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Vibrational resonance is a resonant dynamics induced by a high-frequency periodic force at the low-frequency of the input periodic signal, and the input periodic signal is enhanced by a high-frequency signal. In this paper, a linear time-delayed feedback bistable system with an asymmetric double-well potential driven by both low-frequency and high-frequency periodic forces is constructed. Based on this model, the vibrational resonance phenomenon is investigated. Making use of the method of separating slow motion from fast motion under the conditions of Ω>>ω (Ω is the frequency of the high-frequency signal and ω is the one of the low-frequency signal), equivalent equations to the slow motion and the fast motion are obtained. Neglecting the nonlinear factors, the analytical expression of the response amplitude Q can be obtained, and the effects of the time-delay parameter α and the asymmetric parameter r on the vibrational resonance are discussed in detail. Moreover, the locations at which the vibrational resonance occurs, are obtained by means of solving the condition for a resonance to occur. A major consequence of time-delayed feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delayed parameter, i.e. in Q-α plot, α can induce the Q which is periodic with the periods of the high-frequency signal and the low-frequency signal. The locations at which the vibrational resonance occurs are not changed by the asymmetric parameter r. However, the resonance amplitude is enhanced with increasing r. Specifically, the resonance amplitude is greatly enhanced when r>0.15. On the other hand, in the symmetric case (r=0), BVR at which the vibrational resonance occurs is periodic with the periods of high-frequency signal and low-frequency signal as α increases, which is shown in BVR-α (B is the amplitude of the high-frequency signal) plot. In Q-Ω plot, Q is presented by multi-resonance at the small values of B and Ω, but Q tends to a fixed value at the small values of B and the large values of Ω. We believe that the above theoretical observations will stimulate the experimental study of vibrational resonance in nonlinear oscillators and electronic circuits with time-delayed feedback.
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Keywords:
- time delay /
- asymmetric bistable system /
- vibrational resonance
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[2] Gitterman M 2001 J. Phys. A 34 L355
[3] Zaikin A A, López L, Baltanás J P, Kurths J, Sanjuán M A F 2002 Phys. Rev. E 66 011106
[4] Baltanás J P, López L, Blechman I I, Landa P S, Zaikin A, Kurths J, Sanjuán M A F 2003 Phys.Rev.E 67 066119
[5] Chizhevsky V N, Smeu E, Giacomelli G 2003 Phys. Rev. Lett. 91 220602
[6] Chizhevsky V N, Giacomelli G 2006 Phys. Rev. E 73 022103
[7] Chizhevsky V N, Giacomelli G 2008 Phys. Rev. E 77 051126
[8] Yao C G, Liu Y, Zhan M 2011 Phys. Rev. E83 061122
[9] Gandhimathi V M, Rajasekar S, Kurths J 2006 Phys. Lett. A 360 279
[10] Gandhimathi V M, Rajasekar S 2007 Phys. Scr. 76 693
[11] Yang J H, Liu X B 2010 Chaos 20 033124
[12] Yang J H, Zhu H 2012 Chaos 22 013112
[13] Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[14] Zhang L, Xie T T, Luo M K 2014 Acta Phys. Sin. 63 010506 (in Chinese) [张路, 谢天婷, 罗懋康 2014 物理学报 63 010506]
[15] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuán M A F 2009 Phys. Rev. E 80 046608
[16] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuán M A F 2009 Chaos 19 043128
[17] Yang J H, Liu H G, Chen G 2012 Acta Phys. Sin. 61 180503 (in Chinese) [杨建华, 刘后广, 程刚 2012 物理学报 61 180503]
[18] Wang C J 2011 Chin. Phys. Lett. 28 090504
[19] Deng B, Wang J, Wei X L 2009 Chaos 19 013117
[20] Deng B, Wang J, Wei X L, Yu H T, Li H Y 2014 Phys. Rev. E 89 062916
[21] Yang L J, Liu W H, Yi Ming, Wang C J, Zhu Q M, Zhan X, Jia Y 2012 Phys. Rev. E 86 016209
[22] Wang C J, Yang K L 2012 Chin. J. Phys. 50 607
[23] Jeevarathinam C, Rajasekar S, Sanjuán M A F 2013 Ecol. Complex. 15 33
[24] Ramana Reddy D V, Sen A, Johnston G L 1998 Phys. Rev. Lett. 80 5109
[25] Jia Z L 2009 Int. J. Theor. Phys. 48 226
[26] Wang C J, Yi M, Yang K L, Yang L J 2012 BMC Syst. Biol. 6 S9
[27] Yang J H, Liu X B 2010 J. Phys. A: Math. Theor. 43 122001
[28] Yang J H, Liu X B 2012 Acta Phys. Sin. 61 010505 (in Chinese) [杨建华, 刘先斌 2012 物理学报 61 010505]
[29] Wang C J, Yang K L, Qu S X 2014 Int. J. Mod. Phys. B 28 1450103
[30] Yang J H, Liu X B 2010 Phys. Scr. 82 025006
[31] Daza A, Wagemakers A, Rajasekar S, Sanjuán M A F 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 411
[32] Hu D L, Yang J H, Liu X B 2014 Comput. Biol. Med. 45 80
[33] Jeevarathinam C, Rajasekar S, Sanjuán M A F 2011 Phys. Rev.E 83 066205
[34] Wang C J, Dai Z C, Mei D C 2011 Commun. Theor. Phys. 56 1041
[35] Wio H S, Bouzat S 1999 Braz. J. Phys. 29 136
[36] Chizhevsky V N 2008 Int. J. Bifurcat. Chaos 18 1767
[37] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuán M A F 2011 Int. J. Bifurcat. Chaos 21 275
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[1] Landa P, McClintock P 2000 J. Phys. A 33 L433
[2] Gitterman M 2001 J. Phys. A 34 L355
[3] Zaikin A A, López L, Baltanás J P, Kurths J, Sanjuán M A F 2002 Phys. Rev. E 66 011106
[4] Baltanás J P, López L, Blechman I I, Landa P S, Zaikin A, Kurths J, Sanjuán M A F 2003 Phys.Rev.E 67 066119
[5] Chizhevsky V N, Smeu E, Giacomelli G 2003 Phys. Rev. Lett. 91 220602
[6] Chizhevsky V N, Giacomelli G 2006 Phys. Rev. E 73 022103
[7] Chizhevsky V N, Giacomelli G 2008 Phys. Rev. E 77 051126
[8] Yao C G, Liu Y, Zhan M 2011 Phys. Rev. E83 061122
[9] Gandhimathi V M, Rajasekar S, Kurths J 2006 Phys. Lett. A 360 279
[10] Gandhimathi V M, Rajasekar S 2007 Phys. Scr. 76 693
[11] Yang J H, Liu X B 2010 Chaos 20 033124
[12] Yang J H, Zhu H 2012 Chaos 22 013112
[13] Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[14] Zhang L, Xie T T, Luo M K 2014 Acta Phys. Sin. 63 010506 (in Chinese) [张路, 谢天婷, 罗懋康 2014 物理学报 63 010506]
[15] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuán M A F 2009 Phys. Rev. E 80 046608
[16] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuán M A F 2009 Chaos 19 043128
[17] Yang J H, Liu H G, Chen G 2012 Acta Phys. Sin. 61 180503 (in Chinese) [杨建华, 刘后广, 程刚 2012 物理学报 61 180503]
[18] Wang C J 2011 Chin. Phys. Lett. 28 090504
[19] Deng B, Wang J, Wei X L 2009 Chaos 19 013117
[20] Deng B, Wang J, Wei X L, Yu H T, Li H Y 2014 Phys. Rev. E 89 062916
[21] Yang L J, Liu W H, Yi Ming, Wang C J, Zhu Q M, Zhan X, Jia Y 2012 Phys. Rev. E 86 016209
[22] Wang C J, Yang K L 2012 Chin. J. Phys. 50 607
[23] Jeevarathinam C, Rajasekar S, Sanjuán M A F 2013 Ecol. Complex. 15 33
[24] Ramana Reddy D V, Sen A, Johnston G L 1998 Phys. Rev. Lett. 80 5109
[25] Jia Z L 2009 Int. J. Theor. Phys. 48 226
[26] Wang C J, Yi M, Yang K L, Yang L J 2012 BMC Syst. Biol. 6 S9
[27] Yang J H, Liu X B 2010 J. Phys. A: Math. Theor. 43 122001
[28] Yang J H, Liu X B 2012 Acta Phys. Sin. 61 010505 (in Chinese) [杨建华, 刘先斌 2012 物理学报 61 010505]
[29] Wang C J, Yang K L, Qu S X 2014 Int. J. Mod. Phys. B 28 1450103
[30] Yang J H, Liu X B 2010 Phys. Scr. 82 025006
[31] Daza A, Wagemakers A, Rajasekar S, Sanjuán M A F 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 411
[32] Hu D L, Yang J H, Liu X B 2014 Comput. Biol. Med. 45 80
[33] Jeevarathinam C, Rajasekar S, Sanjuán M A F 2011 Phys. Rev.E 83 066205
[34] Wang C J, Dai Z C, Mei D C 2011 Commun. Theor. Phys. 56 1041
[35] Wio H S, Bouzat S 1999 Braz. J. Phys. 29 136
[36] Chizhevsky V N 2008 Int. J. Bifurcat. Chaos 18 1767
[37] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuán M A F 2011 Int. J. Bifurcat. Chaos 21 275
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