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)

FullText HTML : translate this paragraph

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 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)

Cited By

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.

DownLoad: Full-Size Img PowerPoint

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$\varSigma _1^{\rm{H}}$.

DownLoad: Full-Size Img PowerPoint

图 6 系统$\varSigma_1^{\rm{H}}$的电路设计

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

DownLoad: Full-Size Img PowerPoint

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

map

[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 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)

[1]	Li Ming-Hua, Yuan Zhen-Zhou, Xu Yan, Tian Jun-Fang. Randomness analysis of lane formation in pedestrian counter flow based on improved lattice gas model. Acta Physica Sinica, 2015, 64(1): 018903. doi: 10.7498/aps.64.018903
[2]	Xie Zheng-Chao, Wang Fei, Yan Jian-Hua, Cen Ke-Fa. Comparative studies of Tikhonov regularization and truncated singular value decomposition in the three-dimensional flame temperature field reconstruction. Acta Physica Sinica, 2015, 64(24): 240201. doi: 10.7498/aps.64.240201
[3]	Zhang Yi. Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems. Acta Physica Sinica, 2013, 62(16): 164501. doi: 10.7498/aps.62.164501
[4]	Sun Ke-Hui, He Shao-Bo, He Yi, Yin Lin-Zi. Complexity analysis of chaotic pseudo-random sequences based on spectral entropy algorithm. Acta Physica Sinica, 2013, 62(1): 010501. doi: 10.7498/aps.62.010501
[5]	Zhang Yi, Jin Shi-Xin. Noether symmetries of dynamics for non-conservative systems with time delay. Acta Physica Sinica, 2013, 62(23): 234502. doi: 10.7498/aps.62.234502
[6]	Bao Bo-Cheng, Yang Ping, Ma Zheng-Hua, Zhang Xi. Dynamics of current controlled switching converters under wide circuit parameter variation. Acta Physica Sinica, 2012, 61(22): 220502. doi: 10.7498/aps.61.220502
[7]	Zhang Guo-Ji, Li Xuan, Liu Qing, Zhang Xia-Yan. Random number generator based on discrete trajectory transform in generalized information domain. Acta Physica Sinica, 2012, 61(6): 060502. doi: 10.7498/aps.61.060502
[8]	Guan Bao-Lu, Zhang Jing-Lan, Ren Xiu-Juan, Guo Shuai, Li Shuo, Chuai Dong-Xu, Guo Xia, Shen Guang-Di. Micro-nano-optical machine system tunable wavelength vertical cavity surface emitting lasers with wide tunable range. Acta Physica Sinica, 2011, 60(3): 034206. doi: 10.7498/aps.60.034206
[9]	Wang Kai, Pei Wen-Jiang, Zhang Yi-Feng, Zhou Si-Yuan, Shao Shuo. Parameter estimate from coupled map lattices based on symbolic vector dynamics. Acta Physica Sinica, 2011, 60(7): 070502. doi: 10.7498/aps.60.070502
[10]	Zhang Yi-Min, Zhang Xu-Fang. Reliability analysis of double random Duffing system. Acta Physica Sinica, 2008, 57(7): 3989-3995. doi: 10.7498/aps.57.3989
[11]	Wang Hong, Ouyang Zheng_Biao, Han Yan-Ling, Meng Qing-Sheng, Luo Xian-Da, Liu Jin-Song. Effect of the strength of randomness on lasing threshold in one-dimensional partially random media. Acta Physica Sinica, 2007, 56(5): 2616-2622. doi: 10.7498/aps.56.2616
[12]	Zhang Yi, Mei Feng-Xiang. Effects of non-conservative forces and nonholonomic constraints on Noether symmetries of a Lagrange system. Acta Physica Sinica, 2004, 53(3): 661-668. doi: 10.7498/aps.53.661
[13]	Li Guo-Hui, Xu De-Ming, Zhou Shi-Ping. Chaos synchronization by using random parametric adaptive control method. Acta Physica Sinica, 2004, 53(2): 379-382. doi: 10.7498/aps.53.379
[14]	Zhang Yi. Effects of non-conservative forces and nonholonomic constraints on Lie symmetrie s of a Hamiltonian system. Acta Physica Sinica, 2003, 52(6): 1326-1331. doi: 10.7498/aps.52.1326
[15]	DAI DONG, MA XI-KUI. CHAOS SYNCHRONIZATION ON USING INTERMITTENT PARAMETRIC ADAPTIVE CONTROL. Acta Physica Sinica, 2001, 50(7): 1237-1240. doi: 10.7498/aps.50.1237
[16]	QU KAI-YANG, JIANG YI. STUDIES ON RANDOMNESS AND NUCLEATION RATE OF SUPERCOOLED WATER FREEZING FROM HOMOGENEOUS NUCLEATION. Acta Physica Sinica, 2000, 49(11): 2214-2219. doi: 10.7498/aps.49.2214
[17]	XING YONG-ZHONG, XU GONG-OU. STOCHASTICITY OF THE EFFECTIVE SUBSPACE TAKEN UP BY A COHERENT STATE IN QUANTUM SYSTEM CORRESPONDING TO CLASSICAL CHAOTIC ONE. Acta Physica Sinica, 1999, 48(5): 769-774. doi: 10.7498/aps.48.769
[18]	ZHANG FEI-ZHOU, WANG JIAO, GU YAN. THE STATISTICAL NON-ERGODICITY OF THE EIGENSTATES OF THE QUANTUM CHAOTIC SYSTEMS AND ITS SEMI-CLASSICAL LIMIT. Acta Physica Sinica, 1999, 48(12): 2169-2179. doi: 10.7498/aps.48.2169
[19]	XU YUN, ZHANG JIAN-XIA, DU SHI-PEI. THE JUMP STOCHASTICITY CHARACTERISTICS OF NONLI-NER FUNCTION IN THE DYNAMIC SYSTEM. Acta Physica Sinica, 1991, 40(1): 33-38. doi: 10.7498/aps.40.33
[20]	FU PAN-MING, YE PEI-XIAN. THE EFFECT OF THE STOCHASTIC PROPERTY OF LASER ON THE DEGENERATE FOUR-WAVE MIXING. Acta Physica Sinica, 1985, 34(6): 737-744. doi: 10.7498/aps.34.737

Catalog

Vol. 70, Issue 23 - 2021-12-05

Metrics

Abstract views: 4934
PDF Downloads: 158
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板