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A memristor is a nonlinear nanoscale-sized element with memory function, and it has an italic type 8 voltage-current relation curve that looks like a pinched hysteresis loop characteristic. The memristor is utilized to construct chaotic circuit, which has attracted the attention of the researchers. At present, most of studies focus on applying one or two memristors to the chaotic circuit. In order to study the multi memristor chaotic circuit, in this work we propose a six-order chaotic circuit with two flux-controlled memristors and a charge-controlled memristor. A corresponding six-order nonlinear dynamic differential equation of the circuit state variables is established. The dynamic properties of the circuit are demonstrated in detail. The analyses of equilibria and equilibrium stability show that the circuit has an equilibrium located in the three-dimensional space which is constituted by memristor internal state variables, and it is found that the equilibrium stability is determined by the circuit parameters and the initial states of three memristors. The Lyapunov exponent spectra and bifurcation diagrams of the circuit imply that it can produce two bifurcation behaviors by adjusting its parameters, which are Hopf bifurcation and anti-period doubling bifurcation. The hyperchaos, transient chaos and intermittency cycle phenomena are found in the same system. The dynamical behavior of this circuit is dependent on the initial state of memristor, showing different orbits such as chaotic oscillation, periodic oscillation and stable sink under different initial states. Finally, the simulation results indicate that some strange attractors like lotus type and superposition type are observed when voltage and electricity signal in observing chaotic attractors are generalized to power and energy signal, respectively. And the attractor production between the energy signals of the memristors are studied. Specially, when different initial conditions of three memristors are used to simulate the circuit, we can find the coexistence phenomenon of chaotic attractors with different topological structures or quasi-periodic limit cycle and chaotic attractor. The six-order chaotic oscillating circuit is mainly composed of three parts:the parallel connection between a flux-controlled memristor and capacitor, the serial connection between a charge-controlled memristor and inductor, and another flux-controlled memristor that is alone and floating, which enriches the application of memristor in high-order chaotic circuit. Compared with most of other chaotic systems, it has many circuit parameters and very complex topological structure, which enhances the complexity of chaotic system and the randomness of the generated signal. It is more difficult to decipher the encrypted information in chaotic secure communication, and thus it has better security performance and safety performance.
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Keywords:
- memristor /
- six-order chaotic circuit /
- floating /
- dynamics
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[2] Tour J M, He T 2008 Nature 453 42
[3] Chua L O 1971 IEEE Trans. Circ. Theory 18 507
[4] Wu A L, Zeng Z G 2012 Neural Networks 36 1
[5] Duan S K, Hu X F, Wang L D, Li C D 2012 Sci. China:Inf. Sci. 55 1446
[6] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[7] Hong Q H, Li Z J, Zeng J F, Zeng Y C 2014 Acta Phys. Sin. 63 180502 (in Chinese)[洪庆辉, 李志军, 曾金芳, 曾以成2014 物理学报 63 180502]
[8] Chua L O 2011 Appl. Phys. A 102 765
[9] Wang L, Yang C H, Wen J, Gai S 2015 J. Mater. Sci. 26 4618
[10] Yuan F, Wang G Y, Wang X W 2016 Chaos 26 073107
[11] Chua L O 2015 Radioengin 24 319
[12] Lin Z, Wang H 2010 IETE Tech. Rev. 27 318
[13] 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]
[14] Itoh M, Chua L O 2008 Int. J. Bifurc. Chaos 18 3183
[15] Muthuswamy B, Kokate P P 2009 IETE Tech. Rev. 26 417
[16] Muthuswamy B, Chua L O 2010 Int. J. Bifurc. Chaos 20 1567
[17] Bao B C, Liu Z, Xu J P 2010 Chin. Phys. B 19 030510
[18] Li Z J, Zeng Y C 2013 Chin. Phys. B 22 040502
[19] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B 23 070503
[20] Bao B C, Shi G D, Xu J P, Pan S H 2011 Sci. China:Tech. Sci. 41 1135 (in Chinese)[包伯成, 史国栋, 许建平, 刘中, 潘赛虎 2011中国科学:技术科学 41 1135]
[21] Buscarino A, Fortuna L, Frasca M, Valentina G L 2012 Int. J. Non. Sci. 22 023136
[22] Hong Q H, Zeng Y C, Li Z J 2013 Acta Phys. Sin. 62 230502 (in Chinese)[洪庆辉, 曾以成, 李志军 2013 物理学报 62 230502]
[23] Benhabib J, Nishimura K 1979 J. Econ. Theory 21 421
[24] Bao B C, Xu J P, Liu Z 2010 Chin. Phys. Lett. 27 070504
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[1] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[2] Tour J M, He T 2008 Nature 453 42
[3] Chua L O 1971 IEEE Trans. Circ. Theory 18 507
[4] Wu A L, Zeng Z G 2012 Neural Networks 36 1
[5] Duan S K, Hu X F, Wang L D, Li C D 2012 Sci. China:Inf. Sci. 55 1446
[6] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[7] Hong Q H, Li Z J, Zeng J F, Zeng Y C 2014 Acta Phys. Sin. 63 180502 (in Chinese)[洪庆辉, 李志军, 曾金芳, 曾以成2014 物理学报 63 180502]
[8] Chua L O 2011 Appl. Phys. A 102 765
[9] Wang L, Yang C H, Wen J, Gai S 2015 J. Mater. Sci. 26 4618
[10] Yuan F, Wang G Y, Wang X W 2016 Chaos 26 073107
[11] Chua L O 2015 Radioengin 24 319
[12] Lin Z, Wang H 2010 IETE Tech. Rev. 27 318
[13] 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]
[14] Itoh M, Chua L O 2008 Int. J. Bifurc. Chaos 18 3183
[15] Muthuswamy B, Kokate P P 2009 IETE Tech. Rev. 26 417
[16] Muthuswamy B, Chua L O 2010 Int. J. Bifurc. Chaos 20 1567
[17] Bao B C, Liu Z, Xu J P 2010 Chin. Phys. B 19 030510
[18] Li Z J, Zeng Y C 2013 Chin. Phys. B 22 040502
[19] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B 23 070503
[20] Bao B C, Shi G D, Xu J P, Pan S H 2011 Sci. China:Tech. Sci. 41 1135 (in Chinese)[包伯成, 史国栋, 许建平, 刘中, 潘赛虎 2011中国科学:技术科学 41 1135]
[21] Buscarino A, Fortuna L, Frasca M, Valentina G L 2012 Int. J. Non. Sci. 22 023136
[22] Hong Q H, Zeng Y C, Li Z J 2013 Acta Phys. Sin. 62 230502 (in Chinese)[洪庆辉, 曾以成, 李志军 2013 物理学报 62 230502]
[23] Benhabib J, Nishimura K 1979 J. Econ. Theory 21 421
[24] Bao B C, Xu J P, Liu Z 2010 Chin. Phys. Lett. 27 070504
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