Chaos Suppression in a Pendulum Equation through Parametric Excitation with Phase Shift for Ultra-Subharmonic Resonance

Main Article Content

Xianwei Chen
Xiangling Fu
Jintao Tan

Abstract

Under ultra-subharmonic resonance, we investigate the chaos suppression of pendulum equation by using Melnikov methods, and get the conditions of suppressing chaos for homoclinic and heteroclinic orbits, respectively. At the same time, we give some numerical simulations including the bifurcation diagrams of system and corresponding phase diagrams, and observe that the chaos behaviors of system may be suppressed to period-n(n ∈ Z+) orbits by adjusting the value of Ψ. Although our results are only necessary, not sufficient. Numerical simulations show that our method is effect in suppressing chaos for this case.

Keywords:
Parametric excitation, chaos, chaos control, Melnikov methods.

Article Details

How to Cite
Chen, X., Fu, X., & Tan, J. (2020). Chaos Suppression in a Pendulum Equation through Parametric Excitation with Phase Shift for Ultra-Subharmonic Resonance. Current Journal of Applied Science and Technology, 39(35), 1-11. https://doi.org/10.9734/cjast/2020/v39i3531048
Section
Original Research Article

References

D’Humieres D, Beasley MR, Huberman BA, Libchaber AF. Chaotic states and routes to chaos in the forced pendulum. Phys. Rev. A. 1982;26: 3483-3492.

Pikovsky A, Rosenblum M, Kurths J. Synchronization: A universal concept in nonlinear sciences. Cambridge University Press; 2001.

Rasband SN. Chaotic Dynamics of nonlinear systems. John Wiley, New York; 1990.

Levi M, Hoppensteadt F, Miranke W. Dynamics of the Josephson junction. Quart. Appl. Math. 1978;35:167-198.

Salerno M. Suppression of phase-locking chaos in long Josephson junctions by biharmonic microwave fields. Phys. Rev. B. 1991;44:27202726.

Salerno M, Samuelsen MR. Stabilization of chaotic phase locked dynamics in long Josephson junctions. Phys. Lett. A. 1994;190:177181.

Amer TS. The dynamical behavior of a rigid body relative equilibrium position. Advances in Mathematical Physics. 2017;13:13.

Bishop SR, Clifford MJ. Zones of chaotic behavior in the parametrically excited pendulum. J. Sound Vib. 1996;181:1421-1427.

Cao HJ, Chi XB. Chen GR. Suppressing or inducing chaos by weak resonant excitations in an externally-forced froude pendulum. Int. J. Bifurcat. Chaos. 2004;14:1115-1120.

Chacon R. Natural symmetries and ´ regularization by means of weak parametric modulations in the forced pendulum. Phys. Rev. E. 1995;52:2330-2337.

Chen LJ, Li JB. Chaotic behavior and subharmonic bifurcations for a rotating predulum equation. Int. J. Bifurcat. Chaos. 2004;14:3477-3488.

Chen XW, Jing ZJ, Fu XL. Chaos control in a pendulum system with excitations. Discrete and Continuous Dynamical Systems Series B. 2015;20:373-383.

Clifford MJ, Bishop SR. Approximating the escape zone for the parametrically excited pendulum. J. Sound Vibr. 1994;172:572- 576.

Clifford MJ, Bishop SR. Rotating periodic orbits of the parametrically excited pendulum. Phys. Lett. A. 1995;201:191-196.

Costa DDA, Savi MA. Nonlinear dynamics of an SMA-pendulum system. Nonlinear Dynamics. 2017; 87: 1617-1627.

Costa DDA, Savi MA. Chaos control of an SMApendulum system using thermal actuation with extended time-delayed feedback approach. Nonlinear Dynamics. 2018;93:571-583.

Fu HX, Qian YH. Study on a multifrequency homotopy analysis method for perioddoubling solutions of nonlinear systems. Int. J. Bifurcation and Chaos. 2018;28:18500449.

Garira W, Bishop SR. Rotating solutions of the parametrically excited pendulum. J. Sound Vibr. 2003;263:233-239.

Hou L, Su XC, Chen YS. Bifurcation models of periodic solution in a duffing system under constant force as well as harmonic excitation. J. Bifurcation and Chaos. 2019;29:1950173.

Jing ZJ, Chan KY, Xu DS, Cao HJ. Bifurcation of periodic solutions and chaos in Josephson system. Discr. Contin. Dyn. Syst.-Series A. 2001;7:573-592.

Jing ZJ, Chao HJ. Bifurcation of periodic orbits in Josephson equation with a phase shift. Int. J. Bifurcation and Chaos. 2001;12:1515-1530.

Liu ZH, Zhu WQ. Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation. Chaos Solit. Fract. 2004;20:593-607.

Ruslan ES, Alexander F, Daniel L. Energy control of a pendulum with quantized feedback. Automatica. 2016;67:171-177.

Wang RQ, Jing ZJ. Chaos control of chaotic pendulum system. Chaos Solitons Fractals. 2004;21:201-207.

Wang YL, Li XH, Shen YJ. Cluster oscillation and bifurcation of fractional-order duffing system with two time scales. Acta Mechanica Sinica. 2020;36:926-932.

Yagasaki K, Uozumi T. Controlling chaos in a pendulum subjected to feedforward and feedback control. Int. J. Bifurcation and Chaos. 1997;7:2827-2835.

Nayfeh AH, Mook DT. Nonlinear oscillations. John Wiley, New York; 1979.

Wiggins S. Global bifurcation and chaos: Analytical methods. Springer-Verlag; 1988.

Jing ZJ, Yang JP. Complex dynamics in pendulum equation with parametric and external excitations (I). Int. J. Bifurcat. Chaos. 2006;10:2887-2902.

Jing ZJ, Yang JP. Complex dynamics in pendulum equation with parametric and external excitations (II). Int. J. Bifurcat. Chaos. 2006;10:3053-3078.

Chen XW, Jing ZJ. Complex dynamics in a pendulum equation with a phase shift. Int. J. Bifurcat. Chaos. 2012;22. DOI: 10.1142/S0218127412503075

Chen XW, Jing ZJ, Fu XL. Chaos control in a pendulum system with excitations and phase shift. Nonlinear Dyn. 2014;78:317-

Yang JP, Jing ZJ. Inhibition of chaos in a pendulum equation. Chaos Solitons Fractals. 2008;35:726-737.

Yang JP, Jing ZJ. Controlling in a pendulum equation with ultra-subharmonic resonances. Chaos Solitons Fractals. 2009;42:1214-1226.

Chacon R, Palmero F, Balibrea F. Taming ´ chaos in a driven Josephson junction. Int. J. Bifurcat. Chaos. 2001;11:1897-1909