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A memristor can be used in chaotic system as a nonlinear term, and thus enhancing the complexity of the chaotic system. Fractal theory is a leading and important branch of nonlinear science, and has been widely studied in many fields in the past few decades. The fractal and chaos are bound tightly and their relevant researches are well-established, but few of them focus on the research of the possibility of combining the fractal and the chaotic system. In order to obtain a multi scroll chaotic attractor, the fractal process is novelty introduced into the memristive chaotic system. In this paper, at first, a new memristive chaotic system is proposed. Then, the dynamic characteristics of the system are discussed from the aspects of symmetry, dissipation, stabilization of equilibrium points, power spectrum, Lyapunov exponent and fractional dimension. A mapping relationship based on classical Julia fractal is established. Through this mapping relationship, a multi-scroll memristive chaotic system based on the Julia fractal is obtained. Moreover, several deformed Julia fractal processes are applied to the memristive chaotic system, and abundant chaotic attractors are obtained. For example, the square term of the Julia fractal expression is multiplied by a coefficient, and according to the difference in coefficient, the resulting chaotic attractors have the same shape but different sizes. The exponent of the square term in the Julia fractal is changed into a variable, and the chaotic attractor of different scroll numbers is obtained according to the difference in power exponent. In addition, a rich multi-scroll chaotic attractor is obtained by using the fractal expression in the form of weighted sum polynomial. Finally, the influence of a complex constant in the fractal process on the system is discussed. The simulation results show that the combination of fractal process and chaotic system can obtain rich chaotic attractors, such as multi-scroll chaotic attractors. In general, compared with the single-scroll chaotic attractor, the multi-scroll chaotic attractor has a higher complexity and more adjustability. In addition, compared with other multi-scroll chaotic system, the proposed multi-scroll chaotic system is easy to adjust the number of the scrolls. To summarize, this work not only provides a new method of generating multi-scroll chaotic attractors, but also makes up for the lack of smoothness of the chaotic system caused by using functional methods.
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Keywords:
- chaotic system /
- memristor /
- dynamic characteristics /
- Julia fractal process
[1] Chua L O 1971 IEEE Trans. Circ. Theor. 18 507
[2] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[3] Zhou J, Huang D 2012 Chin. Phys. B 21 048401
[4] Wang L D, Li H F, Duan S K, Huang T W, Wang H M 2016 Neurccomputing 171 23
[5] Min G Q, Wang L D, Duan S K 2015 Acta Phys. Sin. 64 210507 (in Chinese) [闵国旗, 王丽丹, 段书凯 2015 物理学报 64 210507]
[6] Mandelbrot B B 1967 Science 156 636
[7] Mandelbrot B B 1975 Fractals: Form, Chance and Dimension (San Francisco: WH Freeman and Company) pp35-37
[8] Li H Q, Wang F Q 1999 Fractal Theory and Its Application in Molecular Science (Beijing: Science Press) p33 (in Chinese) [李后强, 汪富泉 1999分形理论及其在分子科学中的应用(北京:科学出版社) 第33页]
[9] Lorenz E N 1963 J. Atmos. Sci. 20 130
[10] Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. Circ. Syst. 33 1072
[11] Chen G R 1999 Int. J. Bifurcat. Chaos 9 1465
[12] Wang L D, Drakakis E, Duan S K, He P F, Liao X F 2012 Int. J. Bifurcat. Chaos 22 1250205
[13] Muthuswamy B, Kokate P P 2009 IETE Tech. Rev. 26 417
[14] Zhou Z W, Su Y L, Wang W D, Yan Y 2017 J. Petrol. Explor. Prod. Technol. 7 487
[15] Bouallegue K 2015 Int. J. Bifurcat. Chaos 25 1530002
[16] Chua L O, Roska T 1993 IEEE Trans. Circ. Syst. I 40 147
[17] Yalcin M, Suykens J, Vandewalle J, Ozoguz S 2002 Int. J. Bifurcat. Chaos 12 23
[18] Tang W K S, Zhong G Q, Chen G, Man K F 2001 IEEE Trans. Circ. Syst. I 48 1369
[19] Zarei A 2015 Nonlinear Dyn. 81 585
[20] More C, Vlad R, Chauveau E 2010 Nonlinear Dyn. 59 45
[21] Huan S M, Li Q D, Yang X S 2012 Nonlinear Dyn. 69 1915
[22] L J H, Yu X H, Chen G R 2003 IEEE Trans. Circ. Syst. I 50 198
[23] Yalcin M, Suykens J, van de Walle J 2005 Chaos Modeling and Control Systems Design (Singapore: World Scientific) p59
[24] L J H, Chen G R, Yu X H, Leung H 2004 IEEE Trans. Circ. Syst. I 51 2476
[25] L J H, Yu S M, Leung H, Chen G R 2006 IEEE Trans. Circ. Syst. I 53 149
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[1] Chua L O 1971 IEEE Trans. Circ. Theor. 18 507
[2] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[3] Zhou J, Huang D 2012 Chin. Phys. B 21 048401
[4] Wang L D, Li H F, Duan S K, Huang T W, Wang H M 2016 Neurccomputing 171 23
[5] Min G Q, Wang L D, Duan S K 2015 Acta Phys. Sin. 64 210507 (in Chinese) [闵国旗, 王丽丹, 段书凯 2015 物理学报 64 210507]
[6] Mandelbrot B B 1967 Science 156 636
[7] Mandelbrot B B 1975 Fractals: Form, Chance and Dimension (San Francisco: WH Freeman and Company) pp35-37
[8] Li H Q, Wang F Q 1999 Fractal Theory and Its Application in Molecular Science (Beijing: Science Press) p33 (in Chinese) [李后强, 汪富泉 1999分形理论及其在分子科学中的应用(北京:科学出版社) 第33页]
[9] Lorenz E N 1963 J. Atmos. Sci. 20 130
[10] Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. Circ. Syst. 33 1072
[11] Chen G R 1999 Int. J. Bifurcat. Chaos 9 1465
[12] Wang L D, Drakakis E, Duan S K, He P F, Liao X F 2012 Int. J. Bifurcat. Chaos 22 1250205
[13] Muthuswamy B, Kokate P P 2009 IETE Tech. Rev. 26 417
[14] Zhou Z W, Su Y L, Wang W D, Yan Y 2017 J. Petrol. Explor. Prod. Technol. 7 487
[15] Bouallegue K 2015 Int. J. Bifurcat. Chaos 25 1530002
[16] Chua L O, Roska T 1993 IEEE Trans. Circ. Syst. I 40 147
[17] Yalcin M, Suykens J, Vandewalle J, Ozoguz S 2002 Int. J. Bifurcat. Chaos 12 23
[18] Tang W K S, Zhong G Q, Chen G, Man K F 2001 IEEE Trans. Circ. Syst. I 48 1369
[19] Zarei A 2015 Nonlinear Dyn. 81 585
[20] More C, Vlad R, Chauveau E 2010 Nonlinear Dyn. 59 45
[21] Huan S M, Li Q D, Yang X S 2012 Nonlinear Dyn. 69 1915
[22] L J H, Yu X H, Chen G R 2003 IEEE Trans. Circ. Syst. I 50 198
[23] Yalcin M, Suykens J, van de Walle J 2005 Chaos Modeling and Control Systems Design (Singapore: World Scientific) p59
[24] L J H, Chen G R, Yu X H, Leung H 2004 IEEE Trans. Circ. Syst. I 51 2476
[25] L J H, Yu S M, Leung H, Chen G R 2006 IEEE Trans. Circ. Syst. I 53 149
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