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 & 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, 31 (2007), 292.  doi: 10.1016/j.chaos.2005.10.004.  Google Scholar

[2]

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

[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.   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

R. Chacón, General results on chaos suppression for biharmonically driven dissipative systems,, Phys. Lett. A, 257 (1999), 293.   Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  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.  doi: 10.1016/0375-9601(95)00255-2.  Google Scholar

[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.  doi: 10.1103/PhysRevA.26.3483.  Google Scholar

[15]

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

[16]

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

[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.  doi: 10.1142/S0218127406016525.  Google Scholar

[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.  doi: 10.1142/S0218127406016653.  Google Scholar

[19]

T. Kapitaniak, Introduction,, Chaos, 15 (2003), 201.   Google Scholar

[20]

P. S. Landa, Regular and Chaotic Oscillations,, Springer-Verlag, (2001).   Google Scholar

[21]

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

[22]

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

[23]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations,, John Wiley, (1979).   Google Scholar

[24]

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

[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, 22 (2004), 653.  doi: 10.1016/j.chaos.2004.02.047.  Google Scholar

[26]

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

[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.   Google Scholar

[28]

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

[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.   Google Scholar

[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, 39 (2009), 1092.  doi: 10.1016/j.chaos.2007.05.007.  Google Scholar

[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, 41 (2009), 772.  doi: 10.1016/j.chaos.2008.03.013.  Google Scholar

[32]

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

[33]

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

[34]

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

[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.  doi: 10.1137/0148013.  Google Scholar

[36]

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

[37]

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

[38]

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

show all references

References:
[1]

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

[2]

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

[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.   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

R. Chacón, General results on chaos suppression for biharmonically driven dissipative systems,, Phys. Lett. A, 257 (1999), 293.   Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  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.  doi: 10.1016/0375-9601(95)00255-2.  Google Scholar

[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.  doi: 10.1103/PhysRevA.26.3483.  Google Scholar

[15]

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

[16]

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

[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.  doi: 10.1142/S0218127406016525.  Google Scholar

[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.  doi: 10.1142/S0218127406016653.  Google Scholar

[19]

T. Kapitaniak, Introduction,, Chaos, 15 (2003), 201.   Google Scholar

[20]

P. S. Landa, Regular and Chaotic Oscillations,, Springer-Verlag, (2001).   Google Scholar

[21]

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

[22]

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

[23]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations,, John Wiley, (1979).   Google Scholar

[24]

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

[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, 22 (2004), 653.  doi: 10.1016/j.chaos.2004.02.047.  Google Scholar

[26]

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

[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.   Google Scholar

[28]

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

[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.   Google Scholar

[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, 39 (2009), 1092.  doi: 10.1016/j.chaos.2007.05.007.  Google Scholar

[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, 41 (2009), 772.  doi: 10.1016/j.chaos.2008.03.013.  Google Scholar

[32]

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

[33]

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

[34]

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

[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.  doi: 10.1137/0148013.  Google Scholar

[36]

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

[37]

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

[38]

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

[1]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[2]

J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653
[3]

Carolina Mendoza, Jean Bragard, Pier Luigi Ramazza, Javier Martínez-Mardones, Stefano Boccaletti. Pinning control of spatiotemporal chaos in the LCLV device. Mathematical Biosciences & Engineering, 2007, 4 (3) : 523-530. doi: 10.3934/mbe.2007.4.523
[4]

Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177
[5]

Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67
[6]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660
[7]

Yubai Liu, Xueshan Gao, Fuquan Dai. Implementation of Mamdami fuzzy control on a multi-DOF two-wheel inverted pendulum robot. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1251-1266. doi: 10.3934/dcdss.2015.8.1251
[8]

Dale McDonald. Sensitivity based trajectory following control damping methods. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 127-143. doi: 10.3934/naco.2013.3.127
[9]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[10]

Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020163
[11]

Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic & Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1
[12]

Rafael Labarca, Solange Aranzubia. A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1745-1776. doi: 10.3934/dcds.2018072
[13]

Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105
[14]

Rubén Capeáns, Juan Sabuco, Miguel A. F. Sanjuán. Partial control of chaos: How to avoid undesirable behaviors with small controls in presence of noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3237-3274. doi: 10.3934/dcdsb.2018241
[15]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139
[16]

Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277
[17]

Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003
[18]

Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 875-891. doi: 10.3934/dcdsb.2009.11.875
[19]

Jean-François Rault. A bifurcation for a generalized Burgers' equation in dimension one. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 683-706. doi: 10.3934/dcdss.2012.5.683
[20]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences