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The dependence of sensitivity on initial conditions is the essence of chaos. And the randomness of chaos originates from the high sensitivity to initial values, which is measured by the Lyapunov exponents. It is found in this paper that the cascade of chaotic systems can considerably improve the Lyapunov exponents of cascade chaos and other dynamic properties. Therefore, in this paper, we study the cascade of chaotic systems and the influence on dynamic performances of the cascade chaos, and we present the definition and conditions of chaotic system cascade. It is proved in theory that the Lyapunov exponent of cascade chaos system is a sum of Lyapunov exponents of cascade subsystems. Appropriate cascade for chaotic systems can increase system parameters and expand parameter regions of chaos mapping and full mapping, thereby enhancing initial condition sensitivity of chaotic map and security of chaotic pseudo-random sequences. For logistic map, cubic map and tent map, the dynamic characteristics of logistic-logistic, logistic-cubic and logistic-tent cascade are investigated in detail, verifying the improvements on dynamic characteristics of cascade chaos systems. The proposed chaotic cascade system can be used to generate better pseudo-random sequences for initial condition sensitivity and security.
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Keywords:
- chaos /
- cascade /
- discrete map /
- Lyapunov exponent
[1] Lorenz E N 1993 The Essence of Chaos (Washington: The University of Washington Press) p25
[2] Persohn K J, Povinelli R J 2012 Chaos, Solitons & Fractals 45 238
[3] Chen S L, Hwang T T, Lin W W 2010 IEEE Trans. Circ. Syst.-II: Express Briefs 57 996
[4] Jongsig Bae, Changha Hwang, Doobae Jun 2012 Statistics and Probability Letters 82 1021
[5] Maier M P S, Peacock-López E 2010 Physics Letters A 374 1028
[6] Sun K H, He S B, Yin L Z, A Di Li D L K 2012 Acta Phys. Sin. 61 130507 (in Chinese) [孙克辉, 贺少波, 尹林子, 阿地力·多力坤 2012 61 130507]
[7] Narendra Singh, Aloka Sinha 2010 Optics and Lasers in Engineering 48 398
[8] Martínez-Ñonthe J A, Castañeda-Solís A, Díaz-Méndez A, Cruz-Irisson M, Vázquez-Medina R 2012 Microelectronic Engineering 90 168
[9] Jovic B, Unsworth C P 2010 Electronics Letters 46 1
[10] Feng C F, Xu X J, Wu Z X Wang Y H 2008 Chinese Physics B 17 1674
[11] Young R M B, Read P L 2008 Physica D 237 2251
[12] Thomas Curtright, Andrzej Veitia 2011 Physics Letters A 375 276
[13] Levinsohn E A, Mendoza S A, Peacock-López E 2012 Chaos, Solitons & Fractals 45 426
[14] Wang X Y, Wang M J 2008 Acta Phys. Sin. 57 731 [王兴元, 王明军 2008 57 731]
[15] Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 10504 [孟继德, 包伯成, 徐强 2011 60 10504]
[16] Wang G Y, Yu J B, Gu T X 2011 Acta Phys. Sin. 50 2307 [王改云, 虞厥邦, 古天祥 2001 50 2307]
[17] Bao B C, Kang Z S, Xu J P 2009 Acta Phys. Sin. 58 1420 [包伯成, 康祝圣, 许建平, 胡文 2009 58 1420]
[18] Wei Y, Nan J, Tang G 2011 Czechoslovak Mathematical Journal 61 1023
[19] Ben Futter, Viktor Avrutin, Michael Schanz 2012 Chaos, Solitons & Fractals 45 465
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[1] Lorenz E N 1993 The Essence of Chaos (Washington: The University of Washington Press) p25
[2] Persohn K J, Povinelli R J 2012 Chaos, Solitons & Fractals 45 238
[3] Chen S L, Hwang T T, Lin W W 2010 IEEE Trans. Circ. Syst.-II: Express Briefs 57 996
[4] Jongsig Bae, Changha Hwang, Doobae Jun 2012 Statistics and Probability Letters 82 1021
[5] Maier M P S, Peacock-López E 2010 Physics Letters A 374 1028
[6] Sun K H, He S B, Yin L Z, A Di Li D L K 2012 Acta Phys. Sin. 61 130507 (in Chinese) [孙克辉, 贺少波, 尹林子, 阿地力·多力坤 2012 61 130507]
[7] Narendra Singh, Aloka Sinha 2010 Optics and Lasers in Engineering 48 398
[8] Martínez-Ñonthe J A, Castañeda-Solís A, Díaz-Méndez A, Cruz-Irisson M, Vázquez-Medina R 2012 Microelectronic Engineering 90 168
[9] Jovic B, Unsworth C P 2010 Electronics Letters 46 1
[10] Feng C F, Xu X J, Wu Z X Wang Y H 2008 Chinese Physics B 17 1674
[11] Young R M B, Read P L 2008 Physica D 237 2251
[12] Thomas Curtright, Andrzej Veitia 2011 Physics Letters A 375 276
[13] Levinsohn E A, Mendoza S A, Peacock-López E 2012 Chaos, Solitons & Fractals 45 426
[14] Wang X Y, Wang M J 2008 Acta Phys. Sin. 57 731 [王兴元, 王明军 2008 57 731]
[15] Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 10504 [孟继德, 包伯成, 徐强 2011 60 10504]
[16] Wang G Y, Yu J B, Gu T X 2011 Acta Phys. Sin. 50 2307 [王改云, 虞厥邦, 古天祥 2001 50 2307]
[17] Bao B C, Kang Z S, Xu J P 2009 Acta Phys. Sin. 58 1420 [包伯成, 康祝圣, 许建平, 胡文 2009 58 1420]
[18] Wei Y, Nan J, Tang G 2011 Czechoslovak Mathematical Journal 61 1023
[19] Ben Futter, Viktor Avrutin, Michael Schanz 2012 Chaos, Solitons & Fractals 45 465
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