Attempt to generalize fractional-order electric elements to complex-order ones

doi:10.1088/1674-1056/26/6/060503

Abstract The complex derivative D^α±jβ, with α, β∈R+ is a generalization of the concept of integer derivative, where α=1, β=0. Fractional-order electric elements and circuits are becoming more and more attractive. In this paper, the complex-order electric elements concept is proposed for the first time, and the complex-order elements are modeled and analyzed. Some interesting phenomena are found that the real part of the order affects the phase of output signal, and the imaginary part affects the amplitude for both the complex-order capacitor and complex-order memristor. More interesting is that the complex-order capacitor can do well at the time of fitting electrochemistry impedance spectra. The complex-order memristor is also analyzed. The area inside the hysteresis loops increases with the increasing of the imaginary part of the order and decreases with the increasing of the real part. Some complex case of complex-order memristors hysteresis loops are analyzed at last, whose loop has touching points beyond the origin of the coordinate system.

Keywords: complex derivative fractional-order elements imaginary and real part memristor

Received: 02 March 2017
Revised: 22 March 2017
Accepted manuscript online:

PACS:	05.45.-a	(Nonlinear dynamics and chaos)
	05.10.-a	(Computational methods in statistical physics and nonlinear dynamics)

Corresponding Authors: Gangquan Si
E-mail: sigangquan@mail.xjtu.edu.cn

Cite this article:

Gangquan Si(司刚全), Lijie Diao(刁利杰), Jianwei Zhu(朱建伟), Yuhang Lei(雷妤航), Yanbin Zhang(张彦斌) Attempt to generalize fractional-order electric elements to complex-order ones 2017 Chin. Phys. B 26 060503

[1]	Machado J A T, Kiryakova V and Mainardi F 2014 Commun. Nonlinear Sci. Numer. Simul. 16 1140
[2]	Jesus I S, Machado J A T and Cunha J B 2008 J. Vib. Control 14 1389
[3]	Das-Gupta D K and Scarpa P C N 1996 IEEE Trans. Dielectr. Electr. Insul. 3 366
[4]	Tarasov V E 2009 Theor Math. Phys. 158 355
[5]	Zhang Y, Tian Q and Chen L 2009 Multibody Syst. Dyn. 21 305
[6]	Povstenko Y Z 2005 J. Therm. Stresses 28 83
[7]	Magin R L 2010 Comput. Math. Appl. 59 1586
[8]	Elliott R J and Hoek J V D 2003 Math. Financ. 13 301
[9]	Tang Y, Gao H, Zhang W and Kurths J 2015 Automatica 53 346
[10]	Tang Y, Xing X, Karimi H R and Kocarev L 2016 IEEE Trans. Ind. Electron. 63 1299
[11]	Pinto C M A and Machado J A T 2011 Nonlinear Dyn. 65 247
[12]	Silva M F, Machado J A T and Barbosa R S 2006 Signal Process 86 2785
[13]	Pinto C A and Machado J A T 2012 Int. J. Bifurcat. 21 3053
[14]	Machado J A T 2013 J. Optimiz. Theory. Appl. 156 2
[15]	Lin X, Zhou S and Li H 2016 Chaos, Solitons, and Fractals 26 1595
[16]	Machado J A T 2013 Commun. Nonlinear Sci. Numer. Simul. 18 264
[17]	Jonscher A K 1987 IEEE Trans. Dielectr. Electr. 22 357
[18]	Westerlund S and Ekstam L 1994 IEEE Trans. Dielectr. Electr. Insul. 1 826
[19]	Westerlund S 1991 Phys. Scr. 43 174
[20]	Sugi M, Hirano Y and Miura Y F 1999 IEICE Trans. Fundanentals 82 1627
[21]	Sugi M, Hirano Y, Miura Y F and Saito K 2002 Colloids and Surfaces A: Physicochemical and Engineering Aspects 198 683
[22]	Yang N N, Liu C X and Wu C J 2012 Chin. Phys. B 21 080503
[23]	Wang F and Ma X 2013 J. Power Electron. 13 1008
[24]	Fouda M E, Soltan A and Radwan A G 2016 Integr. Circ. Sign. 87 301
[25]	Hartley T T, Lorenzo C F and Qammer H K 1995 IEEE Trans. CAS-I 42 485
[26]	Zhang H, Chen D Y, Zhou K and Wang Y C 2015 Chin. Phys. B 24 030203
[27]	Zheng H, Qian J, Chen D Y and Herbert H C Iu 2015 Chin. Phys. B 24 080204
[28]	Zhou R, Zhang R F and Chen D Y 2015 J. Electr. Eng. Technol. 10 1598
[29]	Butzer P L and Westphal U 2015 Apidologie 33 233
[30]	Ortigueira M D, Rodríguez-Germa L and Trujillo J J 2011 Commun. Nonlinear Sci. Numer. Simul. 16 4174
[31]	Barsoukov E and Macdonald J R 2005 Impedance Spectroscopy, (Wiley-Interscience) pp. 1-26
[32]	Macdonald J R 1987 J. Electroanal. Chem. 223 25
[33]	Chua L O 1971 IEEE Trans. Circuit Theory 18 507
[34]	Strukov D, Snider G, Stewart D and Williams R 2008 Nature 453 80
[35]	Prezioso M, Merrikh-Bayat F and Hoskins B D 2015 Nature 521 61
[36]	Wang G Y, Cai B Z, Jin P P, et al. 2016 Chin. Phys. B 25 090502
[37]	Wang G Y, Cai B Z, Jin P P and Hu T L 2016 Chin. Phys. B 25 010503

[1]	Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh-Rose and FitzHugh-Nagumo neurons with two time delays Zhan-Hong Guo(郭展宏), Zhi-Jun Li(李志军), Meng-Jiao Wang(王梦蛟), and Ming-Lin Ma(马铭磷). Chin. Phys. B, 2023, 32(3): 038701.
[2]	Memristor's characteristics: From non-ideal to ideal Fan Sun(孙帆), Jing Su(粟静), Jie Li(李杰), Shukai Duan(段书凯), and Xiaofang Hu(胡小方). Chin. Phys. B, 2023, 32(2): 028401.
[3]	High throughput N-modular redundancy for error correction design of memristive stateful logic Xi Zhu(朱熙), Hui Xu(徐晖), Weiping Yang(杨为平), Zhiwei Li(李智炜), Haijun Liu(刘海军), Sen Liu(刘森), Yinan Wang(王义楠), and Hongchang Long(龙泓昌). Chin. Phys. B, 2023, 32(1): 018502.
[4]	Memristor hyperchaos in a generalized Kolmogorov-type system with extreme multistability Xiaodong Jiao(焦晓东), Mingfeng Yuan(袁明峰), Jin Tao(陶金), Hao Sun(孙昊), Qinglin Sun(孙青林), and Zengqiang Chen(陈增强). Chin. Phys. B, 2023, 32(1): 010507.
[5]	High-performance artificial neurons based on Ag/MXene/GST/Pt threshold switching memristors Xiao-Juan Lian(连晓娟), Jin-Ke Fu(付金科), Zhi-Xuan Gao(高志瑄),Shi-Pu Gu(顾世浦), and Lei Wang(王磊). Chin. Phys. B, 2023, 32(1): 017304.
[6]	Firing activities in a fractional-order Hindmarsh-Rose neuron with multistable memristor as autapse Zhi-Jun Li(李志军), Wen-Qiang Xie(谢文强), Jin-Fang Zeng(曾金芳), and Yi-Cheng Zeng(曾以成). Chin. Phys. B, 2023, 32(1): 010503.
[7]	Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system Sheng-Hao Jia(贾生浩), Yu-Xia Li(李玉霞), Qing-Yu Shi(石擎宇), and Xia Huang(黄霞). Chin. Phys. B, 2022, 31(7): 070505.
[8]	Fabrication and investigation of ferroelectric memristors with various synaptic plasticities Qi Qin(秦琦), Miaocheng Zhang(张缪城), Suhao Yao(姚苏昊), Xingyu Chen(陈星宇), Aoze Han(韩翱泽),Ziyang Chen(陈子洋), Chenxi Ma(马晨曦), Min Wang(王敏), Xintong Chen(陈昕彤), Yu Wang(王宇),Qiangqiang Zhang(张强强), Xiaoyan Liu(刘晓燕), Ertao Hu(胡二涛), Lei Wang(王磊), and Yi Tong(童祎). Chin. Phys. B, 2022, 31(7): 078502.
[9]	Pulse coding off-chip learning algorithm for memristive artificial neural network Ming-Jian Guo(郭明健), Shu-Kai Duan(段书凯), and Li-Dan Wang(王丽丹). Chin. Phys. B, 2022, 31(7): 078702.
[10]	A mathematical analysis: From memristor to fracmemristor Wu-Yang Zhu(朱伍洋), Yi-Fei Pu(蒲亦非), Bo Liu(刘博), Bo Yu(余波), and Ji-Liu Zhou(周激流). Chin. Phys. B, 2022, 31(6): 060204.
[11]	The dynamics of a memristor-based Rulkov neuron with fractional-order difference Yan-Mei Lu(卢艳梅), Chun-Hua Wang(王春华), Quan-Li Deng(邓全利), and Cong Xu(徐聪). Chin. Phys. B, 2022, 31(6): 060502.
[12]	Memristor-based multi-synaptic spiking neuron circuit for spiking neural network Wenwu Jiang(蒋文武), Jie Li(李杰), Hongbo Liu(刘洪波), Xicong Qian(钱曦聪), Yuan Ge(葛源), Lidan Wang(王丽丹), and Shukai Duan(段书凯). Chin. Phys. B, 2022, 31(4): 040702.
[13]	Complex dynamic behaviors in hyperbolic-type memristor-based cellular neural network Ai-Xue Qi(齐爱学), Bin-Da Zhu(朱斌达), and Guang-Yi Wang(王光义). Chin. Phys. B, 2022, 31(2): 020502.
[14]	A novel hyperchaotic map with sine chaotification and discrete memristor Qiankun Sun(孙乾坤), Shaobo He(贺少波), Kehui Sun(孙克辉), and Huihai Wang(王会海). Chin. Phys. B, 2022, 31(12): 120501.
[15]	A spintronic memristive circuit on the optimized RBF-MLP neural network Yuan Ge(葛源), Jie Li(李杰), Wenwu Jiang(蒋文武), Lidan Wang(王丽丹), and Shukai Duan(段书凯). Chin. Phys. B, 2022, 31(11): 110702.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Attempt to generalize fractional-order electric elements to complex-order ones

Cite this article:

Online attention