分数阶混沌系统的Adomian分解法求解及其复杂性分析

贺少波; 孙克辉; 王会海

doi:10.7498/aps.63.030502

摘要

根据分数阶微分定义，采用Adomian分解算法，研究了分数阶简化Lorenz系统的数值解. 研究发现，该算法与预估-校正算法相比，求解结果更准确，所耗计算资源和内存资源更少，求解整数阶系统时较Runge-Kutta算法更准确；利用Adomian算法得到的分数阶简化Lorenz系统出现混沌的最小阶数为1.35，比利用预估-校正算法得到的最小阶2.79更小. 采用相图、分岔图分析了该系统的动力学特性，基于谱熵算法（SE）和C0算法分析了该系统的复杂度. 结果表明，复杂度结果和分岔图一致，说明系统的复杂度同样能反映出系统动力学特性；复杂度随阶数q的增加呈总体减小的趋势，而混沌态时系统参数c变化对系统复杂度影响不大. 为分数阶混沌系统应用于信息加密、保密通信领域提供了理论与实验依据.

关键词:

Abstract

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.

Keywords:

Adomian decomposition algorithm /
fractional-order simplified Lorenz system /
dynamical characteristic /
complexity

作者及机构信息

1.
中南大学物理与电子学院, 长沙 410083

基金项目: 国家自然科学基金（批准号：61161006，61073187）资助的课题.

Authors and contacts

1.
School of Physics and Electronics, Central South University, Changsha 410083, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61161006, 61073187).

参考文献

[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

施引文献

[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

[1]	吴朝俊, 方礼熠, 杨宁宁. 含有偏置电压源的非齐次分数阶忆阻混沌电路动力学分析与实验研究. , 2024, 73(1): 010501. doi: 10.7498/aps.73.20231211
[2]	梁华志, 张靖仪. 临界中性Gauss-Bonnet-anti-de Sitter黑洞复杂度演化. , 2021, 70(3): 030401. doi: 10.7498/aps.70.20201286
[3]	徐鑫鑫, 张毅. 分数阶非保守Lagrange系统的一类新型绝热不变量. , 2020, 69(22): 220401. doi: 10.7498/aps.69.20200488
[4]	余波, 何秋燕, 袁晓. 任意阶标度分形格分抗与非正则格型标度方程. , 2018, 67(7): 070202. doi: 10.7498/aps.67.20171671
[5]	阮静雅, 孙克辉, 牟俊. 基于忆阻器反馈的Lorenz超混沌系统及其电路实现. , 2016, 65(19): 190502. doi: 10.7498/aps.65.190502
[6]	娄正坤, 孙涛, 贺威, 杨建华. 基础激励下分数阶线性系统的响应特性分析. , 2016, 65(8): 084501. doi: 10.7498/aps.65.084501
[7]	高鹏, 王超, 支亚, 李旸, 王立玢, 丛正. 铝合金焊缝电涡流磁场信号的非线性特征提取及分类方法研究. , 2014, 63(13): 134103. doi: 10.7498/aps.63.134103
[8]	艾星星, 孙克辉, 贺少波, 王会海. 简化Lorenz多涡卷混沌吸引子的设计与应用. , 2014, 63(12): 120511. doi: 10.7498/aps.63.120511
[9]	刘泉, 李佩玥, 章明朝, 隋永新, 杨怀江. 一类具有Markov性质的混沌系统的构造. , 2013, 62(17): 170505. doi: 10.7498/aps.62.170505
[10]	陈卫东, 刘要龙, 朱奇光, 陈颖. 基于改进雁群PSO算法的模糊自适应扩展卡尔曼滤波的SLAM算法. , 2013, 62(17): 170506. doi: 10.7498/aps.62.170506
[11]	房超, 孙俊, 赖宇阳. 基于复杂性测度的高温气冷堆模拟机运行模式识别及诊断研究. , 2012, 61(17): 170515. doi: 10.7498/aps.61.170515
[12]	孙克辉, 贺少波, 尹林子, 阿地力·多力坤. 模糊熵算法在混沌序列复杂度分析中的应用. , 2012, 61(13): 130507. doi: 10.7498/aps.61.130507
[13]	王发强, 马西奎. 电感电流连续模式下Boost变换器的分数阶建模与仿真分析. , 2011, 60(7): 070506. doi: 10.7498/aps.60.070506
[14]	陈小军, 李赞, 白宝明, 蔡觉平. 一种确定混沌伪随机序列复杂度的模糊关系熵测度. , 2011, 60(6): 064215. doi: 10.7498/aps.60.064215
[15]	孙克辉, 贺少波, 盛利元. 基于强度统计算法的混沌序列复杂度分析. , 2011, 60(2): 020505. doi: 10.7498/aps.60.020505
[16]	孙克辉, 杨静利, 丁家峰, 盛利元. 单参数Lorenz混沌系统的电路设计与实现. , 2010, 59(12): 8385-8392. doi: 10.7498/aps.59.8385
[17]	孙克辉, 谈国强, 盛利元. TD-ERCS离散混沌伪随机序列的复杂性分析. , 2008, 57(6): 3359-3366. doi: 10.7498/aps.57.3359
[18]	庄建军, 宁新宝, 邹鸣, 孙飙, 杨希. 两种熵测度在量化射击运动员短时心率变异性信号复杂度上的一致性. , 2008, 57(5): 2805-2811. doi: 10.7498/aps.57.2805
[19]	何亮, 杜磊, 庄奕琪, 李伟华, 陈建平. 金属互连电迁移噪声的多尺度熵复杂度分析. , 2008, 57(10): 6545-6550. doi: 10.7498/aps.57.6545
[20]	常福宣, 陈　进, 黄　薇. 反常扩散与分数阶对流-扩散方程. , 2005, 54(3): 1113-1117. doi: 10.7498/aps.54.1113

计量

文章访问数: 7741
PDF下载量: 1173
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板