﻿ 一个新五维超混沌电路及其在保密通讯中应用 A New Five Dimensional Hyper-Chaotic Circuit and Its Application in Secure Communication

Open Journal of Circuits and Systems
Vol.05 No.01(2016), Article ID:17165,11 pages
10.12677/OJCS.2016.51002

A New Five Dimensional Hyper-Chaotic Circuit and Its Application in Secure Communication

Zhichao Long, Dazhu Ma

School of Science, Hubei University for Nationalities, Enshi Hubei

Received: Feb. 29th, 2016; accepted: Mar. 15th, 2016; published: Mar. 18th, 2016

ABSTRACT

The dynamic behavior of a hyper-chaotic system is much more difficult to be predicted than that of a normal chaos system. Therefore, it becomes very useful in the secure communication. This paper constructed a new five dimensional hyper-chaotic circuit based on Chen system when the two state variables and an inverse controller are introduced. First, the stability of the fixed points and dynamic behavior of the phase space of the new system are discussed, and three positive Lyapunov exponents are found. Modular circuit of the system is designed. The results of circuit simulation are in agreement with the numerical simulation. Then chaos synchronism of the system is achieved with drive-response synchronization method. Numerical simulation of the secure communication process for the square wave signal is given, and chaotic masking method is used to realize the secure communication circuit with square wave voltage signal of the system. Finally, two ways to deal with secure communication are discussed; one is the chaotic signal mixed with the image, and the other is chaotic signal added to the digital image. It is shown that the latter is better than the former in the effect on secure communication, and is more suitable for information reversion.

Keywords:Chaotic Circuit, Hyperchaos, Lyapunov Exponent, Secure Communication

1. 引言

2. 控制Chen系统到五维超混沌系统

2.1. 模型

Chen系统 [33] 的状态方程为：

(1)

(2)

2.2. 系统基本特性

2.2.1. 定点稳定性

(3)

(4)

(5)

(6)

Figure 1. Phase space structure

2.2.2. 相体积

(7)

(8)

2.2.3. 李雅普诺夫指数

3. 五维超混沌系统的电路实验

Figure 2. Lyapunov exponents of the system

nx → x，ny → y，nz → z，nw → w，nu → u，压缩方程为：

(9)

(10)

4. 系统在保密通讯中的应用探究

4.1. 混沌同步

Figure 3. 5D hyperchaos circuit diagram

Figure 4. Circuit simulation results. The left refers x-y plane, the right refers x-z plane

Figure 5. Circuit simulation results. The left is y-z plane, the right is the z-u plane

(11)

(12)

4.2. 保密通讯数值模拟

Figure 6. Errors over with time

Figure 7. Numerical simulation results of square wave type signal

Figure 8. Encryption and decryption of chaos signal mixed with image

Figure 9. Encryption and decryption of chaos signal mixed with digital image

4.3. 保密通讯电路实现

Figure 10. The principle diagram of the chaotic secret communication

Figure 11. Circuit simulation results of square wave type signal

5. 结论

A New Five Dimensional Hyper-Chaotic Circuit and Its Application in Secure Communication[J]. 电路与系统, 2016, 05(01): 10-20. http://dx.doi.org/10.12677/OJCS.2016.51002

1. 1. 刘崇新. 分数阶混沌电路及应用[M]. 西安: 西安交通大学出版社, 2011: 1-20.

2. 2. Li, T.Y. and Yorke, J.A. (1975) Period Three Implies Chaos. American Mathematical Monthly, 82, 985-992. http://dx.doi.org/10.2307/2318254

3. 3. Wu, X., Huang, T.Y and Zhang, H. (2006) Lyapunov Indices with Two Nearby Trajectories in a Curved Spacetime. Physical Review D, 74, Article ID: 083001. http://dx.doi.org/10.1103/PhysRevD.74.083001

4. 4. Wu, X. and Xie, Y. (2007) Revisit on “Ruling Out Chaos in Compact Binary Systems”. Physical Review D, 76, Article ID: 124004. http://dx.doi.org/10.1103/PhysRevD.76.124004

5. 5. Wu, X. and Xie, Y. (2008) Resurvey of Order and Chaos in Spinning Compact Binaries. Physical Review D, 77, Article ID: 103012. http://dx.doi.org/10.1103/PhysRevD.77.103012

6. 6. Huang, G., Ni, X. and Wu, X. (2014) Chaos in Two Black Holes with Next-to-Leading Order Spin-Spin Interactions. The European Physical Journal C, 74, 1-8. http://dx.doi.org/10.1140/epjc/s10052-014-3012-2

7. 7. Huang, G. and Wu, X. (2014) Dynamics of the Post-Newtonian Circular Restricted Three-Body Problem with Compact Objects. Physical Review D, 89, Article ID: 124034. http://dx.doi.org/10.1103/PhysRevD.89.124034

8. 8. Wu, X., Mei, L. and Huang, G. (2015) Analytical and Numerical Studies on Differences between Lagrangian and Hamiltonian Approaches at the Same Post-Newtonian Order. Physical Review D, 91, Article ID: 024042. http://dx.doi.org/10.1103/physrevd.91.024042

9. 9. Mei, L., Ju, M. and Wu, X. (2013) Dynamics of Spin Effects of Compact Binaries. Monthly Notices of the Royal Astronomical Society, 435, 2246-2255. http://dx.doi.org/10.1093/mnras/stt1441

10. 10. Wu, X. and Huang, G. (2015) Ruling out Chaos in Comparable Mass Compact Binary Systems with One Body Spinning. Monthly Notices of the Royal Astronomical Society, 452, 3167-3178. http://dx.doi.org/10.1093/mnras/stv1485

11. 11. Wu, X. and Huang, T. (2003) Computation of Lyapunov Exponents in General Relativity. Physics Letters A, 313, 77-81. http://dx.doi.org/10.1016/S0375-9601(03)00720-5

12. 12. 伍歆, 黄天衣. 判定轨道混沌的几个指标[J]. 天文学进展, 2006, 23(4): 318-330.

13. 13. Benettin, G., Galgani, L. and Giorgilli, A. (1980) Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them. Part 1: Theory. Meccanica, 15, 9-20. http://dx.doi.org/10.1007/bf02128236

14. 14. Froeschle, C., Lega, E. and Gonczi, R. (1997) Fast Lyapunov Indicators. Application to Asteroidal Motion. Celestial Mechanics and Dynamical Astronomy, 67, 41-62. http://dx.doi.org/10.1023/A:1008276418601

15. 15. Ott, E., Grebogi, C. and Yorke, J.A. (1990) Controlling Chaos. Physical Review Letters, 64, 1196-1199. http://dx.doi.org/10.1103/PhysRevLett.64.1196

16. 16. Hübler, A. (1998) Adaptive Control of Chaotic Systems with Uncertainties. International Journal of Bifurcation & Chaos, 8, 2041-2046. http://dx.doi.org/10.1142/S0218127498001698

17. 17. 余思敏. 混沌系统与混沌电路原理设计及其在通讯中的应用[M]. 西安: 西安电子科技大学出版社, 2011: 4-50.

18. 18. Huang, G.Q. and Wu, X. (2012) Analysis of New Four-Dimensional Chaotic Circuits with Experimental and Numerical Methods. International Journal of Bifurcation and Chaos, 22, 1250042-1250055. http://dx.doi.org/10.1142/S0218127412500423

19. 19. Luo, X.S., Fang, J.Q. and Wang, L.H. (1999) A New Strategy of Chaos Control and a United Mechanism for Several Kinds of Chaos Control Methods. Acta Physica Sinica, 8, 895-901.

20. 20. Boccaletti, S., Grebogi, C., Lai, Y., Mancini, H. and Maza, D. (2000) The Control of Chaos: Theory and Applications. Physics Reports, 329, 103-197. http://dx.doi.org/10.1016/S0370-1573(99)00096-4

21. 21. Rössler, O.E. (1979) An Equation for Hyperchaos. Physics Letters A, 71, 155-157. http://dx.doi.org/10.1016/0375-9601(79)90150-6

22. 22. Thamilmaran, K., Lakshmanan, M. and Venkatesan, A. (2004) A Hyperchaos in a Modified Canonical Chua’s Circuit. International Journal of Bifurcation and Chaos, 14, 221-243. http://dx.doi.org/10.1142/S0218127404009119

23. 23. Yeh, K., Chen, C.W. and Hsiwng, T.K. (2005) Fuzzy Control for Seismically Excited Bridges with Sliding Bearing Isolation. Advanced Intelligent Computing Theories and Applications, 4681, 473-483.

24. 24. Edward, O. (1993) Chaos in Dynamical Systems. Cambridge University Press, Cambridge.

25. 25. 谌龙, 彭海军, 王德石. 一类多涡卷混沌系统构造方法研究[J]. 物理学报, 2008, 57(6): 3337-3341.

26. 26. Chen, G., Moiola, J.L. and Wanf, H.O. ( 2000) Bifurcation Control: Theories, Methods, and Applica-tions. International Journal of Bifurcation and Chaos, 10, 511-548. http://dx.doi.org/10.1142/S0218127400000360

27. 27. 王光义, 郑艳, 刘敬彪. 一个超混沌Lorenz吸引子及其电路实现[J]. 物理学报, 2007(6): 3113-08.

28. 28. Chen, G. and Ueta, T. (1999) Yet Another Chaotic Attractor. Interna-tional Journal of Bifurcation and Chaos, 9, 1465- 1466. http://dx.doi.org/10.1142/S0218127499001024

29. 29. Li, Y., Tang, W.K.S. and Chen, G. (2005) Hyperchaos Evolved from the Generalized Lorenz Equation. International Journal of Circuit Theory and Applications, 33, 234-251. http://dx.doi.org/10.1002/cta.318

30. 30. Li, Y.X., Tang, W.K.S. and Chen, G.R. (2005) Generating Hyper-Chaos via State Feedback Control. International Journal of Bifurcation and Chaos, 15, 3367-3375. http://dx.doi.org/10.1142/S0218127405013988

31. 31. Chen, A., Lu, J., Lü, J. and Yu, S. (2006) Generating hyperchaotic Lü attractor via state feedback control. Physica A, 364, 103-110. http://dx.doi.org/10.1016/j.physa.2005.09.039

32. 32. Wang, G., Zhang, X., Zheng, Y. and Li, Y. (2006) A new modified hyperchaotic Lü system. Physica A, 371, 260-272. http://dx.doi.org/10.1016/j.physa.2006.03.048

33. 33. Wang, B. and Guan, Z.H. (2010) Chaos Synchronization in General Complex Dynamical Networks with Coupling Delays. Nonlinear Analysis Real World Applications, 11, 1925-1932. http://dx.doi.org/10.1016/j.nonrwa.2009.04.020