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Lyapunov指数是系统是否进入混沌态的判据之一，其大小描述了系统混沌态的发达程度. 为了研究级联混沌系统Lyapunov指数的特性，揭示级联混沌系统中子系统之间的扰动机理，首先从伪噪声扰动的角度，建立了子系统间的扰动模型，研究了有、无外噪声影响的Lyapunov指数差异，指出子系统之间的扰动可视为伪噪声对其的影响；然后，在理论上证明了级联系统的Lyapunov指数等于有前级扰动时的各个子系统的Lyapunov指数的代数和，而不等于各个（独立）子系统的Lyapunov指数的代数和. 并以Logistic映射为例，设计了9种级联验证方案. 研究中发现了一些新的特性和现象：级联系统Lyapunov指数存在着随级次增加反而减小的“过犹不及”和“失之毫厘，差之千里”的现象；即使各个（独立）子系统均是混沌的，其级联后的系统也有不是混沌的情况；反之，即使各个（独立）子系统均不是混沌的，其级联后的系统也有混沌的情况；而且，级联后的系统是否是混沌的，与其构成级联系统的子系统的序有关. 最后，指出了级联级次对级联系统存在着利、弊两种影响，由此揭示了变参级联混沌系统存在的潜在风险. 研究结果为系统安全性、密钥（混乱度）质量的科学评价提供了重要的理论依据.
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关键词:
- 噪声 /
- 级联混沌 /
- 子系统 /
- Lyapunov 指数
Lyapunov index is one of criteria for testing whether the system is in a chaotic state, and its value represents the developed level of system chaotic state. To study the Lyapunov index characteristic of cascade chaotic system and reveal disturbance mechanism among subsystems in cascade chaotic system, the following researches are carried out. First, the disturbance model among subsystems is constructed from the viewpoint of pseudo noise disturbance, Lyapunov index difference between without and with external noise influence is investigated. Then the conclusion that disturbances among subsystems can be considered as pseudo noise influence is drawn. Second, the conclusion is proved that cascade system Lyapunov index is not the algebraic sum of each independent subsystems, but the one of each subsystems which consist of pre disturbances. Then taking the logistic representation for example, nine cascade systems are designed to prove this conclusion. And some novel characteristics and phenomena are found from the above investigations. They are (a) the phenomenon of “more is less”, that is, Lyapunov index will decrease with the increase of cascade levels, and the phenomenon of “A miss is as good as mile”; (b) even each independent subsystems is chaotic, the cascade system needs not to be chaotic; conversely, even each independent subsystems is not chaotic, the cascade system may be chaotic; (c) whether the cascade system is chaotic is associated with the order of subsystem. Finally, it is pointed out that cascade level has the influences of pros and cons on cascade system, thus revealing the latent hazard of parametric cascade chaotic system. The research result can provide important theoretic foundation for system security and the scientific evaluation of encryption keys.-
Keywords:
- noise /
- cascaded chaos /
- subsystem /
- Lyapunov exponent
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[22] Stachowiak T, Szydlowski M 2011 Physica D 240 1221
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[26] Shao K Y, Ma Y J, Wang T T, Liu Y H, Yang L, Gao H Y 2013 Acta Phys. Sin. 62 020514 (in Chinese) [邵克勇, 马永晶, 王婷婷, 刘远红, 杨莉, 高宏宇 2013 物理学报 62 020514]
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[1] Alvarez G 2005 Chaos Soliton. Fract. 26 7
[2] Wang X Y, Teng L 2012 Chin. Phys. B 21 020504
[3] Tong X J 2013 Commun. Ninlinear Sci. Numer. Simul. 18 1725
[4] Ahmed A, Abd E, Li L, Wang N, Han Q, Niu X M 2013 Sign. Process. 93 2986
[5] Jin J X, Qiu S S 2010 Acta Phys. Sin. 59 792 (in Chinese) [晋建秀, 丘水生 2010 物理学报 59 792]
[6] Zhu C X, Sun K H 2012 Acta Phys. Sin. 61 120503 (in Chinese) [朱从旭, 孙克辉 2012 物理学报 61 120503]
[7] Yuan Z X, Huang G H 2012 Chin. Phys. B 21 010502
[8] Wang X Y, He G X 2012 Chin. Phys. B 21 060502
[9] Luo Y L, Du M H 2013 Chin. Phys. B 22 080503
[10] Zhang C X, Yu S M 2010 Acta Phys. Sin. 59 3017 (in Chinese) [张朝霞, 禹思敏 2010 物理学报 59 3017]
[11] Jin J G, Chen C, Wei M J, Xia L C, Di Z G, Jia C R 2012 Comput. Engineer. 38 95 (in Chinese) [金建国, 陈晨, 魏明军,夏丽春, 贾春荣 2012 计算机工程 38 95]
[12] Li N, Li J F, Liu Y P 2010 Acta Phys. Sin. 59 5954 (in Chinese) [李农, 李建芬, 刘宇平 2010 物理学报 59 5954]
[13] Hu J F, Guo J B 2008 Acta Phys. Sin. 57 1477 (in Chinese) [胡进峰, 郭静波 2008 物理学报 57 1477]
[14] Chen Z, Zeng Y C, Fu Z J 2008 Acta Phys. Sin. 57 46 (in Chinese) [陈争, 曾以成, 付志坚 2008 物理学报 57 46]
[15] Huang F, Guan Z H 2005 Chaos Soliton. Fract. 23 851
[16] Yao C G, Zhao Q, Yu J 2013 Phys. Lett. A 377 370
[17] Choi S Y, Lee E K 1995 Phys. Lett. A 205 173
[18] Wang G Y, Yuan F 2013 Acta Phys. Sin. 62 020506 (in Chinese) [王光义, 袁方 2013 物理学报 62 020506]
[19] Cao H F, Zhang R X 2012 Acta Phys. Sin. 61 020508 (in Chinese) [曹鹤飞, 张若洵 2012 物理学报 61 020508]
[20] Jin J G, Lin R, Zhang Q L, Hou G Q, Di Z G, Jia C R 2009 Comput. Engineer. 35 137 (in Chinese) [金建国, 林瑞, 张庆凌, 侯国强, 邸志刚, 贾春荣 2009 计算机工程 35 137]
[21] Sekikawa M, Inaba N, Tsubouchi T, Aihara K 2012 Physica D 241 1169
[22] Stachowiak T, Szydlowski M 2011 Physica D 240 1221
[23] David R, Lai Y C 2000 Phys. Lett. A 270 308
[24] Zang H Y, Fan X B, Min L Q, Han D D 2012 Acta Phys. Sin. 61 200508 (in Chinese) [臧鸿雁, 范修斌, 闵乐泉, 韩丹丹 2012 物理学报 61 200508]
[25] Zhou X Y, Qiao X H, Zhu L, Liu S F 2013 Acta Phys. Sin. 62 190504 (in Chinese) [周小勇, 乔晓华, 朱雷, 刘素芬 2013 物理学报 62 190504]
[26] Shao K Y, Ma Y J, Wang T T, Liu Y H, Yang L, Gao H Y 2013 Acta Phys. Sin. 62 020514 (in Chinese) [邵克勇, 马永晶, 王婷婷, 刘远红, 杨莉, 高宏宇 2013 物理学报 62 020514]
[27] Jin J G, Wei M J, Di Z G, Xu G L, Jia C R, Zhao H W 2011 Comput. Engineer. 37 12 (in Chinese) [金建国, 魏明军, 邸志刚, 许广利, 贾春荣, 赵宏微 2011 计算机工程 37 12]
[28] Hao B L 1993 Advanced Series in Nonlinear Science Staring With Parbolas-An Introduction to Chaotic Dynamics (Vol.1) (Shanghai: Shanghai Scientific and Technolgical Education Press) pp122-125 (in Chinese) [郝柏林 1993 从抛物 线谈起–-混沌动力学引论 (第一版) (上海: 上海科学技术出版社) 第122–125页]
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