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In this paper a method is proposed to estimate the unknown parameters of nonlinear map based on the second-order discrete variational method. On the basis of adjoint equations and gradient expressions of cost functional derived from variational method, the second-order adjoint equations and the exact formulas for the Hessian-vector product are educed using the second-order discrete variational method. A new algorithm is given for estimating the unknown parameters of nonlinear maps. The numerical simulations show that all unknown physical parameters of Hyperhenón map and two-dimensional parabolic map are estimated successfully and precisely. Simulation results also demonstrate the validity and advantages of second-order discrete variational method of estimating the parameters of the discrete chaotic systems.
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Keywords:
- nonlinear map /
- parameter estimation /
- second-order discrete variational method /
- adjoint equation
[1] Liu F C, Liang X M 2005 Acta Phys. Sin. 54 4584 (in Chinese) [刘福才, 梁晓明 2005 54 4584]
[2] Wang X Y, Wu X J 2006 Acta Phys. Sin. 55 605 (in Chinese) [王兴元, 武相军 2006 55 605]
[3] 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 55 1651]
[4] Maybhate A, Amritkar R E 1999 Phys. Rev. E 59 284
[5] Parlitz U 1996 Phys. Rev. Lett. 76 1232
[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 49 830]
[7] Guan X P, Peng H P, Li L X, Wang Y Q 2001 Acta Phys . Sin. 50 26 (in Chinese) [关新平, 彭海朋, 李丽香, 王益群2001 50 26]
[8] Cao X Q, Song J Q, Zhang W M, Zhao J, Zhang L L 2011 Acta Phys. Sin. 60 070511 (in Chinese) [曹小群, 宋君强, 张卫民, 赵军, 张理论 2011 60 070511]
[9] Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高飞, 童恒庆2006 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 56 51]
[11] Wang J Y, Huang D X 2008 Acta Phys. Sin. 57 2755 (in Chinese) [王钧炎, 黄德先 2008 57 2755]
[12] Long W, Jiao J J 2012 Acta Phys. Sin. 61 110507 (in Chinese) [龙文, 焦建军2012 61 110507]
[13] Cao X Q, Zhang W M, Song J Q, Zhu X Q, Zhao J 2012 Acta Phys. Sin. 61 020507 (in Chinese) [曹小群, 张卫民, 宋君强, 朱小谦, 赵军 2012 61 020507]
[14] 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)]
[15] He J H 2008 Int. J. Modern. Phys. B. 22 3487
[16] He J H 2000 Appl. Math. Mech. 21 797
[17] He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309
[18] He J H, Lee E W M 2009 Phys. Lett. A 373 1644
[19] Chen E D 2005 Ph. D. Dissertation (Dalian:Dalian University of Science and Technology) (in Chinese) [陈尔东 2005 博士学位论文 (大连:大连理工大学)]
[20] Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 010504 (in Chinese) [孟继德, 包伯成, 徐强 2011 60 010504]
[21] Kelley C T 1999 Iterative Methods for Optimization (Philadelphia:SIAM) p95
[22] Nocedal J, Wright S J 2006 Numerical Optimization (2nd Ed.) (Verlag, Berlin, Heidelberg, New York:Springer) p198
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[1] Liu F C, Liang X M 2005 Acta Phys. Sin. 54 4584 (in Chinese) [刘福才, 梁晓明 2005 54 4584]
[2] Wang X Y, Wu X J 2006 Acta Phys. Sin. 55 605 (in Chinese) [王兴元, 武相军 2006 55 605]
[3] 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 55 1651]
[4] Maybhate A, Amritkar R E 1999 Phys. Rev. E 59 284
[5] Parlitz U 1996 Phys. Rev. Lett. 76 1232
[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 49 830]
[7] Guan X P, Peng H P, Li L X, Wang Y Q 2001 Acta Phys . Sin. 50 26 (in Chinese) [关新平, 彭海朋, 李丽香, 王益群2001 50 26]
[8] Cao X Q, Song J Q, Zhang W M, Zhao J, Zhang L L 2011 Acta Phys. Sin. 60 070511 (in Chinese) [曹小群, 宋君强, 张卫民, 赵军, 张理论 2011 60 070511]
[9] Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高飞, 童恒庆2006 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 56 51]
[11] Wang J Y, Huang D X 2008 Acta Phys. Sin. 57 2755 (in Chinese) [王钧炎, 黄德先 2008 57 2755]
[12] Long W, Jiao J J 2012 Acta Phys. Sin. 61 110507 (in Chinese) [龙文, 焦建军2012 61 110507]
[13] Cao X Q, Zhang W M, Song J Q, Zhu X Q, Zhao J 2012 Acta Phys. Sin. 61 020507 (in Chinese) [曹小群, 张卫民, 宋君强, 朱小谦, 赵军 2012 61 020507]
[14] 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)]
[15] He J H 2008 Int. J. Modern. Phys. B. 22 3487
[16] He J H 2000 Appl. Math. Mech. 21 797
[17] He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309
[18] He J H, Lee E W M 2009 Phys. Lett. A 373 1644
[19] Chen E D 2005 Ph. D. Dissertation (Dalian:Dalian University of Science and Technology) (in Chinese) [陈尔东 2005 博士学位论文 (大连:大连理工大学)]
[20] Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 010504 (in Chinese) [孟继德, 包伯成, 徐强 2011 60 010504]
[21] Kelley C T 1999 Iterative Methods for Optimization (Philadelphia:SIAM) p95
[22] Nocedal J, Wright S J 2006 Numerical Optimization (2nd Ed.) (Verlag, Berlin, Heidelberg, New York:Springer) p198
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