A novel variable-order fractional chaotic map and its dynamics

doi:10.1088/1674-1056/ad1a93

中国物理B ›› 2024, Vol. 33 ›› Issue (3): 30503-030503.doi: 10.1088/1674-1056/ad1a93

A novel variable-order fractional chaotic map and its dynamics

Zhouqing Tang(唐周青)¹, Shaobo He(贺少波)², Huihai Wang(王会海)^1,†, Kehui Sun(孙克辉)³, Zhao Yao(姚昭)³, and Xianming Wu(吴先明)⁴

1 School of Electronic Information, Central South University, Changsha 410083, China;
2 School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China;
3 School of Physics, Central South University, Changsha 410083, China;
4 School of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang 550025, China

收稿日期:2023-11-21 修回日期:2023-12-27 接受日期:2024-01-04 出版日期:2024-02-22 发布日期:2024-02-29
通讯作者: Huihai Wang E-mail:wanghuihai_csu@csu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 62071496, 61901530, and 62061008) and the Natural Science Foundation of Hunan Province of China (Grant No. 2020JJ5767).

A novel variable-order fractional chaotic map and its dynamics

Zhouqing Tang(唐周青)¹, Shaobo He(贺少波)², Huihai Wang(王会海)^1,†, Kehui Sun(孙克辉)³, Zhao Yao(姚昭)³, and Xianming Wu(吴先明)⁴

1 School of Electronic Information, Central South University, Changsha 410083, China;
2 School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China;
3 School of Physics, Central South University, Changsha 410083, China;
4 School of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang 550025, China

Received:2023-11-21 Revised:2023-12-27 Accepted:2024-01-04 Online:2024-02-22 Published:2024-02-29
Contact: Huihai Wang E-mail:wanghuihai_csu@csu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 62071496, 61901530, and 62061008) and the Natural Science Foundation of Hunan Province of China (Grant No. 2020JJ5767).

摘要/Abstract

摘要： In recent years, fractional-order chaotic maps have been paid more attention in publications because of the memory effect. This paper presents a novel variable-order fractional sine map (VFSM) based on the discrete fractional calculus. Specially, the order is defined as an iterative function that incorporates the current state of the system. By analyzing phase diagrams, time sequences, bifurcations, Lyapunov exponents and fuzzy entropy complexity, the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map. The results reveal that the variable order has a good effect on improving the chaotic performance, and it enlarges the range of available parameter values as well as reduces non-chaotic windows. Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values. Moreover, the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation, which proves the potential applications in the field of information security.

关键词: chaos, fractional difference, variable order, multistability, complexity

Abstract: In recent years, fractional-order chaotic maps have been paid more attention in publications because of the memory effect. This paper presents a novel variable-order fractional sine map (VFSM) based on the discrete fractional calculus. Specially, the order is defined as an iterative function that incorporates the current state of the system. By analyzing phase diagrams, time sequences, bifurcations, Lyapunov exponents and fuzzy entropy complexity, the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map. The results reveal that the variable order has a good effect on improving the chaotic performance, and it enlarges the range of available parameter values as well as reduces non-chaotic windows. Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values. Moreover, the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation, which proves the potential applications in the field of information security.

Key words: chaos, fractional difference, variable order, multistability, complexity

中图分类号: (Low-dimensional chaos)

05.45.Ac

05.45.Pq (Numerical simulations of chaotic systems) 05.45.Tp (Time series analysis)

Zhouqing Tang(唐周青), Shaobo He(贺少波), Huihai Wang(王会海), Kehui Sun(孙克辉), Zhao Yao(姚昭), and Xianming Wu(吴先明). A novel variable-order fractional chaotic map and its dynamics[J]. 中国物理B, 2024, 33(3): 30503-030503.

参考文献

[1] Lawnik M, Moysis L and Volos C 2022 Electronics 11 3156
[2] Yan X P, Wang X Y and Xian Y J 2022 Chin. Phys. B 31 080504
[3] Zhou Y, Li C L, Li W, Li H M, Feng W and Qian K 2021 Nonlinear Dyn. 103 2043
[4] Gao X Y, Sun B, Cao Y H, Banerjee S and Mou J 2023 Chin. Phys. B 32 030501
[5] Bai B C, Zhao Q H, Yu X H, Wu H G and Xu Q 2023 Chaos Solitons Fractals 173 113748
[6] El-Latif A, Abd-El-Atty B, Amin M and Iliyasu A M 2020 Sci. Rep. 10 1930
[7] Rybin V, Butusov D, Rodionova E, Karimov T, Ostrovskii V and Tutueva A 2022 Int. J. Bifurcat. Chaos 32 2250136
[8] Li H Z, Hua Z Y, Bao H, Zhu L, Chen M and Bao B C 2020 IEEE Trans. Ind. Electron. 68 9931
[9] Li Y X, Li C B, Zhao Y B and Liu S C 2022 Chaos 32 021104
[10] Wu J Y, Li C B, Xu Y, Jiang Y C and Xu Y X 2023 IEEE Trans. Circuits Syst. I Regul. Pap. 70 378
[11] Yu F, Qian S, Chen X, Huang Y Y, Cai S, Jin J and Du S C 2021 Complexity 2021 6683284
[12] Li C Q, Li S J, Asim M, Nunez J, Alvarez G and Chen G R 2009 Image Vis. Comput. 27 1371
[13] Hartley T T, Lorenzo C F and Qammer H K 1995 IEEE Trans. Circuits Syst. I Fundam. Theor. Appl. 42 485
[14] Xu C J, Liao M X, Li P L, Yao L Y, Qin Q W and Shang Y L 2021 Fractal Fract. 5 257
[15] Lu J G and Chen G R 2006 Chaos Solitons Fractals 27 685
[16] Jumarie G 2009 Appl. Math. Lett. 22 378
[17] Odibat Z and Baleanu D 2020 Appl. Numer. Math. 156 94
[18] Liu Y 2016 Indian J. Phys. 90 313
[19] Ji Y D, Lai L, Zhong S C and Zhang L 2018 Commun. Nonlinear Sci. Numer. Simul. 57 352
[20] Tarasov V E and Zaslavsky G M 2008 J. Phys. A Math. Theor. 41 435101
[21] Edelman M 2020 Chaos Solitons Fractals 137 109774
[22] Li Y Q, He X and Zhang W 2020 Chaos Solitons Fractals 137 109774
[23] Zhou P, Ma J and Tang J 2020 Nonlinear Dyn. 100 2353
[24] Liu Z Y, Xia T C and Wang J B 2018 Chin. Phys. B 27 030502
[25] He S B, Banerjee S and Sun K H 2018 Chaos Solitons Fractals 115 14
[26] Patnaik S, Hollkamp J P and Semperlotti F 2020 Proc. Royal Soc. A 476 20190498
[27] Yao Z, Ma J, Yao Y G and Wang C N 2019 Nonlinear Dyn. 96 205
[28] Zhang H M, Liu F W, Phanikumar M S and Meerschaert M M 2013 Comput. Math. Appl. 66 693
[29] Yang X J and Machado J T 2017 Physica A 481 276
[30] Wu G C, Deng Z G, Baleanu D and Zeng D Q 2019 Chaos 29 083103
[31] Huang L L, Park J H, Wu G C and Mo Z W 2020 J. Comput. Appl. Math. 370 112633
[32] Gary H L and Zhang N F 1988 Math. Comput. 50 513
[33] Atici F and Eloe P 2009 Proc. Am. Math. Soc. 137 981
[34] Anastassiou G A 2009 arXiv: 0911.3370v1 [math.CA]
[35] Abdeljawad T 2011 Comput. Math. Appl. 62 1602
[36] Chen F L, Luo X N and Zhou Y 2011 Adv. Differ. Equ. 2011 713201
[37] Tavazoei M S and Haeri M 2009 Automatica 45 1886
[38] Diblík J, Fečkan M and Pospíšil M 2015 Appl. Math. Comput. 257 230
[39] Hua Z Y, Zhou B H and Zhou Y C 2018 IEEE Trans. Ind. Electron. 66 1273
[40] Karoun R C, Ouannas A, Horani M A and Grassi G 2022 Fractal Fract. 6 575
[41] Li C B, Li Z N, Jiang Y C, Lei T F and Wang X 2023 Symmetry 15 1564
[42] Wolf A, Swift J B, Harry L and Vastano J A 1985 Physica D 16 285
[43] Chen W T, Zhuang J, Yu W X and Wang Z Z 2009 Med. Eng. Phys. 31 61
[44] Dong C Y, Rajagopal K, He S B, Jafari S and Sun K H 2021 Results Phys. 31 105010
[45] Edelman M 2014 arXiv:1404.4906v4 [nlin.CD]
[46] Dong C Y, Sun K H, He S B and Wang H H 2021 Chaos 31 083132
[47] Rossler O 1979 Phys. Lett. A 71 155
[48] Natiq H, Banerjee S, Ariffin M and Said M 2019 Chaos 29 011103
[49] Jafari S, Sprott J C, Pham V, Golpayegani S M R H and Jafari A H 2014 Int. J. Bifurcat. Chaos 24 1450134
[50] Peng Y X, Sun K H and He S B 2020 Int. J. Bifurcat. Chaos 30 2050058
[51] Chen C, Sun K H and He S B 2019 Eur. Phys. J. Plus 134 410

A novel variable-order fractional chaotic map and its dynamics

A novel variable-order fractional chaotic map and its dynamics

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