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The pitchfork bifurcation and vibrational resonance are investigated in this paper. Based on the method of separating slow motion from fast motion, the equivalent equation to the slow motion is obtained. Then, the pitchfork bifurcation is studied. The results show that the amplitude of the high-frequency signal can induce the subcritical pitchfork bifurcation, while both the frequency of the high-frequency signal and the value of the fractional-order can induce supercritical pitchfork bifurcation. The pattern of the vibrational resonance depends on the pitchfork bifurcation. The vibrational resonance presents double-resonance pattern when the pitchfork bifurcation occurs. Or else, the vibrational resonance presents single-resonance pattern. The analytical predications are in good agreement with the numerical calculation results, which verifies the validity of the theoretical results.
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Keywords:
- subcritical pitchfork bifurcation /
- supercritical pitchfork bifurcation /
- fractional damping /
- vibrational resonance
[1] Landa P S, McClintock 2000 J. Phys. A 33 L433
[2] Gitterman M 2001 J. Phys. A 34 L355
[3] Blekhman I I, Landa P S 2004 Int. J. Non-Linear Mech. 39 421
[4] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Phys. Rev. E 80 046608
[5] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Chaos 19 043128
[6] Baltanas J P, Lopez L, Blechman I I, Landa P S, Zaikin A, Kurths J, Sanjuan M A F 2003 Phys. Rev. E 67 066119
[7] Chizhevsky V N, Smeu E, Giacomelli G 2003 Phys. Rev. Lett. 91 220602
[8] Chizhevsky V N, Giacomelli G 2006 Phys. Rev. E 73 22103
[9] Yang J H, Liu X B 2010 J. Phys. A 43 122001
[10] Yang J H, Liu X B 2010 Chaos 20 033124
[11] Yang J H, Liu X B 2010 Phys. Scr. 82 025006
[12] Yang J H, Liu X B 2011 Phys. Scr. 83 065008
[13] Jeevarathinam C, Rajasekar S, Sanjuan M A F 2011 Phys. Rev. E 83 066205
[14] Lin M, Huang Y M 2007 Acta Phys. Sin. 56 6173 (in Chinese) [林敏, 黄咏梅 2007 56 6173]
[15] Deng B, Wang J, Wei X, Tsang K M, Chan W L 2010 Chaos 20 013113
[16] Qin Y M, Wang J, Men C, Deng B, Wei X L 2011 Chaos 21 023133
[17] Yu H, Wang J, Sun J, Yu H 2012 Chaos 22 033105
[18] Sun J, Deng B, Liu C, Yu H, Wang J, Wei X, Zhao J 2013 Appl. Math. Model. 37 6311
[19] Yang J H, Zhu H 2012 Chaos 22 013112
[20] Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[21] Monje C A, Chen Y, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer)
[22] Blekhman I I 2000 Vibrational Mechanics (Singapore: World Scientific)
[23] Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer-Verlag)
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[1] Landa P S, McClintock 2000 J. Phys. A 33 L433
[2] Gitterman M 2001 J. Phys. A 34 L355
[3] Blekhman I I, Landa P S 2004 Int. J. Non-Linear Mech. 39 421
[4] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Phys. Rev. E 80 046608
[5] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Chaos 19 043128
[6] Baltanas J P, Lopez L, Blechman I I, Landa P S, Zaikin A, Kurths J, Sanjuan M A F 2003 Phys. Rev. E 67 066119
[7] Chizhevsky V N, Smeu E, Giacomelli G 2003 Phys. Rev. Lett. 91 220602
[8] Chizhevsky V N, Giacomelli G 2006 Phys. Rev. E 73 22103
[9] Yang J H, Liu X B 2010 J. Phys. A 43 122001
[10] Yang J H, Liu X B 2010 Chaos 20 033124
[11] Yang J H, Liu X B 2010 Phys. Scr. 82 025006
[12] Yang J H, Liu X B 2011 Phys. Scr. 83 065008
[13] Jeevarathinam C, Rajasekar S, Sanjuan M A F 2011 Phys. Rev. E 83 066205
[14] Lin M, Huang Y M 2007 Acta Phys. Sin. 56 6173 (in Chinese) [林敏, 黄咏梅 2007 56 6173]
[15] Deng B, Wang J, Wei X, Tsang K M, Chan W L 2010 Chaos 20 013113
[16] Qin Y M, Wang J, Men C, Deng B, Wei X L 2011 Chaos 21 023133
[17] Yu H, Wang J, Sun J, Yu H 2012 Chaos 22 033105
[18] Sun J, Deng B, Liu C, Yu H, Wang J, Wei X, Zhao J 2013 Appl. Math. Model. 37 6311
[19] Yang J H, Zhu H 2012 Chaos 22 013112
[20] Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[21] Monje C A, Chen Y, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer)
[22] Blekhman I I 2000 Vibrational Mechanics (Singapore: World Scientific)
[23] Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer-Verlag)
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