Figure 1.
Phase portrait of system (2) for $ \alpha = 0.1 $ .
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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
| 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 |
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.
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
R. Chacón, F. Palmero and F. Balibrea,
Taming chaos in a driven Josephson junction, Int. J. Bifurcat. Chaos, 11 (2001), 1897-1909.
|
| [7] |
R. Chacón,
Relative effectiveness of weak periodic excitations in suppressing homoclinic heteroclinic chaos, Eur. Phys. J. B, 65 (2002), 207-210.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [25] |
T. Kapitaniak,
Introduction, Chaos Solitons Fractals, 15 (2003), 201-203.
|
| [26] |
M. Lakshman and K. Murall, Chaos in Nonlinear Oscillations–Controlling and Synchronization, , Singapore: World Scientific, 1996. |
| [27] |
P. S. Landa, Regular and Chaotic Oscillations, Spring-Verlag, 2001. |
| [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. |
| [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. |
| [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. |
| [31] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, 2001.
doi: 10.1017/CBO9780511755743.
|
| [32] |
S. N. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley, New York, 1990. |
| [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. |
| [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. |
| [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. |
| [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. |
| [37] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990.
doi: 10.1007/978-1-4757-4067-7. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
R. Chacón, F. Palmero and F. Balibrea,
Taming chaos in a driven Josephson junction, Int. J. Bifurcat. Chaos, 11 (2001), 1897-1909.
|
| [7] |
R. Chacón,
Relative effectiveness of weak periodic excitations in suppressing homoclinic heteroclinic chaos, Eur. Phys. J. B, 65 (2002), 207-210.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [25] |
T. Kapitaniak,
Introduction, Chaos Solitons Fractals, 15 (2003), 201-203.
|
| [26] |
M. Lakshman and K. Murall, Chaos in Nonlinear Oscillations–Controlling and Synchronization, , Singapore: World Scientific, 1996. |
| [27] |
P. S. Landa, Regular and Chaotic Oscillations, Spring-Verlag, 2001. |
| [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. |
| [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. |
| [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. |
| [31] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, 2001.
doi: 10.1017/CBO9780511755743.
|
| [32] |
S. N. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley, New York, 1990. |
| [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. |
| [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. |
| [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. |
| [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. |
| [37] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990.
doi: 10.1007/978-1-4757-4067-7. |
| [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. |
| [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. |
| [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. |
| [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. |
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 $ .
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 $ .
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 $ .
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 $ .
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 $ .
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 $ .
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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