A class of two-dimensional rational maps with self-excited and hidden attractors

doi:10.1088/1674-1056/ac4025

中国物理B ›› 2022, Vol. 31 ›› Issue (3): 30503-030503.doi: 10.1088/1674-1056/ac4025

A class of two-dimensional rational maps with self-excited and hidden attractors

Li-Ping Zhang(张丽萍)^1,2, Yang Liu(刘洋)³, Zhou-Chao Wei(魏周超)⁴, Hai-Bo Jiang(姜海波)^2,†, and Qin-Sheng Bi(毕勤胜)¹

1 Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China;
2 School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China;
3 College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4;
4 QF, UK;
4 School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

收稿日期:2021-06-13 修回日期:2021-10-31 接受日期:2021-12-05 出版日期:2022-02-22 发布日期:2022-02-17
通讯作者: Hai-Bo Jiang E-mail:yctcjhb@126.com
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11672257, 11772306, 11972173, and 12172340) and the 5th 333 High-level Per sonnel Training Project of Jiangsu Province of China (Grant No. BRA2018324).

A class of two-dimensional rational maps with self-excited and hidden attractors

Li-Ping Zhang(张丽萍)^1,2, Yang Liu(刘洋)³, Zhou-Chao Wei(魏周超)⁴, Hai-Bo Jiang(姜海波)^2,†, and Qin-Sheng Bi(毕勤胜)¹

1 Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China;
2 School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China;
3 College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4;
4 QF, UK;
4 School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Received:2021-06-13 Revised:2021-10-31 Accepted:2021-12-05 Online:2022-02-22 Published:2022-02-17
Contact: Hai-Bo Jiang E-mail:yctcjhb@126.com
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11672257, 11772306, 11972173, and 12172340) and the 5th 333 High-level Per sonnel Training Project of Jiangsu Province of China (Grant No. BRA2018324).

摘要/Abstract

摘要： This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan—Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

关键词: two-dimensional rational map, hidden attractors, multi-stability, a line of fixed points, chaotic attractor

Abstract: This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan—Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

Key words: two-dimensional rational map, hidden attractors, multi-stability, a line of fixed points, chaotic attractor

中图分类号: (Low-dimensional chaos)

05.45.Ac

05.45.Pq (Numerical simulations of chaotic systems)

Li-Ping Zhang(张丽萍), Yang Liu(刘洋), Zhou-Chao Wei(魏周超),Hai-Bo Jiang(姜海波), and Qin-Sheng Bi(毕勤胜). A class of two-dimensional rational maps with self-excited and hidden attractors[J]. 中国物理B, 2022, 31(3): 30503-030503.

参考文献

[1] Leonov G A and Kuznetsov N V 2013 Int. J. Bifurc. Chaos 23 1330002
[2] Leonov G A, Kuznetsov N V and Vagaitsev V I 2011 Phys. Lett. A 375 2230
[3] Leonov G A, Kuznetsov N V and Vagaitsev V I 2012 Physica D 241 1482
[4] Leonov G A, Kuznetsov N V, Kiseleva M A, Solovyeva E P and Zaretskiy A M 2014 Nonlinear Dyn. 77 277
[5] Leonov G A, Kuznetsov N V, Kuznetsova O A, Seldedzhi S M and Vagaitsev V I 2011 WSEAS Trans. Syst. Contr. 6 54
[6] Pham V T, Volos C and Kapitaniak T 2017 Systems with Hidden Attractors (Switzerland:Springer)
[7] Pisarchik A N and Feudel U 2014 Phys. Rep. 540 167
[8] Li C B and Sprott J C 2014 Int. J. Bifurc. Chaos 24 1450034
[9] Liu Y and Páez-Chávez J 2017 Nonlin. Dyn. 88 1289
[10] Tang Y X, Khalaf A J M, Rajagopal K, Pham V T, Jafari S and Tian Y 2018 Chin. Phys. B 27 040502
[11] Wei Z C, Li Y Y, Sang B, Liu Y J and Zhang W 2019 Int. J. Bifurc. Chaos 29 1950095
[12] Mira C, Gardini L, Barugola A and Cathala J C 1996 Chaotic Dynamics in Two-Dimensional Noninvertible Maps (Singapore:World Scientific)
[13] Sprott J C 1993 Strange Attractors:Creating Patterns in Chaos (New York:M&T books)
[14] Sprott J C 2010 Elegant Chaos (Singapore:World Scientific) pp. 24-30
[15] Jiang H B, Liu Y, Wei Z C and Zhang L P 2016 Nonlin. Dyn. 85 2719
[16] Jiang H B, Liu Y, Wei Z C and Zhang L P 2016 Int. J. Bifurc. Chaos 26 1650206
[17] Jiang H B, Liu Y, Wei Z C and Zhang L P 2019 Int. J. Bifurc. Chaos 29 1950094
[18] Huynh V V, Ouannas A, Wang X, Pham V T, Nguyen X Q and Alsaadi F E 2019 Entropy 21 279
[19] Luo A C J 2020 Bifurcation and Stability in Nonlinear Discrete Systems (Singapore:Springer)
[20] Ouannas A, Wang X, Khennaoui A A, Bendoukha S, Pham V T and Alsaadi F E 2018 Entropy 20 720
[21] Hadjabi F, Ouannas A, Shawagfeh N, Khennaoui A A and Grassi G 2020 Symmetry 12 756
[22] Ouannas A, Khennaoui A A, Bendoukha S, Vo T P, Pham V T and Huynh V V 2018 Appl. Sci. 8 2640
[23] Khennaoui A A, Ouannas A, Bendoukha S, Grassi G, Wang X, Pham V T and Alsaadi F E 2019 Adv. Differ. Equ. 2019 412
[24] Liu Z Y, Xia T C and Wang J B 2018 Chin. Phys. B 27 030502
[25] Ouannas A, Khennaoui A A, Momani S, Pham V T and Reyad R 2020 Chin. Phys. B 29 050504
[26] Dudkowski D, Prasad A and Kapitaniak T 2016 Chaos 26 103103
[27] Dudkowski D, Prasad A and Kapitaniak T 2017 Int. J. Bifurc. Chaos 27 1750063
[28] Danca M F and Fečan M 2019 Commun. Nonlinear Sci. Numer. Simul. 74 1
[29] Danca M F and Lampart M 2021 Chaos Solit. Fract. 142 110371
[30] Zhang L P, Jiang H B, Liu Y, Wei Z C and Bi Q S 2021 Int. J. Bifurc. Chaos 31 2150047
[31] Zhang L P, Liu Y, Wei Z C, Jiang H B and Bi Q S 2020 Chin. Phys. B 29 060501
[32] Bao B C, Li H Z, Zhu L Zhang X and Chen M 2020 Chaos 30 033107
[33] Kong S X, Li C B, Jiang H B, Lai Q and Jiang X W 2021 Chaos 31 043121
[34] Rulkov N 2002 Phys. Rev. E 65 041922
[35] Lu J, Wu X, Lu J and Kang L S 2004 Chaos Solit. Fract. 22 311
[36] Chang L, Lu J and Deng X 2005 Chaos Solit. Fract. 24 1135
[37] Elhadj Z and Sprott J C 2011 Int. J. Bifurc. Chaos 21 155
[38] Elhadj Z and Sprott J C 2011 Int. J. Open Problems Compt. Math 4 1
[39] Somarakis C and Baras J S 2013 Int. J. Bifurc. Chaos 23 1330021
[40] Chen G, Kudryashova E V, Kuznetsov N V and Leonov G A 2016 Int. J. Bifurc. Chaos 26 1650126
[41] Ouannas A, Khennaoui A A, Bendoukha S, Wang Z and Pham V T 2020 J. Syst. Sci. Complex. 33 584
[42] Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (2ed) (New York:Springer-Verlag)
[43] Leonov G A, Kuznetsov N V and Mokaev T N 2015 Commun. Nonlinear Sci. Numer. Simul. 28 166
[44] Wolf A, Swift J B, Swinney H L and Vastano J A 1985 Phys. D 16 285

A class of two-dimensional rational maps with self-excited and hidden attractors

A class of two-dimensional rational maps with self-excited and hidden attractors

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Sheng-Hao Jia(贾生浩), Yu-Xia Li(李玉霞), Qing-Yu Shi(石擎宇), and Xia Huang(黄霞). Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system[J]. 中国物理B, 2022, 31(7): 70505-070505.
[2]	Li-Ping Zhang(张丽萍), Yang Liu(刘洋), Zhou-Chao Wei(魏周超), Hai-Bo Jiang(姜海波), Wei-Peng Lyu(吕伟鹏), and Qin-Sheng Bi(毕勤胜). Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor[J]. 中国物理B, 2022, 31(10): 100503-100503.
[3]	张丽萍, 刘洋, 魏周超, 姜海波, 毕勤胜. A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors[J]. 中国物理B, 2020, 29(6): 60501-060501.
[4]	Jay Prakash Singh, Binoy Krishna Roy, 魏周超. A new four-dimensional chaotic system with first Lyapunov exponent of about 22, hyperbolic curve and circular paraboloid types of equilibria and its switching synchronization by an adaptive global integral sliding mode control[J]. 中国物理B, 2018, 27(4): 40503-040503.
[5]	唐妍霞, Abdul Jalil M Khalaf, Karthikeyan Rajagopal, Viet-Thanh Pham, Sajad Jafari, 田野. A new nonlinear oscillator with infinite number of coexisting hidden and self-excited attractors[J]. 中国物理B, 2018, 27(4): 40502-040502.
[6]	J P Singh, V T Pham, T Hayat, S Jafari, F E Alsaadi, B K Roy. A new four-dimensional hyperjerk system with stable equilibrium point, circuit implementation, and its synchronization by using an adaptive integrator backstepping control[J]. 中国物理B, 2018, 27(10): 100501-100501.
[7]	鲍江宏, 陈丹丹. Coexisting hidden attractors in a 4D segmented disc dynamo with one stable equilibrium or a line equilibrium[J]. 中国物理B, 2017, 26(8): 80201-080201.
[8]	胡晓宇, 刘崇新, 刘凌, 姚亚鹏, 郑广超. Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system[J]. 中国物理B, 2017, 26(11): 110502-110502.
[9]	孙常春. Generation of countless embedded trumpet-shaped chaotic attractors in two opposite directions from a new three-dimensional system with no equilibrium point[J]. , 2014, 23(9): 90502-090502.
[10]	孙常春, 赵恩良, 徐启程. Generation of a novel spherical chaotic attractor from a new three-dimensional system[J]. 中国物理B, 2014, 23(5): 50505-050505.
[11]	余飞, 王春华, 尹晋文, 徐浩. Novel four-dimensional autonomous chaotic system generating one-, two-, three- and four-wing attractors[J]. 中国物理B, 2011, 20(11): 110505-110505.
[12]	董高高, 郑松, 田立新, 杜瑞瑾, 孙梅. The analysis of complex behaviours of a novel three dimensional autonomous system[J]. 中国物理B, 2010, 19(7): 70514-070514.
[13]	SaraDadras, HamidRezaMomeni. Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system[J]. 中国物理B, 2010, 19(6): 60506-060506.
[14]	董恩增, 陈在平, 陈增强, 袁著祉. A novel four-wing chaotic attractor generated from a three-dimensional quadratic autonomous system[J]. 中国物理B, 2009, 18(7): 2680-2689.
[15]	祁伟, 汪映海. Synchronization of time-delay chaotic systems on small-world networks with delayed coupling[J]. 中国物理B, 2009, 18(4): 1404-1408.