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Chaos control in a pendulum system with excitations
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 |
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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