Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system

doi:10.1088/1674-1056/ab9dea

Abstract

A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.

Keywords: grid multi-scroll chaotic attractor Jerk system specific form of the sine function circuit implementation

Received: 19 April 2020
Revised: 14 June 2020
Accepted manuscript online: 18 June 2020

PACS:	82.40.Bj	(Oscillations, chaos, and bifurcations)
	05.45.Pq	(Numerical simulations of chaotic systems)
	05.45.-a	(Nonlinear dynamics and chaos)

Corresponding Authors: ^†Corresponding author. E-mail: fengxiao@nwpu.edu.cn

Cite this article:

Peng-Fei Ding(丁鹏飞), Xiao-Yi Feng(冯晓毅)†, and Cheng-Mao Wu(吴成茂) Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system 2020 Chin. Phys. B 29 108202

Fig. 1.

The waveform of the different sine functions: (a) f(x) with b = 0.5 and n₁ = n₂ = 2, (b) g(x) with b = 0.5, (c) h(x) with A = 6 and a = 0.5 π, (d) p(x) with a = 1, b = 0.1, c = 3, and d = 0. The a, c, and d are real constants, and x, y, and z are state variables of the system (3). The b, n₁, and n₂ are real constants in Eq. (4). The number of scrolls generated by the system (3) with suitable parameters can be adjusted by the parameters n₁ and n₂.

Fig. 2.

Different number of scroll chaotic attractors are generated by system (3) with a = c = d = 0.3 and b = 0.5: (a) 3-scroll chaotic attractor with n₁ = 1 and n₂ = 2, (b) 5-scroll chaotic attractor with n₁ = 2 and n₂ = 3.

Fig. 3.

Grid multi-scroll chaotic attractors for a = 1, b = 0.5, c = 0.3, d = 0.5, and A = 1: (a) 6 × 3 grid multi-scroll chaotic attractors with n₁ = 3, n₂ = 3, and M = 1; (b) 5 × 4 grid multi-scroll chaotic attractors with n₁ = 2, n₂ = 3, and M = 2.

Fig. 4.

The equilibrium point distribution of the 6 × 3 grid multi-scroll chaotic attractors.

Fig. 5.

The system (9) with Eqs. (4) and (10), and d ∈ (0,1): (a) Lyapunov exponents; (b) bifurcation diagram.

Fig. 6.

The specific form of the sine function f(x) with b = 0.5, n₁ = n₂ = 3. (a) Electronic circuit diagram; (b) simulation result with the unit of horizontal ordinate 2 s/Div, and the unit of vertical ordinate 500 mV/Div.

Fig. 7.

The specific form of the sine function f(x) with b = 0.5, n₁ = 2, n₂ = 3. (a) Electronic circuit diagram; (b) simulation result with the unit of horizontal ordinate 2 s/Div, and the unit of vertical ordinate 500 mV/Div.

Fig. 8.

The sign function f₁(y) of Eq. (10) with A = 1, M = 1. (a) Electronic circuit diagram; (b) circuit simulation result with the unit of horizontal ordinate 1 V/Div, and the unit of vertical ordinate 1 V/Div.

Fig. 9.

The sign function f₁(y) of Eq. (11) with A = 1, M = 1. (a) Electronic circuit diagram; (b) circuit simulation result with the unit of horizontal ordinate 1 V/Div, and the unit of vertical ordinate 2 V/Div.

Fig. 10.

Grid multi-scroll chaotic attractor circuit.

Fig. 11.

Circuit simulation results: (a) 6 × 3 grid multi-scroll chaotic attractors, (b) 5 × 4 grid multi-scroll chaotic attractors.

Fig. 12.

Hardware circuits experimental results. (a) Hardware circuits connection diagram; (b) experimental results of the 6 × 3 grid multi-scroll chaotic attractors; (c) experimental results of the 5 × 4 grid multi-scroll chaotic attractors.

[1]	Wang G Y, He S L 2003 IEEE Transactions on Circuits and Systems-I: Fundamental Theory & Applications 50 945 DOI: 10.1109/TCSI.2003.812606
[2]	Xiang X Q, Shi B C 2010 Chaos 20 013104 DOI: 10.1063/1.3279568
[3]	Jin T, Zhang H 2011 Science China-Information Sciences 54 2324 DOI: 10.1007/s11432-011-4308-6
[4]	Ma S S, Lu M, Ding J F, Huang W, Yuan H 2015 Science China-Information Sciences 58 102401 DOI: 10.1007/s11432-015-5344-4
[5]	Kacar S 2016 Optik 127 9551 DOI: 10.1016/j.ijleo.2016.07.044
[6]	Vaidyanathan S, Akgul A, Kacar S, Cavusoglu U 2018 Euro. Phys. J. P. 133 46 DOI: 10.1140/epjp/i2018-11872-8
[7]	Jiang Y J, Tang S Y 2018 Multimedia Systems 24 355 DOI: 10.1007/s00530-017-0565-6
[8]	Xiong Z L, Qu S C, Luo J 2019 Complexity 2019 3827201 DOI: 10.1155/2019/3827201
[9]	Liu J X, Wang Z X, Shu M L, Zhang F F, Leng S, Sun X H 2019 Complexity 2019 7242791 DOI: 10.1155/2019/7242791
[10]	Yu F, Zhang Z N, Liu L, Shen H, Huang Y Y, Shi C Q, Cai S, Song Y, Du S C, Xu Q 2020 Complexity 2020 5859273 DOI: 10.1155/2020/5859273
[11]	Chang D, Li Z J, Wang M J, Zeng Y C 2018 Aeu-International Journal of Electronics and Communications 88 20 DOI: 10.1016/j.aeue.2018.03.007
[12]	Liu L Z, Zhang J Q, Xu G X, Liang L S, Wang M S 2014 Acta Phys. Sin. 63 010501 in Chinese DOI: 10.7498/aps.63.010501
[13]	Yu F, Qian S, Chen X, Liu L, Shi C Q, Cai S, Song Y, Wang C H 2020 International Journal of Bifurcation and Chaos 2020 DOI: 10.1142/S0218127420501412
[14]	Xie E Y, Li C Q, Yu S M, Lü J H 2017 Signal Processing 132 150 DOI: 10.1016/j.sigpro.2016.10.002
[15]	Zhang L M, Sun K H, Liu W H, He S B 2017 Chin. Phys. B 26 100504 DOI: 10.1088/1674-1056/26/10/100504
[16]	Liu Z Y, Xia T C, Wang J B 2018 Chin. Phys. B 27 030502 DOI: 10.1088/1674-1056/27/3/030502
[17]	Asgari-Chenaghlu M, Balafar M-A, Feizi-Derakhshi M R 2019 Signal Processing 157 1 DOI: 10.1016/j.sigpro.2018.11.010
[18]	Wang S C, Wang C H, Xu C 2019 Optics and Lasers in Engineering 128 105995 DOI: 10.1016/j.optlaseng.2019.105995
[19]	Zhou M J, Wang C H 2020 Signal Processing 171 107484 DOI: 10.1016/j.sigpro.2020.107484
[20]	Xu C, Sun J R, Wang C H 2020 International Journal of Bifurcation and Chaos 30 2050060 DOI: 10.1142/S0218127420500601
[21]	Lin H R, Wang C H 2020 Applied Mathematics and Computation 369 124840 DOI: 10.1016/j.amc.2019.124840
[22]	Lin H R, Wang C H, Tan Y M 2020 Nonlinear Dynamics 99 2369 DOI: 10.1007/s11071-019-05408-5
[23]	Yu F, Liu L, Qian S, Li L X, Huang Y Y, Shi C Q, Cai S, Wu X M, Du S C, Wan Q Z 2020 Complexity 2020 8034196 DOI: 10.1155/2020/8034196
[24]	Yu F, Liu L, He B Y, Huang Y Y, Shi C Q, Cai S, Song Y, Du S C, Wan Q Z 2019 Complexity 2019 4047957 DOI: 10.1155/2019/4047957
[25]	Jin J, Cui L 2019 Complexity 4106398 DOI: 10.1155/2019/4106398
[26]	Yu F, Liu L, Shen H, Liu L, Zhang Z N, Huang Y Y, Shi C Q, Cai S, Wu X M, Du S C, Wang Q Z 2020 Complexity 2020 5904607 DOI: 10.1155/2020/5904607
[27]	Yu F, Shen H, Liu L, Zhang Z N, Huang Y Y, He B Y, Cai S, Song Y, Yin B, Du S C, Xu Q 2020 Complexity 2020 5212601 DOI: 10.1155/2020/5212601
[28]	Suykens J A K, Vandewalle J 1993 IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 40 861 DOI: 10.1109/81.251829
[29]	Zhong G Q, Man K F, Chen G R 2002 International Journal of Bifurcation and Chaos 12 2907 DOI: 10.1142/S0218127402006230
[30]	Wang F Q, Liu C X 2007 Chin. Phys. 16 942 DOI: 10.1088/1009-1963/16/4/014
[31]	Zhang C X, Yu S M 2009 Chin. Phys. B 18 119 DOI: 10.1088/1674-1056/18/1/019
[32]	Sanchez-Lopez C 2011 Applied Mathematics and Computation 217 4350 DOI: 10.1016/j.amc.2010.11.009
[33]	Dalia Pano-Azucena A, de Jesus Rangel-Magdaleno J, Tlelo-Cuautle E, de Jesus Quintas-Valles A 2017 Nonlinear Dynamics 87 2203 DOI: 10.1007/s11071-016-3184-4
[34]	Yalcin M E, Suykens J A K, Vandewalle J, Ozoguz S 2002 International Journal of Bifurcation and Chaos 12 23 DOI: 10.1142/S0218127402004164
[35]	Lü J H, Chen G R, Yu X H, Leung H 2004 IEEE Transactions on Circuits and Systems-I: Regular Papers 51 2476 DOI: 10.1109/TCSI.2004.838151
[36]	Zhang G T, Wang F Q 2018 Chin. Phys. B 27 018201 DOI: 10.1088/1674-1056/27/1/018201
[37]	Chen S B, Zeng Y C, Xu M L, Chen J S 2011 Acta Phys. Sin. 60 020507 in Chinese DOI: 10.7498/aps.60.020507
[38]	Zhang C X, Yu S M 2016 Chin. Phys. B 25 050503 DOI: 10.1088/1674-1056/25/5/050503
[39]	Li C L, Yu S M, Luo X S 2012 Acta Phys. Sin. 61 110502 in Chinese DOI: 10.7498/aps.61.110502
[40]	Zhang X, Wang C H, Yao W, Lin H R 2019 Nonlinear Dynamics 97 2159 DOI: 10.1007/s11071-019-05113-3
[41]	Zhang X, Wang C H 2019 International Journal of Bifurcation and Chaos 29 1950117 DOI: 10.1142/S0218127419501177
[42]	Zhang X, Wang C H 2019 IEEE Access 7 16336 DOI: 10.1109/ACCESS.2019.2894853
[43]	Deng Q L, Wang C H 2019 Chaos 29 093112 DOI: 10.1063/1.5116732
[44]	Ji Q B, Zhou Y, Yang Z Q, Meng X Y 2015 Chin. Phys. Lett. 32 050501 DOI: 10.1088/0256-307X/32/5/050501
[45]	Van Ha N, Song H J 2015 Chin. Phys. Lett. 32 038201 DOI: 10.1088/0256-307X/32/3/038201
[46]	Sato Y D 2013 Chin. Phys. Lett. 30 128201 DOI: 10.1088/0256-307X/30/12/128201
[47]	Zhou J C, Song H J 2013 Chin. Phys. Lett. 30 020501 DOI: 10.1088/0256-307X/30/2/020501
[48]	Nam S G, Nguyen Van H, Song H J 2014 Chin. Phys. Lett. 31 060502 DOI: 10.1088/0256-307X/31/6/060502
[49]	Van Ha N, Song H J 2013 Chin. Phys. Lett. 30 060501 DOI: 10.1088/0256-307X/30/6/060501
[50]	Li X H, Bi Q S 2013 Chin. Phys. Lett. 30 070503 DOI: 10.1088/0256-307X/30/7/070503
[51]	Niu S, Shuai J W, Qi H 2017 Acta Phys. Sin. 66 238701 in Chinese DOI: 10.7498/aps.66.238701
[52]	Zhang H, Guo X X, Xiang S Y 2018 Acta Phys. Sin. 67 204202 in Chinese DOI: 10.7498/aps.67.20181038
[53]	Li H M, Fan Y Y, Sun H Y, Zhang J, Jia M 2012 Acta Phys. Sin. 61 029501 in Chinese DOI: 10.7498/aps.61.029501
[54]	Zhang R, Peng M, Zhang Z D, Bi Q S 2018 Chin. Phys. B 27 110501 DOI: 10.1088/1674-1056/27/11/110501
[55]	Han Q S, Chen D Y, Zhang H 2017 Chin. Phys. B 26 128202 DOI: 10.1088/1674-1056/26/12/128202
[56]	Zhang L, Tang J S, Han Q 2018 Chin. Phys. B 27 094702 DOI: 10.1088/1674-1056/27/9/094702
[57]	Wang D G, Zhou C H, Zhang X P 2017 Chin. Phys. B 26 128709 DOI: 10.1088/1674-1056/26/12/128709
[58]	Gao X, Chen D Y, Zhang H, Xu B B, Wang X Y 2018 Chin. Phys. B 27 128202 DOI: 10.1088/1674-1056/27/12/128202
[59]	Dong L F, Yue H, Fan W L, Li Y Y, Yang Y J, Xiao H 2011 Acta Phys. Sin. 60 065206 in Chinese DOI: 10.7498/aps.60.065206
[60]	Dong L F, Li S F, Fan W L 2011 Acta Phys. Sin. 60 065205 in Chinese DOI: 10.7498/aps.60.065205
[61]	Wang F Q, Liu C X 2006 Chin. Phys. 15 2878 DOI: 10.1088/1009-1963/15/12/019
[62]	Chen L, Shi Y D, Wang D S 2010 Chin. Phys. B 19 100503 DOI: 10.1088/1674-1056/19/10/100503
[63]	Luo X H, Tu Z W, Liu X R, Cai C, Liang Y L, Gong P 2010 Chin. Phys. B 19 070510 DOI: 10.1088/1674-1056/19/7/070510
[64]	Yalcin M E 2007 Chaos Solitons Fract. 34 1659 DOI: 10.1016/j.chaos.2006.04.058
[65]	Wang C H, Luo X W, Wan Z 2014 Optik 125 6716 DOI: 10.1016/j.ijleo.2014.07.084
[66]	Ai W, Sun K H, Fu Y L 2018 International Journal of Modern Physics C 29 1850049 DOI: 10.1142/S0129183118500493
[67]	Ma J, Wu X Y, Chu R T, Zhang L P 2014 Nonlinear Dynamics 76 1951 DOI: 10.1007/s11071-014-1260-1
[68]	Zhang J X, Tang W S 2009 Chaos Solitons & Fractals 42 2181 DOI: 10.1016/j.chaos.2009.03.158
[69]	He S B, Sun K H, Wang H H, Ai X X, Xu Y X 2016 Journal of Applied Analysis and Computation 6 1180 DOI: 10.11948/2016078
[70]	Yu S M, Lü J H, Leung H, Chen G R 2005 IEEE Transactions on Circuits and Systems-I: Regular Papers 52 1459 DOI: 10.1109/TCSI.2005.851717
[71]	Tang W K S, Zhong G Q, Chen G R, Man K F 2001 IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 48 1369 DOI: 10.1109/81.964432
[72]	Luo X H 2009 Chin. Phys. B 18 3304 DOI: 10.1088/1674-1056/18/8/034
[73]	Hadef S, Boukabou A 2014 Journal of the Franklin Institute-Engineering and Applied Mathematics 351 2728 DOI: 10.1016/j.jfranklin.2014.01.015
[74]	Li F, Ma J 2016 Plos One 11 e0154282
[75]	Hu X Y, Liu C X, Liu L, Ni J K, Li S L 2016 Nonlinear Dynamics 86 1725 DOI: 10.1007/s11071-016-2989-5
[76]	Hu X Y, Liu C X, Liu L, Yao Y P, Zheng G C 2017 Chin. Phys. B 26 110502 DOI: 10.1088/1674-1056/26/11/110502

[1]	Analysis and implementation of new fractional-order multi-scroll hidden attractors Li Cui(崔力), Wen-Hui Luo(雒文辉), and Qing-Li Ou(欧青立). Chin. Phys. B, 2021, 30(2): 020501.
[2]	A novel methodology for constructing a multi-wing chaotic and hyperchaotic system with a unified step function switching control Chao-Xia Zhang(张朝霞), Si-Min Yu(禹思敏). Chin. Phys. B, 2016, 25(5): 050503.
[3]	Hopf bifurcation analysis and circuit implementation for a novel four-wing hyper-chaotic system Xue Wei (薛薇), Qi Guo-Yuan (齐国元), Mu Jing-Jing (沐晶晶), Jia Hong-Yan (贾红艳), Guo Yan-Ling (郭彦岭). Chin. Phys. B, 2013, 22(8): 080504.
[4]	Forming and implementing a hyperchaotic system with rich dynamics Liu Wen-Bo(刘文波), Wallace K. S. Tang(邓榤生), and Chen Guan-Rong(陈关荣) . Chin. Phys. B, 2011, 20(9): 090510.
[5]	A novel one equilibrium hyper-chaotic system generated upon Lü attractor Jia Hong-Yan(贾红艳), Chen Zeng-Qiang(陈增强), and Yuan Zhu-Zhi(袁著祉). Chin. Phys. B, 2010, 19(2): 020507.
[6]	Circuit implementation and multiform intermittency in a hyper-chaotic model extended from the Lorenz system Cang Shi-Jian(仓诗建), Chen Zeng-Qiang(陈增强), and Wu Wen-Juan(吴文娟). Chin. Phys. B, 2009, 18(5): 1792-1800.
[7]	Design and implementation of a novel multi-scroll chaotic system Zhang Chao-Xia(张朝霞) and Yu Si-Min (禹思敏). Chin. Phys. B, 2009, 18(1): 119-129.
[8]	A new Rösslor hyperchaotic system and its realization with systematic circuit parameter design Wang Guang-Yi (王光义), He Hai-Lian (何海莲). Chin. Phys. B, 2008, 17(11): 4014-4021.
[9]	Analysis and implementation of a new hyperchaotic system Wang Guang-Yi(王光义),Liu Jing-Biao(刘敬彪), and Zheng Xin(郑欣). Chin. Phys. B, 2007, 16(8): 2278-2284.
[10]	Hyperchaos evolved from the Liu chaotic system Wang Fa-Qiang (王发强), Liu Chong-Xin (刘崇新). Chin. Phys. B, 2006, 15(5): 963-968.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system

Cite this article:

Online attention