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February  2021, 26(2): 847-860. doi: 10.3934/dcdsb.2020144

Chaos control in a special pendulum system for ultra-subharmonic resonance

Xianwei Chen 1,, , Xiangling Fu 1, and  Zhujun Jing 2,

1. 

School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China
2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Xianwei Chen

Received  January 2019 Revised  December 2019 Published  February 2021 Early access  May 2020

Fund Project: This work is supported by the Province Natural Science Foundation of Hunan (No. 2018JJ2110)

In this paper, we study the chaos control of pendulum system with vibration of suspension axis for ultra-subharmonic resonance by using Melnikov methods, and give a necessary condition for controlling heteroclinic chaos and homoclinic chaos, respectively. We give some bifurcation diagrams by numerical simulations, which indicate that the chaos behaviors for ultra-subharmonic resonance may be inhibited to periodic orbits by adjusting phase-difference of parametric excitation, and prove that results obtained are very effective in inhibiting chaos for ultra-subharmonic resonance.

Keywords: Pendulum equation, homoclinic chaos, heteroclinic chaos, chaos control, Melnikov methods.
Mathematics Subject Classification: Primary: 34G20, 34H05; Secondary: 34H10.
Citation: Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144
References:
[1]

T. S. Amer, The dynamical behavior of a rigid body relative equilibrium position, Advances in Mathematical Physics, 2017 (2017), Art. ID 8070525, 13pages. doi: 10.1155/2017/8070525.  Google Scholar

[2]

S. R. Bishop and M. J. Clifford, Zones of chaotic behavior in the parametrically exicited pendulum, J. Sound Vibration, 181 (1996), 142-147.  doi: 10.1006/jsvi.1996.0011.  Google Scholar

[3]

Y. Braiman and I. Goldhirsch, Taming chaotic dynamics with weak periodic perturbation, Phys. Rev. Lett., 66 (1991), 2545-2548.  doi: 10.1103/PhysRevLett.66.2545.  Google Scholar

[4]

H. J. Cao, X. B. Chi and G. R. Chen, Suppressing or inducing chaos by weak resonant excitations in an externally-forced froude pendulum, Int. J. Bifurcat. Chaos, 14 (2004), 1115-1120.  doi: 10.1142/S0218127404009673.  Google Scholar

[5]

R. Chacón, Natural symmetries and regularization by means of weak parametric modulations in the forced pendulum, Phys. Rev. E, 52 (1995), 2330-2337.  doi: 10.1103/PhysRevE.52.2330.  Google Scholar

[6]

R. Chacón, F. Palmero and F. Balibrea, Taming chaos in a driven Josephson junction, Int. J. Bifurcat. Chaos, 11 (2001), 1897-1909.   Google Scholar

[7]

R. Chacón, Relative effectiveness of weak periodic excitations in suppressing homoclinic heteroclinic chaos, Eur. Phys. J. B, 65 (2002), 207-210.   Google Scholar

[8]

L. J. Chen and J. B. Li, Chaotic behavior and subharmonic bifurcations for a rotating predulum equation, Int. J. Bifurcation Chaos, 14 (2004), 3477-3488.  doi: 10.1142/S0218127404011478.  Google Scholar

[9]

X. W. Chen and Z. J. Jing, Complex dynamics in a pendulum equation with a phase shift, Int. J. Bifurcat. Chaos, 22 (2012), 1250307, 40 pp. doi: 10.1142/S0218127412503075.  Google Scholar

[10]

X. W. Chen, Z. J. Jing and X. L. Fu, Chaos control in a pendulum system with excitations and phase shift, Nonlinear Dyn., 78 (2014), 317-327.  doi: 10.1007/s11071-014-1441-y.  Google Scholar

[11]

X. W. Chen, Z. J. Jing and X. L. Fu, Chaos control in a pendulum system with excitations, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 373-383.  doi: 10.3934/dcdsb.2015.20.373.  Google Scholar

[12]

M. J. Clifford and S. R. Bishop, Approximating the escape zone for the parametrically excited pendulum, J. Sound Vibr., 172 (1994), 572-576.  doi: 10.1006/jsvi.1994.1199.  Google Scholar

[13]

M. J. Clifford and S. R. Bishop, Rotating periodic orbits of parametrically excited pendulum, Phys. Lett. A, 201 (1995), 191-196.  doi: 10.1016/0375-9601(95)00255-2.  Google Scholar

[14]

D. D. A. Costa and M. A. Savi, Nonlinear dynamics of an SMA-pendulum system, Nonlinear Dynamics, 87 (2017), 1617-1627.  doi: 10.1007/s11071-016-3137-y.  Google Scholar

[15]

D. D. A. Costa and M. A. Savi, Chaos control of an SMA–pendulum system using thermal actuation with extended time-delayed feedback approach, Nonlinear Dynamics, 93 (2018), 571-583.  doi: 10.1007/s11071-018-4210-5.  Google Scholar

[16]

D. D'Humieres, M. R. Beasley, B. A. Huberman and A. F. Libchaber, Chaotic states and routes to chaos in the forced pendulum, Phys. Rev. A, 26 (1982), 3483-3492.  doi: 10.1103/PhysRevA.26.3483.  Google Scholar

[17]

W. X. Ding, H. Q. She, W. Huang and C. X. Yu, Controlling chaos in a discharge plasma, Phys. Rev. Lett., 72 (1994), 96-99.  doi: 10.1103/PhysRevLett.72.96.  Google Scholar

[18]

W. L. Ditto, S. N. Rauseo and M. L. Spano, Experimental control of chaos, Controlling Chaos, (1996), 105–107. doi: 10.1016/B978-012396840-1/50035-7.  Google Scholar

[19]

X. L. Fu, J. Deng and Z. J. Jing, Complex dynamics in physical pendulum equation with suspension axis vibrations, Acta Mathematica Applicatae Sinica, English series, 26 (2010), 55-78.  doi: 10.1007/s10255-008-8276-6.  Google Scholar

[20]

W. Garira and S. R. Bishop, Rotating solutions of the parametrically excited pendulum, J. Sound Vibr., 263 (2003), 233-239.  doi: 10.1016/S0022-460X(02)01435-9.  Google Scholar

[21]

Z. J. Jing, K. Y. Chan, D. S. Xu and H. J. Cao, Bifurcation of periodic solutions and chaos in Josephson system, Discr. Contin. Dyn. Syst.-Series A, 7 (2001), 573-592.  doi: 10.3934/dcds.2001.7.573.  Google Scholar

[22]

Z. J. Jing and H. J. Chao, Bifurcation of periodic orbits in Josephson equation with a phase shift, Int. J. Bifurcation and Chaos, 12 (2002), 1515-1530.  doi: 10.1142/S0218127402005261.  Google Scholar

[23]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (Ⅰ), Int. J. Bifurcat. Chaos, 16 (2006), 2887-2902.  doi: 10.1142/S0218127406016525.  Google Scholar

[24]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (Ⅱ), Int. J. Bifurcat. Chaos, 16 (2006), 3053-3078.  doi: 10.1142/S0218127406016653.  Google Scholar

[25]

T. Kapitaniak, Introduction, Chaos Solitons Fractals, 15 (2003), 201-203.   Google Scholar

[26]

M. Lakshman and K. Murall, Chaos in Nonlinear Oscillations–Controlling and Synchronization, , Singapore: World Scientific, 1996. Google Scholar

[27]

P. S. Landa, Regular and Chaotic Oscillations, Spring-Verlag, 2001. Google Scholar

[28]

M. Levi, F. Hoppensteadt and W. Miranke, Dynamics of the Josephson junction, Quart. Appl. Math., 36 (1978), 167-198.  doi: 10.1090/qam/484023.  Google Scholar

[29]

Z. H. Liu and W. Q. Zhu, Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation, Chaos Solit. Fract., 20 (2004), 593-607.  doi: 10.1016/j.chaos.2003.08.010.  Google Scholar

[30]

R. Lima and M. Pettine, Suppression of chaos by resonant parametric perturbations, Phys. Rev. A, 41 (1990), 726-733.  doi: 10.1103/PhysRevA.41.726.  Google Scholar

[31] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[32]

S. N. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley, New York, 1990.  Google Scholar

[33]

E. S. Ruslan, F. Alexander and L. Daniel, Energy control of a pendulum with quantized feedback, Automatica, 67 (2016), 171-177.  doi: 10.1016/j.automatica.2016.01.019.  Google Scholar

[34]

M. Salerno, Suppression of phase-locking chaos in long Josephson junctions by biharmonic microwave fields, Phys. Rev. B, 44 (1991), 2720-2726.  doi: 10.1103/PhysRevB.44.2720.  Google Scholar

[35]

M. Salerno and M. R. Samuelsen, Stabilization of chaotic phase locked dynamics in long Josephson junctions, Phys. Lett. A, 190 (1994), 177-181.  doi: 10.1016/0375-9601(94)90073-6.  Google Scholar

[36]

R. Q. Wang and Z. J. Jing, Chaos control of chaotic pendulum system, Chaos, Solitons and Fractals, 21 (2004), 201-207.  doi: 10.1016/j.chaos.2003.10.011.  Google Scholar

[37]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990. doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[38]

K. Yagasaki and T. Uozumi, Controlling chaos in a pendulum subjected to feedforward and feedback control, Int. J. Bifurcation and Chaos, 7 (1997), 2827-2835.  doi: 10.1142/S0218127497001904.  Google Scholar

[39]

J. P. Yang and Z. J. Jing, Inhibition of chaos in a pendulum equation, Chaos, Solitons and Fractals, 35 (2008), 726-737.  doi: 10.1016/j.chaos.2006.05.065.  Google Scholar

[40]

J. P. Yang and Z. J. Jing, Control of chaos in a three-well duffing system, Chaos, Solitons and Fractals, 41 (2009), 1311-1328.  doi: 10.1016/j.chaos.2008.05.018.  Google Scholar

[41]

J. P. Yang and Z. J. Jing, Controlling in a pendulum equation with ultra-subharmonic resonances, Chaos, Solitons and Fractals, 42 (2009), 1214-1226.  doi: 10.1016/j.chaos.2009.03.035.  Google Scholar

show all references

References:
[1]

T. S. Amer, The dynamical behavior of a rigid body relative equilibrium position, Advances in Mathematical Physics, 2017 (2017), Art. ID 8070525, 13pages. doi: 10.1155/2017/8070525.  Google Scholar

[2]

S. R. Bishop and M. J. Clifford, Zones of chaotic behavior in the parametrically exicited pendulum, J. Sound Vibration, 181 (1996), 142-147.  doi: 10.1006/jsvi.1996.0011.  Google Scholar

[3]

Y. Braiman and I. Goldhirsch, Taming chaotic dynamics with weak periodic perturbation, Phys. Rev. Lett., 66 (1991), 2545-2548.  doi: 10.1103/PhysRevLett.66.2545.  Google Scholar

[4]

H. J. Cao, X. B. Chi and G. R. Chen, Suppressing or inducing chaos by weak resonant excitations in an externally-forced froude pendulum, Int. J. Bifurcat. Chaos, 14 (2004), 1115-1120.  doi: 10.1142/S0218127404009673.  Google Scholar

[5]

R. Chacón, Natural symmetries and regularization by means of weak parametric modulations in the forced pendulum, Phys. Rev. E, 52 (1995), 2330-2337.  doi: 10.1103/PhysRevE.52.2330.  Google Scholar

[6]

R. Chacón, F. Palmero and F. Balibrea, Taming chaos in a driven Josephson junction, Int. J. Bifurcat. Chaos, 11 (2001), 1897-1909.   Google Scholar

[7]

R. Chacón, Relative effectiveness of weak periodic excitations in suppressing homoclinic heteroclinic chaos, Eur. Phys. J. B, 65 (2002), 207-210.   Google Scholar

[8]

L. J. Chen and J. B. Li, Chaotic behavior and subharmonic bifurcations for a rotating predulum equation, Int. J. Bifurcation Chaos, 14 (2004), 3477-3488.  doi: 10.1142/S0218127404011478.  Google Scholar

[9]

X. W. Chen and Z. J. Jing, Complex dynamics in a pendulum equation with a phase shift, Int. J. Bifurcat. Chaos, 22 (2012), 1250307, 40 pp. doi: 10.1142/S0218127412503075.  Google Scholar

[10]

X. W. Chen, Z. J. Jing and X. L. Fu, Chaos control in a pendulum system with excitations and phase shift, Nonlinear Dyn., 78 (2014), 317-327.  doi: 10.1007/s11071-014-1441-y.  Google Scholar

[11]

X. W. Chen, Z. J. Jing and X. L. Fu, Chaos control in a pendulum system with excitations, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 373-383.  doi: 10.3934/dcdsb.2015.20.373.  Google Scholar

[12]

M. J. Clifford and S. R. Bishop, Approximating the escape zone for the parametrically excited pendulum, J. Sound Vibr., 172 (1994), 572-576.  doi: 10.1006/jsvi.1994.1199.  Google Scholar

[13]

M. J. Clifford and S. R. Bishop, Rotating periodic orbits of parametrically excited pendulum, Phys. Lett. A, 201 (1995), 191-196.  doi: 10.1016/0375-9601(95)00255-2.  Google Scholar

[14]

D. D. A. Costa and M. A. Savi, Nonlinear dynamics of an SMA-pendulum system, Nonlinear Dynamics, 87 (2017), 1617-1627.  doi: 10.1007/s11071-016-3137-y.  Google Scholar

[15]

D. D. A. Costa and M. A. Savi, Chaos control of an SMA–pendulum system using thermal actuation with extended time-delayed feedback approach, Nonlinear Dynamics, 93 (2018), 571-583.  doi: 10.1007/s11071-018-4210-5.  Google Scholar

[16]

D. D'Humieres, M. R. Beasley, B. A. Huberman and A. F. Libchaber, Chaotic states and routes to chaos in the forced pendulum, Phys. Rev. A, 26 (1982), 3483-3492.  doi: 10.1103/PhysRevA.26.3483.  Google Scholar

[17]

W. X. Ding, H. Q. She, W. Huang and C. X. Yu, Controlling chaos in a discharge plasma, Phys. Rev. Lett., 72 (1994), 96-99.  doi: 10.1103/PhysRevLett.72.96.  Google Scholar

[18]

W. L. Ditto, S. N. Rauseo and M. L. Spano, Experimental control of chaos, Controlling Chaos, (1996), 105–107. doi: 10.1016/B978-012396840-1/50035-7.  Google Scholar

[19]

X. L. Fu, J. Deng and Z. J. Jing, Complex dynamics in physical pendulum equation with suspension axis vibrations, Acta Mathematica Applicatae Sinica, English series, 26 (2010), 55-78.  doi: 10.1007/s10255-008-8276-6.  Google Scholar

[20]

W. Garira and S. R. Bishop, Rotating solutions of the parametrically excited pendulum, J. Sound Vibr., 263 (2003), 233-239.  doi: 10.1016/S0022-460X(02)01435-9.  Google Scholar

[21]

Z. J. Jing, K. Y. Chan, D. S. Xu and H. J. Cao, Bifurcation of periodic solutions and chaos in Josephson system, Discr. Contin. Dyn. Syst.-Series A, 7 (2001), 573-592.  doi: 10.3934/dcds.2001.7.573.  Google Scholar

[22]

Z. J. Jing and H. J. Chao, Bifurcation of periodic orbits in Josephson equation with a phase shift, Int. J. Bifurcation and Chaos, 12 (2002), 1515-1530.  doi: 10.1142/S0218127402005261.  Google Scholar

[23]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (Ⅰ), Int. J. Bifurcat. Chaos, 16 (2006), 2887-2902.  doi: 10.1142/S0218127406016525.  Google Scholar

[24]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (Ⅱ), Int. J. Bifurcat. Chaos, 16 (2006), 3053-3078.  doi: 10.1142/S0218127406016653.  Google Scholar

[25]

T. Kapitaniak, Introduction, Chaos Solitons Fractals, 15 (2003), 201-203.   Google Scholar

[26]

M. Lakshman and K. Murall, Chaos in Nonlinear Oscillations–Controlling and Synchronization, , Singapore: World Scientific, 1996. Google Scholar

[27]

P. S. Landa, Regular and Chaotic Oscillations, Spring-Verlag, 2001. Google Scholar

[28]

M. Levi, F. Hoppensteadt and W. Miranke, Dynamics of the Josephson junction, Quart. Appl. Math., 36 (1978), 167-198.  doi: 10.1090/qam/484023.  Google Scholar

[29]

Z. H. Liu and W. Q. Zhu, Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation, Chaos Solit. Fract., 20 (2004), 593-607.  doi: 10.1016/j.chaos.2003.08.010.  Google Scholar

[30]

R. Lima and M. Pettine, Suppression of chaos by resonant parametric perturbations, Phys. Rev. A, 41 (1990), 726-733.  doi: 10.1103/PhysRevA.41.726.  Google Scholar

[31] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[32]

S. N. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley, New York, 1990.  Google Scholar

[33]

E. S. Ruslan, F. Alexander and L. Daniel, Energy control of a pendulum with quantized feedback, Automatica, 67 (2016), 171-177.  doi: 10.1016/j.automatica.2016.01.019.  Google Scholar

[34]

M. Salerno, Suppression of phase-locking chaos in long Josephson junctions by biharmonic microwave fields, Phys. Rev. B, 44 (1991), 2720-2726.  doi: 10.1103/PhysRevB.44.2720.  Google Scholar

[35]

M. Salerno and M. R. Samuelsen, Stabilization of chaotic phase locked dynamics in long Josephson junctions, Phys. Lett. A, 190 (1994), 177-181.  doi: 10.1016/0375-9601(94)90073-6.  Google Scholar

[36]

R. Q. Wang and Z. J. Jing, Chaos control of chaotic pendulum system, Chaos, Solitons and Fractals, 21 (2004), 201-207.  doi: 10.1016/j.chaos.2003.10.011.  Google Scholar

[37]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990. doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[38]

K. Yagasaki and T. Uozumi, Controlling chaos in a pendulum subjected to feedforward and feedback control, Int. J. Bifurcation and Chaos, 7 (1997), 2827-2835.  doi: 10.1142/S0218127497001904.  Google Scholar

[39]

J. P. Yang and Z. J. Jing, Inhibition of chaos in a pendulum equation, Chaos, Solitons and Fractals, 35 (2008), 726-737.  doi: 10.1016/j.chaos.2006.05.065.  Google Scholar

[40]

J. P. Yang and Z. J. Jing, Control of chaos in a three-well duffing system, Chaos, Solitons and Fractals, 41 (2009), 1311-1328.  doi: 10.1016/j.chaos.2008.05.018.  Google Scholar

[41]

J. P. Yang and Z. J. Jing, Controlling in a pendulum equation with ultra-subharmonic resonances, Chaos, Solitons and Fractals, 42 (2009), 1214-1226.  doi: 10.1016/j.chaos.2009.03.035.  Google Scholar

Figure 1.  Phase portrait of system (2) for $ \alpha = 0.1 $.
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Figure 2.  The chaotic attractor of system (1) for $ \alpha = 0.1, \; \omega = 1.5, \; \delta = 0.38, \; f_1 = 1.381 $, $ \gamma = 0.01 $ and $ f_0 = 0 $.
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Figure 3.  The bifurcation diagram of system (1) in ($ \Psi $, x) plane for $ \alpha = 0.1 $, $ f_1 = 1.381 $, $ f_0 = 0.2 $, $ \delta = 0.38 $, $ \Omega = 0.75 $ and $ \omega = 1.5 $.
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Figure 4.  The bifurcation diagram of system (1) in ($ \Psi $, x) plane for $ \alpha = 0.1 $, $ f_1 = 1.381 $, $ f_0 = 0.2 $, $ \delta = 0.38 $, $ \Omega = 0.5 $ and $ \omega = 1.5 $.
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Figure 5.  The bifurcation diagram of system (1) in ($ \Psi $, x) plane for $ \alpha = 0.1 $, $ f_1 = 1.381 $, $ f_0 = 0.4 $, $ \delta = 0.38 $, $ \Omega = 1 $ and $ \omega = 1.5 $.
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Figure 6.  The bifurcation diagram of system (1) in ($ \Psi $, x) plane for $ \alpha = 0.1 $, $ f_1 = 1.381 $, $ f_0 = 2 $, $ \delta = 0.38 $, $ \Omega = 0.75 $ and $ \omega = 1.5 $.
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Figure 7.  The bifurcation diagram of system (1) in ($ \Psi $, x) plane for $ \alpha = 0.1 $, $ f_1 = 1.381 $, $ \delta = 0.38 $, $ \Omega = 0.5 $ and $ \omega = 1.5 $: (a) $ f_0 = 1 $; (b) $ f_0 = 4 $.
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Figure 8.  The bifurcation diagram of system (1) in ($ \Psi $, x) plane for $ \alpha = 0.1 $, $ f_1 = 1.381 $, $ \delta = 0.38 $, $ \Omega = 1 $ and $ \omega = 1.5 $: (a) $ f_0 = 2 $; (b) $ f_0 = 2.5 $.
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