang S X 2006 Acta Phys. Sin. 55 1651 (in Chinese) [吴忠强、谭拂晓、王绍仙 2006 物理学报 55 1651]

    [5]

    Tao C H, Lu J A, Lü J H 2002 Acta Phys. Sin. 51 1497 ( in Chinese) [陶朝海、陆君安、吕金虎 2002 物理学报 51 1497]

    [6]

    He M F, Mu Y M, Zhao L Z 2000 Acta Phys. Sin. 49 830 ( in Chinese) [贺明峰、穆云明、赵立中2000 物理学报 49 830 ]

    [7]

    Guan X P, Peng H P, Li L X, Wang X Q 2001 Acta Phys. Sin. 50 26 (in Chinese) [关新平、彭海朋、李丽香、王益群2001 物理学报 50 26 ]

    [8]

    Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴 栋、马西奎、李富才、尤 勇 2002 物理学报 51 2459]

    [9]

    Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高 飞、童恒庆 2006 物理学报 55 577]

    [10]

    Li L X, Peng H P, Yang Y X, Wang X D 2007 Acta Phys. Sin. 56 51 (in Chinese) [李丽香、彭海朋、 杨义先、王向东 2007 物理学报 56 51]

    [11]

    Wang J Y, Huang D X 2008 Acta Phys. Sin. 57 2755 (in Chinese) [王钧炎、黄德先 2008 物理学报 57 2755]

    [12]

    Huang S X, Wu R S 2001 Mathematical Physics Problems in Atmosphere Science (Beijing: Meteorology Press) (in Chinese) [黄思训、伍荣生 2001 大气科学中的数学物理问题(北京:气象出版社)]

    [13]

    Liu R W, Zhang H B, Chen L Q 2006 Chin. Phys. 15 249

    [14]

    Li G C, Mei F X 2006 Chin. Phys. 15 2496

    [15]

    Fu J L, Dai G D 2007 Chin. Phys. 16 570

    [16]

    Shi S Y, Fu J L, Chen L Q 2008 Chin. Phys. B 17 385

    [17]

    Cai G L, Huang J J 2006 Acta Phys. Sin. 55 3997 (in Chinese) [蔡国梁、黄娟娟 2006 物理学报 55 3997]

    [18]

    He J H 2008 Int. J. Modern. Phys. B 22 3487

    [19]

    He J H 1997 Int. J. Turbo. Jet-Eng. 14 23

    [20]

    He J H 2000 Appl. Math. Mech. 21 797

    [21]

    He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309

    [22]

    He J H, Lee E W M 2009 Phys. Lett. A 373 1644

    [23]

    Zheng C B, Liu B, Wang Z J, Zheng S K 2009 Int. J. Nonlin. Sci. Numer. 10 1369

    [24]

    Zheng C B, Liu B, Wang Z J, Zheng S K 2009 Int. J. Nonlin. Sci. Numer. 10 1523

  • [1]

    Liu F C, Liang XM 2005 Acta Phys. Sin. 54 4584 (in Chinese) [刘福才、梁晓明 2005 物理学报 54 4584]

    [2]

    Li L X, Peng H P, Lu H B, Guan X P 2001 Acta Phys. Sin. 50 629 (in Chinese) [李丽香、彭海朋、卢辉斌、关新平 2001 物理学报 50 629]

    [3]

    Wang X Y, Wu X J 2006 Acta Phys. Sin. 55 605 (in Chinese) [王兴元、武相军 2006 物理学报55 605]

    [4]

    Wu Z Q, Tan F X, Wang S X 2006 Acta Phys. Sin. 55 1651 (in Chinese) [吴忠强、谭拂晓、王绍仙 2006 物理学报 55 1651]

    [5]

    Tao C H, Lu J A, Lü J H 2002 Acta Phys. Sin. 51 1497 ( in Chinese) [陶朝海、陆君安、吕金虎 2002 物理学报 51 1497]

    [6]

    He M F, Mu Y M, Zhao L Z 2000 Acta Phys. Sin. 49 830 ( in Chinese) [贺明峰、穆云明、赵立中2000 物理学报 49 830 ]

    [7]

    Guan X P, Peng H P, Li L X, Wang X Q 2001 Acta Phys. Sin. 50 26 (in Chinese) [关新平、彭海朋、李丽香、王益群2001 物理学报 50 26 ]

    [8]

    Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴 栋、马西奎、李富才、尤 勇 2002 物理学报 51 2459]

    [9]

    Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高 飞、童恒庆 2006 物理学报 55 577]

    [10]

    Li L X, Peng H P, Yang Y X, Wang X D 2007 Acta Phys. Sin. 56 51 (in Chinese) [李丽香、彭海朋、 杨义先、王向东 2007 物理学报 56 51]

    [11]

    Wang J Y, Huang D X 2008 Acta Phys. Sin. 57 2755 (in Chinese) [王钧炎、黄德先 2008 物理学报 57 2755]

    [12]

    Huang S X, Wu R S 2001 Mathematical Physics Problems in Atmosphere Science (Beijing: Meteorology Press) (in Chinese) [黄思训、伍荣生 2001 大气科学中的数学物理问题(北京:气象出版社)]

    [13]

    Liu R W, Zhang H B, Chen L Q 2006 Chin. Phys. 15 249

    [14]

    Li G C, Mei F X 2006 Chin. Phys. 15 2496

    [15]

    Fu J L, Dai G D 2007 Chin. Phys. 16 570

    [16]

    Shi S Y, Fu J L, Chen L Q 2008 Chin. Phys. B 17 385

    [17]

    Cai G L, Huang J J 2006 Acta Phys. Sin. 55 3997 (in Chinese) [蔡国梁、黄娟娟 2006 物理学报 55 3997]

    [18]

    He J H 2008 Int. J. Modern. Phys. B 22 3487

    [19]

    He J H 1997 Int. J. Turbo. Jet-Eng. 14 23

    [20]

    He J H 2000 Appl. Math. Mech. 21 797

    [21]

    He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309

    [22]

    He J H, Lee E W M 2009 Phys. Lett. A 373 1644

    [23]

    Zheng C B, Liu B, Wang Z J, Zheng S K 2009 Int. J. Nonlin. Sci. Numer. 10 1369

    [24]

    Zheng C B, Liu B, Wang Z J, Zheng S K 2009 Int. J. Nonlin. Sci. Numer. 10 1523

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  • Received Date:  08 October 2010
  • Accepted Date:  18 October 2010
  • Published Online:  15 July 2011

Estimating parameters of chaotic system with variational method

  • 1. School of Computer Science, National University of Defense Technology, Changsha 410073, China

Abstract: In this paper a method is presented to estimate the unknown parameters of chaotic system based on the variational principle, which can be applied to all chaotic systems governed by the following equation:x= F(x,θ). Firstly,the equation of the chaotic system is included into the objective functional. Secondly, the universal formulas of the adjoint equation for chaotic systems and the functional gradient for unknown parameters are derived using the variational principle. Thirdly, the algorithm to estimate unknown parameters of chaotic system is designed according to above formulas. Finally, all unknown parameters of the typical Lorenz chaotic system and the hyperchaotic Chen system are estimated separately. Numerical simulations show that the effectiveness and the feasibility of the proposed method to estimate unknown parameters of chaotic systems.

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