American Institute of Mathematical Sciences

Advanced
March  2015, 20(2): 373-383. doi: 10.3934/dcdsb.2015.20.373

Chaos control in a pendulum system with excitations

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

1. 

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

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

Received  January 2014 Revised  August 2014 Published  January 2015

This paper is devoted to investigate the problem of controlling chaos for a pendulum system with parametric and external excitations. By using Melnikov methods, the criteria of controlling chaos are obtained. Numerical simulations are given to illustrate the effect of the chaos control for this system, suppression of homoclinic chaos is more effective than suppression of heteroclinic chaos, and the chaotic motions can be suppressed to period-motions by adjusting parameters of chaos-suppressing excitation. Finally, we calculate the maximum Lyapunov exponents (LE) in parameter-plane and observe the frequency of chaos-suppressing excitation also play an important role in the process of chaos control.
Keywords: bifurcation, Melnikov methods., Pendulum equation, chaos control.
Mathematics Subject Classification: Primary: 34G20, 34H05; Secondary: 34H1.
Citation: Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373
References:
[1]

A. Alasty and H. Salarieh, Nonlinear feedback control of chaotic pendulum in presence of saturation effect, Chaos, Solitons and Fractals, 31 (2007), 292-304. doi: 10.1016/j.chaos.2005.10.004.

[2]

G. L. Baker, Control of the chaotic driven pendulum, Am. J. Phys., 63 (1995), 832-838. doi: 10.1119/1.17808.

[3]

S. R. Bishop and M. J. Clifford, Zones of chaotic behavior in the parametrically exicited pendulum, J. of Sound and Vibration, 18 (2006), 1421-1427.

[4]

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.

[5]

H. J. Cao and G. R. Chen, Global and local control of homoclinic and heteroclinic bifurcation, Int. J. Bifurcat. Chaos, 15 (2005), 2411-2432. doi: 10.1142/S0218127405013393.

[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, General results on chaos suppression for biharmonically driven dissipative systems, Phys. Lett. A, 257 (1999), 293-300.

[8]

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.

[9]

G. R. Chen, J. Moiola and H. O. Wang, Bifurcation control: Theories, methods and applications, Int. J. Bifurat. Chaos, 10 (2000), 511-548. doi: 10.1142/S0218127400000360.

[10]

G. R. Chen and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, 1998.

[11]

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

[12]

M. J. Clifford and S. R. Bishop, Approximating the escape zone for the parametrically excited pendulum, Journal of Sound and Vibration, 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'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.

[15]

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.

[16]

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

[17]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (I), Int. J. Bifurcat. Chaos, 10 (2006), 2887-2902. doi: 10.1142/S0218127406016525.

[18]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (II), Int. J. Bifurcat. Chaos, 10 (2006), 3053-3078. doi: 10.1142/S0218127406016653.

[19]

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

[20]

P. S. Landa, Regular and Chaotic Oscillations, Springer-Verlag, 2001.

[21]

M. Lakshman and K. Murall, Chaos in Nonlinear Oscillations-Controlling and Synchronization, World Scientific, Singapore, 1996. doi: 10.1142/9789812798701.

[22]

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

[23]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley, New York, 1979.

[24]

E. Ott, N. Grebogi and J. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196-1199. doi: 10.1103/PhysRevLett.64.1196.

[25]

F. H. I. Perira-Pinto, A. M. Ferreira and M. A. Savi, Chaos control in a nonlinear pendulum using a semi-continuous method, Chaos, Solitons and Fractals, 22 (2004), 653-668. doi: 10.1016/j.chaos.2004.02.047.

[26]

S. Rajasekar, Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator, Pramana J. Phys., 41 (1993), 295-309. doi: 10.1007/BF02847395.

[27]

V. Ravichandran, S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A. F. Sanjuan, Role of asymmetries in the chaotic dynamics of the double-well Duffing oscillator, Pramana J. Phys., 72 (2009), 927-937.

[28]

T. Shinbrot, E. Ott, N. Grebogi and J. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Lett., 65 (1990), 3215-3218. doi: 10.1103/PhysRevLett.65.3215.

[29]

M. S. Siewe, C. Tchawoua and S. Rajasekar, Homoclinic bifurcation and chaos in $\Phi^6$-rayleigh oscillator with three wells driven by an amplitude modulated force, Int. J. Bifurcation Chaos, 21 (2011), 1583-1593.

[30]

M. S. Siewe, H. J. Cao and M. A. F. Sanjuan, Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh-Duffing oscillator, Chaos, Solitons and Fractals, 39 (2009), 1092-1099. doi: 10.1016/j.chaos.2007.05.007.

[31]

M. S. Siewe, H. J. Cao and M. A. F. Sanjuan, On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potentia, Chaos, Solitons and Fractals, 41 (2009), 772-782. doi: 10.1016/j.chaos.2008.03.013.

[32]

D. J. Sudor and S. R. Bishop, Inverted dynamics of a tilted parametric pendulum, European Journal of Mechanics Alsolids, 18 (1996), 517-526. doi: 10.1016/S0997-7538(99)00135-7.

[33]

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.

[34]

S. Wiggins, Global Bifurcation and Chaos: Analytical Methods, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1042-9.

[35]

S. Wiggins, On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations, SIAM. J. Appl. Math., 48 (1988), 262-285. doi: 10.1137/0148013.

[36]

K. Yagasaki, Dynamics a pendulum subjected to feedforward and feedback control, JSME. Int. J., 41 (1998), 545-554.

[37]

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

[38]

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.

show all references

References:
[1]

A. Alasty and H. Salarieh, Nonlinear feedback control of chaotic pendulum in presence of saturation effect, Chaos, Solitons and Fractals, 31 (2007), 292-304. doi: 10.1016/j.chaos.2005.10.004.

[2]

G. L. Baker, Control of the chaotic driven pendulum, Am. J. Phys., 63 (1995), 832-838. doi: 10.1119/1.17808.

[3]

S. R. Bishop and M. J. Clifford, Zones of chaotic behavior in the parametrically exicited pendulum, J. of Sound and Vibration, 18 (2006), 1421-1427.

[4]

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.

[5]

H. J. Cao and G. R. Chen, Global and local control of homoclinic and heteroclinic bifurcation, Int. J. Bifurcat. Chaos, 15 (2005), 2411-2432. doi: 10.1142/S0218127405013393.

[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, General results on chaos suppression for biharmonically driven dissipative systems, Phys. Lett. A, 257 (1999), 293-300.

[8]

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.

[9]

G. R. Chen, J. Moiola and H. O. Wang, Bifurcation control: Theories, methods and applications, Int. J. Bifurat. Chaos, 10 (2000), 511-548. doi: 10.1142/S0218127400000360.

[10]

G. R. Chen and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, 1998.

[11]

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

[12]

M. J. Clifford and S. R. Bishop, Approximating the escape zone for the parametrically excited pendulum, Journal of Sound and Vibration, 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'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.

[15]

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.

[16]

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

[17]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (I), Int. J. Bifurcat. Chaos, 10 (2006), 2887-2902. doi: 10.1142/S0218127406016525.

[18]

Z. J. Jing and J. P. Yang, Complex dynamics in pendulum equation with parametric and external excitations (II), Int. J. Bifurcat. Chaos, 10 (2006), 3053-3078. doi: 10.1142/S0218127406016653.

[19]

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

[20]

P. S. Landa, Regular and Chaotic Oscillations, Springer-Verlag, 2001.

[21]

M. Lakshman and K. Murall, Chaos in Nonlinear Oscillations-Controlling and Synchronization, World Scientific, Singapore, 1996. doi: 10.1142/9789812798701.

[22]

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

[23]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley, New York, 1979.

[24]

E. Ott, N. Grebogi and J. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196-1199. doi: 10.1103/PhysRevLett.64.1196.

[25]

F. H. I. Perira-Pinto, A. M. Ferreira and M. A. Savi, Chaos control in a nonlinear pendulum using a semi-continuous method, Chaos, Solitons and Fractals, 22 (2004), 653-668. doi: 10.1016/j.chaos.2004.02.047.

[26]

S. Rajasekar, Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator, Pramana J. Phys., 41 (1993), 295-309. doi: 10.1007/BF02847395.

[27]

V. Ravichandran, S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A. F. Sanjuan, Role of asymmetries in the chaotic dynamics of the double-well Duffing oscillator, Pramana J. Phys., 72 (2009), 927-937.

[28]

T. Shinbrot, E. Ott, N. Grebogi and J. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Lett., 65 (1990), 3215-3218. doi: 10.1103/PhysRevLett.65.3215.

[29]

M. S. Siewe, C. Tchawoua and S. Rajasekar, Homoclinic bifurcation and chaos in $\Phi^6$-rayleigh oscillator with three wells driven by an amplitude modulated force, Int. J. Bifurcation Chaos, 21 (2011), 1583-1593.

[30]

M. S. Siewe, H. J. Cao and M. A. F. Sanjuan, Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh-Duffing oscillator, Chaos, Solitons and Fractals, 39 (2009), 1092-1099. doi: 10.1016/j.chaos.2007.05.007.

[31]

M. S. Siewe, H. J. Cao and M. A. F. Sanjuan, On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potentia, Chaos, Solitons and Fractals, 41 (2009), 772-782. doi: 10.1016/j.chaos.2008.03.013.

[32]

D. J. Sudor and S. R. Bishop, Inverted dynamics of a tilted parametric pendulum, European Journal of Mechanics Alsolids, 18 (1996), 517-526. doi: 10.1016/S0997-7538(99)00135-7.

[33]

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.

[34]

S. Wiggins, Global Bifurcation and Chaos: Analytical Methods, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1042-9.

[35]

S. Wiggins, On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations, SIAM. J. Appl. Math., 48 (1988), 262-285. doi: 10.1137/0148013.

[36]

K. Yagasaki, Dynamics a pendulum subjected to feedforward and feedback control, JSME. Int. J., 41 (1998), 545-554.

[37]

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

[38]

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.

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