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The classical attractors, defined as self-excited attractors, such as Lorenz attractor, Rssler attractor, Chua's attractor and many other well-known attractors, are all excited from unstable index-2 saddle-foci, namely, an attractor with an attraction basin corresponds to an unstable equilibrium. A new type of attractors, defined as hidden attractors, was first found and reported in 2011, whose attraction basin does not intersect with small neighborhoods of the equilibria of the system. Due to the existences of hidden attractors, some particular dynamical systems associated with line equilibrium, or no equilibrium, or stable equilibrium have attracted much attention recently. Additionally, by introducing memristors into existing oscillating circuits or substituting nonlinear resistors in classical chaotic circuits with memristors, a variety of memristor based chaotic and hyperchaotic circuits are simply established and has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we present a novel memristive system with no equilibrium, from which an interesting and striking phenomenon of coexistence of the behaviors of hidden multiple attractors and the corresponding multistability is perfectly demonstrated by numerical simulations and experimental measurements. According to a newly proposed circuit realization scheme, a new type of four-dimensional memristive self-oscillated system is easily implemented by directly replacing a linear coupling resistor in an existing three-dimensional self-oscillated system circuit with a voltage-controlled memristor. The proposed system has no equilibrium, but can generate various hidden attractors including periodic limit cycle, quasi-periodic limit cycle, chaotic attractor, and coexisting attractors and so on. Based on bifurcation diagram, Lyapunov exponent spectra, and phase portraits, complex hidden dynamics with respect to a system parameter of the memristive self-oscillated system are studied. Specially, when different initial conditions are used, the system displays the coexistence phenomenon of chaotic attractors with different topological structures or quasi-periodic limit cycle and chaotic attractor, as well as the phenomenon of multiple attractors of quasi-periodic limit cycle and chaotic attractors with multiple topological structures. The results imply that some coexisting hidden multiple attractors reflecting the emergences of multistability can be observed in the proposed memristive self-oscillated system, which are well illustrated by several conventional dynamical analysis tools. Based on PSIM circuit simulation model, the memristive self-oscillated system is easily made in at a hardware level on a breadboard and two kinds of dynamical behaviors of coexisting hidden multiple attractors are captured in hardware experiments. Hardware experimental measurements are consistent with numerical simulations, which demonstrates that the proposed memristive self-oscillated system has very abundant and complex hidden dynamical characteristics.
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[2] Rssler O E 1976 Phys. Lett. A 57 397
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[4] L J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659
[5] Liu W B, Chen G R 2003 Int. J. Bifurcation Chaos 13 261
[6] L J H, Chen G R, Cheng D 2004 Int. J. Bifurcation Chaos 14 1507
[7] Liu C X, Liu T, Liu L, Liu K 2004 Chaos, Solitions Fractals 22 1031
[8] Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295
[9] Bao B C, Liu Z, Xu J P 2009 J. Sys. Eng. Electron. 20 1179
[10] Yu S M, L J H, Yu X H, Chen G R 2012 IEEE Trans. Circuits Syst. I: Regular Papers 59 1015
[11] Bao B C, Zhou G H, Xu J P, Liu Z 2010 Int. J. Bifurcation Chaos 20 2203
[12] Peng Z P, Wang C H, Lin Y, Luo X W 2014 Acta Phys. Sin. 63 240506 (in Chinese) [彭再平, 王春华, 林愿, 骆小文 2014 物理学报 63 240506]
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[16] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B 23 070503
[17] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[18] 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 物理学报 63 010502]
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[22] Wu H G, Bao B C, Chen M 2014 Chin. Phys. B 23 118401
[23] Wang X Y, Fitch A L, Iu H H C, Sreeramb V, Qi W G 2012 Chin. Phys. B 21 108501
[24] Corinto F, Ascoli A 2012 Electron. Lett. 48 824
[25] Bao B C, Yu J J, Hu F W, Liu Z 2014 Int. J. Bifurcation Chaos 24 1450143
[26] Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230
[27] Leonov G A, Kuznetsov N V, Mokaev T N 2015 Commu. Nonlinear Sci. Numer. Simul. 28 166
[28] Li C B, Sprott J C 2014 Int. J. Bifurcation Chaos 24 1450034
[29] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255
[30] Wei Z C 2011 Phys. Lett. A 376 102
[31] Sharma P R, Shrimali M D, Prasad A, Leonov G A, Kuznetsov N V 2015 Int. J. Bifurcation Chaos 25 1550061
[32] Zhao H T, Lin Y P, Dai Y X 2014 Int. J. Bifurcation Chaos 24 1450080
[33] Dang X Y, Li C B, Bao B C, Wu H G 2015 Chin. Phys. B 24 050503
[34] Xu Q, Lin Y, Bao B C, Chen M 2016 Chaos, Solitions Fractals 83 186
[35] Bao B C, Li Q D, Wang N, Xu Q 2016 Chaos 26 043111
[36] Pisarchik A N, Feudel U 2014 Phys. Rep. 540 167
[37] Patel M S, Patel U, Sen A, Sethia G C, Hens C, Dana S K 2014 Phys. Rev. E 89 022918
[38] Bao B C, Xu Q, Bao H, Chen M 2016 Electron. Lett. 52 1008
[39] Kuznetsov A P, Kuznetsov S P, Stankevich N V 2010 Commun. Nonlinear Sci. Numer. Simul. 15 1676
[40] Kuznetsov A P, Kuznetsov S P, Mosekilde E, Stankevich N V 2013 Eur. Phys. J. Spec. Top. 222 2391
[41] Kuznetsov A P, Kuznetsov S P, Mosekilde E, Stankevich N V 2015 J. Phys. A 48 125101
[42] Chen M, Yu J J, Bao B C 2015 Electron. Lett. 51 462
[43] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
[44] Bao B C, Wang C L, Wu H G, Qiao X H 2014 Acta Phys. Sin. 63 240504 (in Chinese) [包伯成, 王春丽, 武花干, 乔晓华 2014 物理学报 63 240504]
[45] Kengne J, Tabekoueng Z N, Tamba V K, Negou A N 2015 Chaos 25 103126
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[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Rssler O E 1976 Phys. Lett. A 57 397
[3] Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465
[4] L J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659
[5] Liu W B, Chen G R 2003 Int. J. Bifurcation Chaos 13 261
[6] L J H, Chen G R, Cheng D 2004 Int. J. Bifurcation Chaos 14 1507
[7] Liu C X, Liu T, Liu L, Liu K 2004 Chaos, Solitions Fractals 22 1031
[8] Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295
[9] Bao B C, Liu Z, Xu J P 2009 J. Sys. Eng. Electron. 20 1179
[10] Yu S M, L J H, Yu X H, Chen G R 2012 IEEE Trans. Circuits Syst. I: Regular Papers 59 1015
[11] Bao B C, Zhou G H, Xu J P, Liu Z 2010 Int. J. Bifurcation Chaos 20 2203
[12] Peng Z P, Wang C H, Lin Y, Luo X W 2014 Acta Phys. Sin. 63 240506 (in Chinese) [彭再平, 王春华, 林愿, 骆小文 2014 物理学报 63 240506]
[13] Bao B C 2013 An Introduction to Chaotic Circuits (Beijing: Science Press) p68 (in Chinese) [包伯成 2013 混沌电路导论 (北京: 科学出版社) 第68页]
[14] Chua L O 2012 Proc. IEEE 100 1920
[15] Wang G Y, He J L, Yuan F, Peng C J 2013 Chin. Phys. Lett. 30 110506
[16] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B 23 070503
[17] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[18] 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 物理学报 63 010502]
[19] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[20] Kim H, Sah M P, Yang C, Cho S, Chua L O 2012 IEEE Trans. Circuits Syst. I: Regular Papers 59 2422
[21] Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin. Phys. B 20 120502
[22] Wu H G, Bao B C, Chen M 2014 Chin. Phys. B 23 118401
[23] Wang X Y, Fitch A L, Iu H H C, Sreeramb V, Qi W G 2012 Chin. Phys. B 21 108501
[24] Corinto F, Ascoli A 2012 Electron. Lett. 48 824
[25] Bao B C, Yu J J, Hu F W, Liu Z 2014 Int. J. Bifurcation Chaos 24 1450143
[26] Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230
[27] Leonov G A, Kuznetsov N V, Mokaev T N 2015 Commu. Nonlinear Sci. Numer. Simul. 28 166
[28] Li C B, Sprott J C 2014 Int. J. Bifurcation Chaos 24 1450034
[29] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255
[30] Wei Z C 2011 Phys. Lett. A 376 102
[31] Sharma P R, Shrimali M D, Prasad A, Leonov G A, Kuznetsov N V 2015 Int. J. Bifurcation Chaos 25 1550061
[32] Zhao H T, Lin Y P, Dai Y X 2014 Int. J. Bifurcation Chaos 24 1450080
[33] Dang X Y, Li C B, Bao B C, Wu H G 2015 Chin. Phys. B 24 050503
[34] Xu Q, Lin Y, Bao B C, Chen M 2016 Chaos, Solitions Fractals 83 186
[35] Bao B C, Li Q D, Wang N, Xu Q 2016 Chaos 26 043111
[36] Pisarchik A N, Feudel U 2014 Phys. Rep. 540 167
[37] Patel M S, Patel U, Sen A, Sethia G C, Hens C, Dana S K 2014 Phys. Rev. E 89 022918
[38] Bao B C, Xu Q, Bao H, Chen M 2016 Electron. Lett. 52 1008
[39] Kuznetsov A P, Kuznetsov S P, Stankevich N V 2010 Commun. Nonlinear Sci. Numer. Simul. 15 1676
[40] Kuznetsov A P, Kuznetsov S P, Mosekilde E, Stankevich N V 2013 Eur. Phys. J. Spec. Top. 222 2391
[41] Kuznetsov A P, Kuznetsov S P, Mosekilde E, Stankevich N V 2015 J. Phys. A 48 125101
[42] Chen M, Yu J J, Bao B C 2015 Electron. Lett. 51 462
[43] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
[44] Bao B C, Wang C L, Wu H G, Qiao X H 2014 Acta Phys. Sin. 63 240504 (in Chinese) [包伯成, 王春丽, 武花干, 乔晓华 2014 物理学报 63 240504]
[45] Kengne J, Tabekoueng Z N, Tamba V K, Negou A N 2015 Chaos 25 103126
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