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Because of simple schematic structure and complex dynamical behaviors, the Chua's system is considered as a paradigm for chaos research. Despite a great many of studies relating to the Chua's system, most of them focus on its positive parameter space. This is explained by the fact that the implementation of the Chua's circuit with negative parameters needs resistors, inductances and/or capacitors with negative values, and thus leads to physical impossibility. In order to extend the parameter space of the Chua's system to its negative side, where all system parameters are negative, an equivalent realization of the Chua's circuit is developed with off-the-shelf electronic components by an electronic analogy method. Recently, the research of fractional-order chaotic systems has received considerable interest. However, the theoretical and experimental studies of the fractional-order Chua's system with negative parameters are still lacking. In this study, we set up a model of the fractional-order Chua's system in negative parameter space. The stability of all equilibrium points is investigated with the fractional-order stability theory. Based on the Grnwald-Letnikov derivative, the dynamical behaviors dependent on the control parameter and the fractional orders are investigated by standard nonlinear analysis techniques including phase portraits, the largest Lyapunov exponents, and bifurcation diagrams. In order to further verify the dynamic behaviors of the fractional-order Chua's system with negative parameters, an experimental implementation of the Chua's circuit with negative parameters based on an electronic analogy is performed with off-the-shelf electronic components such as operational amplifiers, resistors and capacitors. The experimental tests are conducted on the resulting circuit. A period-doubling bifurcation route to chaos is successfully observed and some typical phase diagrams are captured by an oscilloscope, which are well consistent with theoretical analyses and numerical simulations. The numerical simulations and the experimental results show that the fractional-order Chua's system in negative parameter space can still exhibit rich dynamical behaviors. But it is worth noting that the classical double-scroll chaotic attractor emerging in a conventional Chua's system cannot be found in this system. This work focuses mainly on the dynamical behaviors of the fractional-order Chua's system with negative parameters, which was not reported previously. Thus the research results of this study will further enrich the dynamical behaviors of the Chua's system, and play a positive role in promoting the chaos-based applications of the Chua's system. Meanwhile, the results obtained in this work lead to the conjecture that there remain some unknown and striking behaviors in the Chua's system with negative parameters, which need further revealing.
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Keywords:
- fractional order /
- Chua's system /
- negative parameter space /
- chaotic circuit
[1] He S B, Sun K H, Banerjee S 2016 Eur. Phys. J. Plus. 131 254
[2] Liu X J, Hong L, Jiang J 2016 Acta Phys. Sin. 65 180502 (in Chinese)[刘晓君, 洪灵, 江俊 2016 65 180502]
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[6] Shao S Y, Min F H, Ma M L, Wang E R 2013 Acta Phys. Sin. 62 130504 (in Chinese)[邵书义, 闵富红, 马美玲, 王恩荣 2013 62 130504]
[7] Xi H L, Yu S M, Zhang R X, Xu L 2014 Optik 125 2036
[8] He S B, Sun K H, Wang H H 2016 Math. Meth. Appl. Sci. 39 2965
[9] Bao B C, Wang N, Chen M, Xu Q, Wang J 2016 Nonlinear Dyn. 84 511
[10] Li Z J, Ma M L, Wang M J, Zeng Y C 2017 Int. J. Electron. Commun. 71 21
[11] Zhang X G, Sun H T, Zhao J L, Liu J Z, Ma Y D, Han T W 2014 Acta Phys. Sin. 63 200503 (in Chinese)[张新国, 孙洪涛, 赵金兰, 刘冀钊, 马义德, 韩廷武 2014 63 200503]
[12] Ma M L, Min F H, Shao S Y, Huang M Y 2014 Acta Phys. Sin. 63 010507 (in Chinese)[马美玲, 闵富红, 邵书义, 黄苗玉 2014 63 010507]
[13] Banerjee T 2012 Nonlinear Dyn. 68 565
[14] Cafagna D, Grassi G 2008 Int. J. Bifurcation Chaos 18 615
[15] Agarwal R P, El-Sayed A M A, Salman S M 2013 Adv. Differ. Equ-NY 1 320
[16] Zhang H, Chen D Y, Zhou K, Wang Y C 2015 Chin. Phys. B 24 030203
[17] Zhu H, Zhou S, Zhang J 2009 Chaos Solitons Fract. 39 1595
[18] Li C P, Deng W H, Xu D 2006 Physica A 36 171
[19] Rocha R, Medrano T R O 2009 Nonlinear Dyn. 56 389
[20] Medrano T R O, Rocha R 2014 Int. J. Bifurcation Chaos 24 1430025
[21] Hartly T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. CAS I 42 485
[22] Zhu H 2007 M. S. Dissertation (Chongqing:Chongqing University) (in Chinese)[朱浩 2007 硕士学位论文 (重庆:重庆大学)]
[23] Hu J B, Zhao L D 2013 Acta Phys. Sin. 62 240504 (in Chinese)[胡建兵, 赵灵冬 2013 62 240504]
[24] Sun K H, Yang J L, Ding J F, Sheng L Y 2010 Acta Phys. Sin. 59 8385 (in Chinese)[孙克辉, 杨静利, 丁家峰, 盛利元 2010 59 8385]
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[1] He S B, Sun K H, Banerjee S 2016 Eur. Phys. J. Plus. 131 254
[2] Liu X J, Hong L, Jiang J 2016 Acta Phys. Sin. 65 180502 (in Chinese)[刘晓君, 洪灵, 江俊 2016 65 180502]
[3] Li C L, Zhang J 2016 Int. J. Syst. Sci. 47 2440
[4] Lin F F, Zeng Z Z 2017 Acta Phys. Sin. 66 090504 (in Chinese)[林飞飞, 曾喆昭 2017 66 090504]
[5] Li Z J, Zeng Y C, Li Z B 2014 Acta Phys. Sin. 63 010502 (in Chinese)[李志军, 曾以成, 李志斌 2014 63 010502]
[6] Shao S Y, Min F H, Ma M L, Wang E R 2013 Acta Phys. Sin. 62 130504 (in Chinese)[邵书义, 闵富红, 马美玲, 王恩荣 2013 62 130504]
[7] Xi H L, Yu S M, Zhang R X, Xu L 2014 Optik 125 2036
[8] He S B, Sun K H, Wang H H 2016 Math. Meth. Appl. Sci. 39 2965
[9] Bao B C, Wang N, Chen M, Xu Q, Wang J 2016 Nonlinear Dyn. 84 511
[10] Li Z J, Ma M L, Wang M J, Zeng Y C 2017 Int. J. Electron. Commun. 71 21
[11] Zhang X G, Sun H T, Zhao J L, Liu J Z, Ma Y D, Han T W 2014 Acta Phys. Sin. 63 200503 (in Chinese)[张新国, 孙洪涛, 赵金兰, 刘冀钊, 马义德, 韩廷武 2014 63 200503]
[12] Ma M L, Min F H, Shao S Y, Huang M Y 2014 Acta Phys. Sin. 63 010507 (in Chinese)[马美玲, 闵富红, 邵书义, 黄苗玉 2014 63 010507]
[13] Banerjee T 2012 Nonlinear Dyn. 68 565
[14] Cafagna D, Grassi G 2008 Int. J. Bifurcation Chaos 18 615
[15] Agarwal R P, El-Sayed A M A, Salman S M 2013 Adv. Differ. Equ-NY 1 320
[16] Zhang H, Chen D Y, Zhou K, Wang Y C 2015 Chin. Phys. B 24 030203
[17] Zhu H, Zhou S, Zhang J 2009 Chaos Solitons Fract. 39 1595
[18] Li C P, Deng W H, Xu D 2006 Physica A 36 171
[19] Rocha R, Medrano T R O 2009 Nonlinear Dyn. 56 389
[20] Medrano T R O, Rocha R 2014 Int. J. Bifurcation Chaos 24 1430025
[21] Hartly T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. CAS I 42 485
[22] Zhu H 2007 M. S. Dissertation (Chongqing:Chongqing University) (in Chinese)[朱浩 2007 硕士学位论文 (重庆:重庆大学)]
[23] Hu J B, Zhao L D 2013 Acta Phys. Sin. 62 240504 (in Chinese)[胡建兵, 赵灵冬 2013 62 240504]
[24] Sun K H, Yang J L, Ding J F, Sheng L Y 2010 Acta Phys. Sin. 59 8385 (in Chinese)[孙克辉, 杨静利, 丁家峰, 盛利元 2010 59 8385]
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