Dynamic study of a new five-dimensional conservative hyperchaotic system with wide parameter range

Zhang Ze-Feng; Huang Li-Lian; Xiang Jian-Hong; Liu Shuai

doi:10.7498/aps.70.20210592

Abstract

Conservative systems have no attractors. Therefore, compared with common dissipative systems, conservative systems have good ergodicity, strong pseudo-randomness and high security performance, thereby making them more suitable for applications in chaotic secure communication and other fields. Owing to these features, a new five-dimensional conservative hyperchaotic system with a wide parameter range is designed. Firstly, the Hamiltonian energy and Casimir energy are analyzed, showing that the new system satisfies the Hamiltonian energy conservation and can generate chaos. Next, the dynamic analysis is carried out, including conservativeness proof, equilibrium point analysis, Lyapunov exponential spectrum, and bifurcation diagrams analysis, thereby proving that the new system has the characteristics of conservative system and can always maintain a hyperchaotic state in a wide parameter range. At the same time, the phase diagram and Poincaré section diagram of the new system in a wide parameter range are compared. The results show that the randomness and ergodicity of the system are enhanced with the increase of parameters. Then, the NIST test shows that the chaotic random sequences generated by the new system in a wide parameter range have strong pseudo-randomness. Finally, the circuit simulation and hardware circuit experiment of the conservative hyperchaotic system are carried out, which proves that the new system has good ergodicity and realizability.

Keywords:

Authors and contacts

1.
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
2.
MIIT Key Laboratory of Advanced Marine Communication and Information Technology, Harbin Engineering University, Harbin 150001, China

Corresponding author: Huang Li-Lian, lilian_huang@163.com ; Xiang Jian-Hong, xiangjianhong@hrbeu.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61203004), the Natural Science Foundation of Heilongjiang Province, China (Grant No. F201220), and the Joint Guidance Project of the Natural Science Foundation of Heilongjiang Province, China (Grant No. LH2020 F022)

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References

[1]	禹思敏, 吕金虎, 李澄清 2016 电子与信息学报 38 735 Yu S M, Lü J H, Li C Q 2016 J. Elec. Info. Tech. 38 735
[2]	Lorenz E N 1963 J. Atmos. Sci. 20 130
[3]	Sprott J C 1994 Phys. Rev. E 50 647 Google Scholar
[4]	Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465 Google Scholar
[5]	Lü J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659 Google Scholar
[6]	Dang X Y, Li C B, Bao B C, Wu H G 2015 Chin. Phys. B 24 050503 Google Scholar
[7]	鲜永菊, 扶坤荣, 徐昌彪 2021 振动与冲击 40 15 Xian Y J, Fu K R, Xu C B 2021 J. Vib. Shock. 40 15
[8]	Wang G Y, Yuan F, Chen G R, Zhang Y 2018 Chaos 28 013125 Google Scholar
[9]	Akhshani A, Akhavan A, Mobaraki A, Lim S C, Hassan Z 2014 Commun. Nonlinear Sci. Numer. Simul. 19 101 Google Scholar
[10]	Kadhim A F, Mhaibes H I 2018 Int. J. Appl. Eng. Res. 13 2141
[11]	Qi G Y 2018 Nonlinear Dyn. 95 2063 Google Scholar
[12]	Vaidyanathan S, Volos C 2015 Arch. Control Sci. 25 333 Google Scholar
[13]	Cang S J, Wu A G, Wang Z H, Chen Z Q 2017 Nonlinear Dyn. 89 2495 Google Scholar
[14]	Dong E Z, Yuan M F, Du S Z, Chen Z Q 2019 Appl. Math. Model. 73 40 Google Scholar
[15]	Gu S Q, Du B X, Wan Y J 2020 Int. J. Bifurcation Chaos 30 2050242 Google Scholar
[16]	Chen M, Wang C, Wu H G, Xu Q, Bao B C 2021 Nonlinear Dyn. 103 643 Google Scholar
[17]	Bouteghrine B, Tanougast C, Sadoudi S 2021 J. Circuits Syst. Comput. 30 2150280 Google Scholar
[18]	Lin Z S, Wang G Y, Wang X Y, Yu S M, Lü J H 2018 Nonlinear Dyn. 94 1003 Google Scholar
[19]	徐昌彪, 钟德, 夏诚, 黎周 2019 振动与冲击 38 125 Xu C B, Zhong D, Xia C, Li Z 2019 J. Vib. Shock. 38 125
[20]	Barboza R 2007 Int. J. Bifurcation Chaos 17 4285 Google Scholar
[21]	贾红艳, 陈增强, 袁著祉 2009 物理学报 58 4469 Google Scholar Jia H Y, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin. 58 4469 Google Scholar
[22]	Liu J M, Zhang W 2013 Optik 124 5528 Google Scholar
[23]	Xian Y J, Xia C, Guo T T, Fu K R, Xu C B 2018 Results Phys. 11 368 Google Scholar
[24]	徐昌彪, 黎周 2019 浙江大学学报(工学版) 53 1552 Google Scholar Xu C B, Li Z 2019 J. Zhejiang Univ. (Eng. Sci.) 53 1552 Google Scholar
[25]	Sprott J C 2011 Int. J. Bifurcation Chaos 21 2391 Google Scholar
[26]	Dong E Z, Jiao X D, Du S Z, Chen Z Q, Qi G Y 2020 Complexity 2020 4627597 Google Scholar
[27]	Lakshmanan M, Rajasekar S 2003 Nonlinear Dynamics (Berlin Heidelberg: Springer-Verlag) pp191–238
[28]	Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285 Google Scholar
[29]	包伯成 2013 混沌电路导论 (北京: 科学出版社) 第45—49页 Bao B C 2013 An Introduction to Chaotic Circuits (Beijing: Science Press) pp45−49 (in Chinese)
[30]	Ramasubramanian K, Sriram M S 2000 Physica D 139 72 Google Scholar
[31]	Rukhin A, Soto J, Nechvatal J, Smid M, Barker E, Leigh S, Levenson M, Vangel M, Banks D, Heckert A, Dray J, Vo S 2021https://csrc.nist.gov/publications/detail/sp/800-22/rev-1 a/final [2021-06-26]
[32]	禹思敏 2011 混沌系统与混沌电路 (西安: 西安电子科技大学出版社) 第293—312页 Yu S M 2011 Chaotic Systems and Chaotic Circuits (Xi’an: Xidian University Press) pp293−312 (in Chinese)

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图 1 数值仿真, 红色部分为系统${\varSigma _1}$, 蓝色部分为系统$\varSigma _1^{\rm{H}}$　(a)时域波形; (b)Casimir功率; (c)${x_1}{\rm{ \text- }}{x_4}{\rm{ \text- }}{x_5}$相图; (d)${x_1}{\rm{ \text- }}{x_4}$相图

Figure 1. Numerical simulation, the red part is system${\varSigma_1}$, the blue part is system$\varSigma _1^{\rm{H}}$: (a) Time domain waveforms; (b) Casimir power; (c)${x_1}{\rm{ \text- }}{x_4}{\rm{\text-}}{x_5}$plane; (d)${x_1}{\rm{ \text- }}{x_4}$plane.

DownLoad: Full-Size Img PowerPoint

图 2 数值仿真　(a)系统$\varSigma_1^{\rm{H}}$的Lyapunov指数谱; (b)系统$\varSigma_1^{\rm{H}}$的分岔图

Figure 2. Numerical simulation: (a) Lyapunov exponent spectrum of system$\varSigma _1^{\rm{H}}$; (b) bifurcation diagram of system$\varSigma _1^{ {{\rm{H}}}}$

DownLoad: Full-Size Img PowerPoint

图 3 系统$\varSigma _1^{\rm{H}}$的相图和Poincaré截面图　(a)$ {\varPi _2} = 2 $时的相图; (b)$ {\varPi _2} = 2 $时的Poincaré截面图; (c)$ {\varPi _2} = 500 $时的相图; (d)$ {\varPi _2} = 500 $时的Poincaré截面图; (e)$ {\varPi _2} = 1000 $时的相图; (f)$ {\varPi _2} = 1000 $时的Poincaré截面图; (g)$ {\varPi _2} = 1500 $时的相图; (h)$ {\varPi _2} = 1500 $时的Poincaré截面图

Figure 3. Phase diagrams and Poincaré maps of system$\varSigma _1^{\rm{H}}$: (a) The phase diagram when$ {\varPi _2} = 2 $; (b) the Poincaré map when$ {\varPi _2} = 2 $; (c) the phase diagram when$ {\varPi _2} = 500 $; (d) the Poincaré map when$ {\varPi _2} = 500 $; (e) the phase diagram when$ {\varPi _2} = 1000 $; (f) the Poincaré map when$ {\varPi _2} = 1000 $; (g) the phase diagram when$ {\varPi _2} = 1500 $; (h) the Poincaré map when $ {\varPi _2} = 1500 $.

DownLoad: Full-Size Img PowerPoint

图 4 系统$\varSigma_1^{\rm{H}}$在$ {\varPi _2} = 2 $的相图　(a)${x_2}{\rm{ \text- }}{x_5}$平面; (b)${x_3}{\rm{ \text- }}{x_5}$平面

Figure 4. Phase diagrams of system$\varSigma_1^{\rm{H}}$when$ {\varPi _2} = 2 $: (a) ${x_2}{\rm{ \text- }}{x_5}$plane; (b) ${x_3}{\rm{ \text- }}{x_5}$plane.

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图 5 系统$\varSigma_1^{\rm{H}}$非重叠模块匹配检验的P-values的直方图

Figure 5. P-values histogram of non-overlapping template matching test of system$\varSigma _1^{\rm{H}}$.

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图 6 系统$\varSigma_1^{\rm{H}}$的电路设计

Figure 6. Circuit design of system$\varSigma_1^{\rm{H}}$.

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图 7 系统$\varSigma _1^{\rm{H}}$的仿真电路　(a)${x_2}{\rm{ \text- }}{x_5}$相图; (b)${x_3}{\rm{ \text- }}{x_5}$相图

Figure 7. Simulation circuit of system$\varSigma _1^{\rm{H}}$: (a) ${x_2}{\rm{ \text- }}{x_5}$plane; (b) ${x_3}{\rm{ \text- }}{x_5}$plane.

DownLoad: Full-Size Img PowerPoint

图 8 系统$\varSigma _1^{\rm{H}}$的实际电路　(a)硬件实验电路; (b)实验结果

Figure 8. Actual circuit of system$\varSigma _1^{\rm{H}}$: (a) Hardware experimental circuit; (b) experimental result.

DownLoad: Full-Size Img PowerPoint

表 1 系统$\varSigma_1^{\rm{H}}$的平衡点和特征值

Table 1. Equilibrium points and characteristic values of system$\varSigma _1^{\rm{H}}$.

系统	平衡点($ {k_1}, {k_4} \in \mathbb{R} $)	特征值($ \sigma , \omega \in {\mathbb{R}^ + } $)	平衡点类型
$\varSigma _1^{\rm{H} }$	$ (0, 0, 0, 0, 0) $	$ (0, {\rm{j}}{\omega _1}, - {\rm{j}}{\omega _1}, {\rm{j}}{\omega _2}, - {\rm{j}}{\omega _2}) $	中心点
		$ (0, {\sigma _1}, - {\sigma _2}, {\sigma _3} + {\rm{j}}\omega , {\sigma _3} - {\rm{j}}\omega ) $	鞍焦点
	$ ({k_1}, 0, 0, 0, 0) $	$ (0, {\sigma _1}, - {\sigma _2}, - {\sigma _3} + {\rm{j}}\omega , - {\sigma _3} - {\rm{j}}\omega ) $	鞍焦点
		$ (0, \sigma + {\rm{j}}{\omega _1}, \sigma - {\rm{j}}{\omega _1}, - \sigma + {\rm{j}}{\omega _2}, - \sigma - {\rm{j}}{\omega _2}) $	鞍焦点
		$ (0, {\sigma _1}, {\sigma _2}, - {\sigma _3}, - {\sigma _4}) $	鞍焦点
	$ (\sqrt 2 , 0, \sqrt 2 {k_4}, {k_4}, 0) $	$ (0, 0, - {\sigma _1}, {\sigma _2} + {\rm{j}}\omega , {\sigma _2} - {\rm{j}}\omega ) $	鞍焦点
		$ (0, \sigma + {\rm{j}}{\omega _1}, \sigma - {\rm{j}}{\omega _1}, - \sigma + {\rm{j}}{\omega _2}, - \sigma - {\rm{j}}{\omega _2}) $	鞍焦点
		$ (0, {\sigma _1}, {\sigma _2}, - {\sigma _3}, - {\sigma _4}) $	鞍焦点
	$ ( - \sqrt 2 , 0, - \sqrt 2 {k_4}, {k_4}, 0) $	$ (0, 0, {\sigma _1}, - {\sigma _2} + {\rm{j}}\omega , - {\sigma _2} - {\rm{j}}\omega ) $	鞍焦点
		$ (0, \sigma + {\rm{j}}{\omega _1}, \sigma - {\rm{j}}{\omega _1}, - \sigma + {\rm{j}}{\omega _2}, - \sigma - {\rm{j}}{\omega _2}) $	鞍焦点

DownLoad: CSV

表 2 系统$\varSigma _1^{\rm{H}}$的NIST测试结果

Table 2. NIST test results of System$\varSigma _1^{\rm{H}}$.

No.	Statistical test	P-value	Proportion
1	Frequency	0.759756	0.99
2	Block frequency	0.494392	1.00
3	Cumulative sums	0.595549	1.00
4	Runs	0.867692	0.99
5	Longest run	0.102526	0.98
6	Rank	0.115387	1.00
7	FFT	0.455937	1.00
8	Nonoverlapping template	0.015598	0.98
9	Overlapping template	0.699313	0.99
10	Universal	0.678686	0.98
11	Approximate entropy	0.574903	1.00
12	Random excursions	0.186566	0.9836
13	Random excursions variant	0.023812	1.00
14	Serial	0.514124	0.99
15	Linear complexity	0.350485	0.99

DownLoad: CSV

[1]	禹思敏, 吕金虎, 李澄清 2016 电子与信息学报 38 735 Yu S M, Lü J H, Li C Q 2016 J. Elec. Info. Tech. 38 735
[2]	Lorenz E N 1963 J. Atmos. Sci. 20 130
[3]	Sprott J C 1994 Phys. Rev. E 50 647 Google Scholar
[4]	Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465 Google Scholar
[5]	Lü J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659 Google Scholar
[6]	Dang X Y, Li C B, Bao B C, Wu H G 2015 Chin. Phys. B 24 050503 Google Scholar
[7]	鲜永菊, 扶坤荣, 徐昌彪 2021 振动与冲击 40 15 Xian Y J, Fu K R, Xu C B 2021 J. Vib. Shock. 40 15
[8]	Wang G Y, Yuan F, Chen G R, Zhang Y 2018 Chaos 28 013125 Google Scholar
[9]	Akhshani A, Akhavan A, Mobaraki A, Lim S C, Hassan Z 2014 Commun. Nonlinear Sci. Numer. Simul. 19 101 Google Scholar
[10]	Kadhim A F, Mhaibes H I 2018 Int. J. Appl. Eng. Res. 13 2141
[11]	Qi G Y 2018 Nonlinear Dyn. 95 2063 Google Scholar
[12]	Vaidyanathan S, Volos C 2015 Arch. Control Sci. 25 333 Google Scholar
[13]	Cang S J, Wu A G, Wang Z H, Chen Z Q 2017 Nonlinear Dyn. 89 2495 Google Scholar
[14]	Dong E Z, Yuan M F, Du S Z, Chen Z Q 2019 Appl. Math. Model. 73 40 Google Scholar
[15]	Gu S Q, Du B X, Wan Y J 2020 Int. J. Bifurcation Chaos 30 2050242 Google Scholar
[16]	Chen M, Wang C, Wu H G, Xu Q, Bao B C 2021 Nonlinear Dyn. 103 643 Google Scholar
[17]	Bouteghrine B, Tanougast C, Sadoudi S 2021 J. Circuits Syst. Comput. 30 2150280 Google Scholar
[18]	Lin Z S, Wang G Y, Wang X Y, Yu S M, Lü J H 2018 Nonlinear Dyn. 94 1003 Google Scholar
[19]	徐昌彪, 钟德, 夏诚, 黎周 2019 振动与冲击 38 125 Xu C B, Zhong D, Xia C, Li Z 2019 J. Vib. Shock. 38 125
[20]	Barboza R 2007 Int. J. Bifurcation Chaos 17 4285 Google Scholar
[21]	贾红艳, 陈增强, 袁著祉 2009 物理学报 58 4469 Google Scholar Jia H Y, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin. 58 4469 Google Scholar
[22]	Liu J M, Zhang W 2013 Optik 124 5528 Google Scholar
[23]	Xian Y J, Xia C, Guo T T, Fu K R, Xu C B 2018 Results Phys. 11 368 Google Scholar
[24]	徐昌彪, 黎周 2019 浙江大学学报(工学版) 53 1552 Google Scholar Xu C B, Li Z 2019 J. Zhejiang Univ. (Eng. Sci.) 53 1552 Google Scholar
[25]	Sprott J C 2011 Int. J. Bifurcation Chaos 21 2391 Google Scholar
[26]	Dong E Z, Jiao X D, Du S Z, Chen Z Q, Qi G Y 2020 Complexity 2020 4627597 Google Scholar
[27]	Lakshmanan M, Rajasekar S 2003 Nonlinear Dynamics (Berlin Heidelberg: Springer-Verlag) pp191–238
[28]	Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285 Google Scholar
[29]	包伯成 2013 混沌电路导论 (北京: 科学出版社) 第45—49页 Bao B C 2013 An Introduction to Chaotic Circuits (Beijing: Science Press) pp45−49 (in Chinese)
[30]	Ramasubramanian K, Sriram M S 2000 Physica D 139 72 Google Scholar
[31]	Rukhin A, Soto J, Nechvatal J, Smid M, Barker E, Leigh S, Levenson M, Vangel M, Banks D, Heckert A, Dray J, Vo S 2021https://csrc.nist.gov/publications/detail/sp/800-22/rev-1 a/final [2021-06-26]
[32]	禹思敏 2011 混沌系统与混沌电路 (西安: 西安电子科技大学出版社) 第293—312页 Yu S M 2011 Chaotic Systems and Chaotic Circuits (Xi’an: Xidian University Press) pp293−312 (in Chinese)

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