分数阶统一混沌系统动力学及其复杂度分析

摘要/Abstract

摘要： 基于Adomian分解算法、Lyapunov指数谱、分岔图和吸引子相图分析了分数阶统一混沌系统的复杂动力学特性,并揭示了系统状态随参数和微分阶数变化的规律以及系统走向混沌的道路。采用C₀算法和SampEn算法计算了分数阶统一混沌系统的复杂度。通过分析与最大Lyapunov指数谱的比较,发现复杂度的计算结果与最大Lyapunov指数谱结果在反应分数阶统一混沌系统的动力学特性方面具有较好的一致性,且C₀算法的分析结果优于SampEn算法的分析结果。最后,设计了基于统一混沌系统的伪随机序列发生器。测试结果表明,其可以通过全部NIST测试项目,这为分数阶统一混沌系统的实际应用奠定了实验基础。

关键词: Adomian分解算法, 分数阶微积分, 复杂度, 统一混沌系统, 伪随机序列

Abstract: Based on Adomian decomposition method (ADM),Lyapunov exponent spectrum,bifurcation diagram and attractor diagram,dynamics of the fractional-order unified chaotic system and the low of system state changing with its parameter and derivative order were analyzed,and the route from period state to chaos were observed.Moreover,complexity of fractional-order unified chaotic system was analyzed by means of C₀ algorithm and Sample entropy algorithm.Through comparative analysis with the maximum Lyapunov exponent spectrum,it shows that the complexity analysis results are well consistent with the results of the maximum Lyapunov exponent spectrum dynamics in reflecting the dynamics of the fractional-order unified chaotic system,and the results of C₀ algorithm is better than the results of Sample entropy algorithm.Finally,a pseudo random sequence generator was designed based on the fractional order unified chao-tic system.The test results show that it can pass all NIST tests,which laid an experimental basis for the practical applications of the fractional order unified chaotic system.

Key words: Adomian decomposition method, complexity, Fractional-order calculus, Pseudo-random sequence, Unified chaotic system

中图分类号:

TP0415

严波, 贺少波. 分数阶统一混沌系统动力学及其复杂度分析[J]. 计算机科学, 2019, 46(11A): 539-543. https://doi.org/

YAN Bo, HE Shao-bo. Dynamics and Complexity Analysis of Fractional-order Unified Chaotic System[J]. Computer Science, 2019, 46(11A): 539-543. https://doi.org/

参考文献

[1]KARAYER H,DEMIRHAN D,BÜYÜK F.Conformable fractional Nikiforov-Uvarov method[J].Communications in Theoretical Physics,2016,66(7):12-18.
[2]FENG Q H,MENG F W.Oscillation results for a fractional order dynamic equation on time scales with conformable fractional derivative[J].Advances in Difference Equations,2018.
[3]SOORKI M N,TAVAZOEI M S.Adaptive robust control offractional-order swarm systems in the presence of model uncertainties and external disturbances[J].IET Control Theory & Applications,2018,12(7):961-969.
[4]HE S B,SUN K H,WANG H H.Solution and dynamics analysis of a fractional-order hyper-chaotic system[J].Mathematical Methods in the Applied Sciences,2016,39(11):2965-2973.
[5]HE S B,SUN K H,MEI X Y,et al.Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative[J].European Physical Journal Plus,2017,132(1):36.
[6]孙克辉,杨静利,丘水生.分数阶混沌系统的仿真方法研究[J].系统仿真学报,2011,23(11):2361-2365.
[7]PAN I,DAS S.Frequency domain design of fractional order PID controller for AVR system using chaotic multi-objective optimization[J].International Journal of Electrical Power & Energy Systems,2013,51(10):106-118.
[8]WANG H H,SUN K H, HE S B.Characteristic analysis and DSP realization of fractional-order simplified Lorenz system based on Adomian decomposition method[J].International Journal of Bifurcation & Chaos,2015,25(6):1550085.
[9]ODIBAT Z,MOMANI S.Modified homotopy perturbationmethod:Application to quadratic Riccati differential equation of fractional order[J].Chaos Solitons & Fractals,2008,36(1):167-174.
[10]LIU C X,HUANG L L.Adomian decomposition method for detection of chaos in the Rucklige system[J].U.P.B.Scientific Bulletin,2015,77(3):299-306.
[11]HE S B,SUN K H,WANG H H.Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system[J].Entropy,2015,17(12):8299-8311.
[12]贺少波,孙克辉,王会海.分数阶混沌系统的Adomian分解法求解及其复杂性分析[J].物理学报,2014,63(3):50-57.
[13]陈小军,李赞,白宝明,等.一种基于模糊熵的混沌伪随机序列复杂度分析方法[J].电子与信息学报,2011,33(5):1198-1203.
[14]孙克辉,贺少波,何毅,等.混沌伪随机序列的谱熵复杂性分析[J].物理学报,2013,62(1):27-34.
[15]孙克辉,贺少波,朱从旭,等.基于C₀算法的混沌系统复杂度特性分析[J].电子学报,2013,41(9):1765-1771.
[16]LI N Q,PAN W,YAN L S,et al.Analysis of nonlinear dynamics and detecting messages embedded in chaotic carriers using sample entropy algorithm [J].Journal of the Optical Society of America B,2011,28(8):2018-2024.
[17]沈恩华,蔡志杰,顾凡及.C₀复杂度的数学基础[J].应用数学和力学,2005,26(9):1083-1090.
[18]RICHMAN J S,MOORMAN J R.Physiological time-series analysis using approximate entropy and sample entropy [J].American Journal of Physiology Heart & Circulatory Physiology,2000,278(6):2039-2049.
[19]XU Y,WANG H,LI Y,et al.Image encryption based on synchronization of fractional chaotic systems[J].Communications in Nonlinear Science & Numerical Simulation,2014,19(10):3735-3744.
[20]MARTÍNEZ-GUERRA R,GARCÍA J J M,PRIETO S M D.Secure communications via synchronization of Liouvillian chaotic systems[J].Journal of the Franklin Institute,2016,353(17):4384-4399.
[21]薛薇,许进康,贾红艳.一个分数阶超混沌系统同步及其保密通信研究[J].系统仿真学报,2016,28(8):1915-1928.
[22]简远鸣.TD-ERCS混沌伪随机序列发生器的DSP实现[D].长沙:中南大学,2008.
[23]丘嵘,王春雷.一个新的伪随机序列发生器的设计与实现[J].杭州电子科技大学学报,2012,32(3):5-8.
[24]马英杰,于航如.Tent混沌伪随机序列发生器设计与实现[J].北京电子科技学院学报,2015(4):61-64.
[25]陈关荣,吕金虎.Lorenz系统族的动力学分析、控制与同步[M].北京:科学出版社,2003.
[26]SHEN Z,LI J.Chaos control for a unified chaotic system using output feedback controllers[J].Mathematics & Computers in Simulation,2016,132:208-219.
[27]LIU L,MIAO S,HU H,et al.Pseudorandom bit generatorbased on non-stationary logistic maps[J].IET Information Security,2016,10(2):87-94.

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