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Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical solution of the fractional-order simplified Lorenz system is investigated. Results show that compared with the Adams-Bashforth-Moulton algorithm, Adomian decomposition algorithm yields more accurate results and needs less computing as well as memory resources. It is even more accurate than Runge-Kutta algorithm when solving the integer order system. The minimum order of the simplified Lorenz system solved by using Adomian decomposition algorithm is 1.35, which is much smaller than 2.79 achieved by the Adams-Bashforth-Moulton algorithm. Dynamical characteristics of the system are studied by the phase diagram, bifurcation analysis, and complexities are calculated by employing the spectral entropy (SE) algorithm and C0 algorithm. Complexity results are consistent with the bifurcation diagrams, for which mean complexity can also reflect the dynamic characteristics of a chaotic system. Complexity decreases with increasing order q, and there are little influences on complexity versus changes of parameter c when the system is chaotic. It provides a theoretical and experimental basis for the application of fractional-order chaotic system in the field of encryption and secure communication.
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Keywords:
- Adomian decomposition algorithm /
- fractional-order simplified Lorenz system /
- dynamical characteristic /
- complexity
[1] Zhang X X, Qiu T S, Sheng H 2013 Acta Phys. Sin. 41 508 (in Chinese) [张旭秀, 邱天爽, 盛虎 2013 41 508]
[2] Zhao L D, Hu J B, Fang J A, Zhang W B 2012 Nonl. Dyn. 70 475
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[5] Daftardar-Gejji V, Bhalekar S 2010 Comp. Math. Appl. 59 1117
[6] Ge Z M, Ou C Y 2007 Chaos. Soli. Frac. 34 262
[7] Chen D, Zhang R, Sprott J C 2012 Nonl. Dyn. 70 1549
[8] Chen D, Liu Y, Ma X 2012 Nonl. Dyn. 67 893
[9] Wang Z, Huang X, Li Y X 2013 Chin. Phys. B 22 010504
[10] Diethelm K 1997 Elec. Trans. Numer. Anal. 5 1
[11] Sun H, Abdelwahab A, 1984 Onaral B IEEE Trans. Auto. Cont. 29 441
[12] Mohammed S T, Mohammad H 2008 Nonl. Anal. 69 1299
[13] Adomian G. 1984 J. Math. Anal. Appl. 102 420
[14] Cafagna D, Grassi G. 2008 Int. J. Bifur. Chaos 18 1845
[15] Cafagna D, Grassi G 2009 Int. J. Bifur. Chaos 19 339
[16] Gottwald G A, Melbourne I 2004 Proc. Roy. Soc. London. A: Math. Phys. Eng. Sci. 460 603
[17] Chen X J, Li Z, Bai B M 2011 J. Elec. Info. Tech. 33 1198 (in Chinese) [陈小军, 李赞, 白宝明 2011 电子与信息学报 33 1198]
[18] Sun K H, He S B, Sheng L Y 2011 Acta Phys. Sin. 60 20505 (in Chinese) [孙克辉, 贺少波, 盛利元 2011 60 20505]
[19] Sun K H, He S B, He Y 2013 Acta Phys. Sin. 62 10501 (in Chinese) [孙克辉, 贺少波, 何毅 2013 62 10501]
[20] Shen E H, Cai Z J, Gu F J 2005 Appl. Math. Mech. 26 1083 (in Chinese) [沈恩华, 蔡志杰, 顾凡及 2005 应用数学和力学 26 1083]
[21] Zhu C X, Zhou Y 2009 Cont. Deci. 24 161 (in Chinese) [朱呈祥, 邹云 2009 控制与决策 24 161]
[22] Liu S D, Shi S Y, Liu S S 2007 Meteor. Sci. Tech 35 15(in Chinese) [刘式达, 时少英, 刘式适 2007气象科技 35 15]
[23] Sun K, Wang X, Sprott J C 2010 Int. J. Bifur. Chaos 20 1209
[24] Abbaoui K, Cherruault Y 1994 Comp. Math. Appl. 28 103
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[1] Zhang X X, Qiu T S, Sheng H 2013 Acta Phys. Sin. 41 508 (in Chinese) [张旭秀, 邱天爽, 盛虎 2013 41 508]
[2] Zhao L D, Hu J B, Fang J A, Zhang W B 2012 Nonl. Dyn. 70 475
[3] Ke T D, Obukhovskii V, Wong N C 2013 Appl. Anal. 92 115
[4] Li C G, Chen G R 2004. Physica A: Stat. Mech. Appl. 341 55
[5] Daftardar-Gejji V, Bhalekar S 2010 Comp. Math. Appl. 59 1117
[6] Ge Z M, Ou C Y 2007 Chaos. Soli. Frac. 34 262
[7] Chen D, Zhang R, Sprott J C 2012 Nonl. Dyn. 70 1549
[8] Chen D, Liu Y, Ma X 2012 Nonl. Dyn. 67 893
[9] Wang Z, Huang X, Li Y X 2013 Chin. Phys. B 22 010504
[10] Diethelm K 1997 Elec. Trans. Numer. Anal. 5 1
[11] Sun H, Abdelwahab A, 1984 Onaral B IEEE Trans. Auto. Cont. 29 441
[12] Mohammed S T, Mohammad H 2008 Nonl. Anal. 69 1299
[13] Adomian G. 1984 J. Math. Anal. Appl. 102 420
[14] Cafagna D, Grassi G. 2008 Int. J. Bifur. Chaos 18 1845
[15] Cafagna D, Grassi G 2009 Int. J. Bifur. Chaos 19 339
[16] Gottwald G A, Melbourne I 2004 Proc. Roy. Soc. London. A: Math. Phys. Eng. Sci. 460 603
[17] Chen X J, Li Z, Bai B M 2011 J. Elec. Info. Tech. 33 1198 (in Chinese) [陈小军, 李赞, 白宝明 2011 电子与信息学报 33 1198]
[18] Sun K H, He S B, Sheng L Y 2011 Acta Phys. Sin. 60 20505 (in Chinese) [孙克辉, 贺少波, 盛利元 2011 60 20505]
[19] Sun K H, He S B, He Y 2013 Acta Phys. Sin. 62 10501 (in Chinese) [孙克辉, 贺少波, 何毅 2013 62 10501]
[20] Shen E H, Cai Z J, Gu F J 2005 Appl. Math. Mech. 26 1083 (in Chinese) [沈恩华, 蔡志杰, 顾凡及 2005 应用数学和力学 26 1083]
[21] Zhu C X, Zhou Y 2009 Cont. Deci. 24 161 (in Chinese) [朱呈祥, 邹云 2009 控制与决策 24 161]
[22] Liu S D, Shi S Y, Liu S S 2007 Meteor. Sci. Tech 35 15(in Chinese) [刘式达, 时少英, 刘式适 2007气象科技 35 15]
[23] Sun K, Wang X, Sprott J C 2010 Int. J. Bifur. Chaos 20 1209
[24] Abbaoui K, Cherruault Y 1994 Comp. Math. Appl. 28 103
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