• Y A Amer Faculty of science, Zagazig University
  • S M Ahmed Faculty of science, El-Arish University,
  • ManarM Dahshan Faculty of science, El-Arish University
  • N M Al Faculty of science, Suez Canal University,




Vibration control, nonlinear oscillation, perturbation technique, sub-harmonic resonance.


Chaotic behavior of6 ï† -Rayleigh oscillator with three wells is investigated. The method of multiple scale method is used
to solve the system up to 3rd order approximation. Effect of parameters is studied numerically; all resonance cases are
studied numerically to obtain the worst case. Stability of the system is investigated using both phase


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Author Biographies

Y A Amer, Faculty of science, Zagazig University

Mathematics Department

ManarM Dahshan, Faculty of science, El-Arish University

Mathematics Department

N M Al, Faculty of science, Suez Canal University,

Mathematics Department


[1] M. SieweSiewe, C.Tchawoua and S. Rajasekar," Homoclinicbifurcation and chaos in Φ6-Rayleigh oscillator with three
wells driven an amplitude modulated force",Tiruchirapalli 620 024, Tamilnadu, India.
[2] C. Chin, A. H. Nayfeh, and W. Lacarbonara, Two-to-one internal resonances in parametrically excited buckled beam,
AIAA paper no 97-1081, 1997.
[3] A. F. EL-Bassiouny, M. M. Kamel and A. Abdel-Khalik, "Two-to-one internal resonances in nonlinear two degree of
freedom system with parametric and external excitations", Mathematics and Computers in Simulation, Vol.63, pp. 45-
56, 2003.
[4] J. Xu and K. W. Chung, "Effect of time delayed position feedback on a van der pol oscillator", Physica D, Vol. 180, pp.
17-39, 2003.
[5] M. Eissa and Y. A. Amer, "Vibration control of a cantilever beam subject to both external and parametric excitation",
Applied Mathematics and Computation, Vol. 152, pp. 611-619, 2004.
[6] S. J. Zhu, Y. F.Zheng and Y. M. Fu, "Analysis of non-linear damping dynamics of a two-degree-of-freedom vibration
system with non-linear and non-linear spring", Journal of Sound and Vibration, Vol. 271, pp. 15-24,2004.
[7] M. Eissa, W. A. A.EL-Ganaini and Y. S. Hamed, " Saturation, stability and resonance of non-linear systems", Physica
A, Vol. 356, pp. 341-358, 2005.
[8] S. EL-Serafi, M. Eissa, H. EL-Sherbiny and T. H. EL-Ghareeb, "Comparison between passive and active control of
non-linear dynamical system", Japan Journal of Ind. And Appl. Math., Vol. 23, Issue2, pp. 139-161, 2006.
[9] S. Chatterjee, " Vibration control by recursive time-delayed acceleration feedback", Journal of Sound and Vibration, Vol. 317, pp. 67-90, 2008.
[10] M. M. Kamel, W. A.A. EL-Ganaini and Y. S. Hamed, "Vibration suppression in muti-tool ultrasonic machining to multi-external and parametric excitation", ActaMehanicaSinica, Vol. 25, pp. 403-415,2009. [11] W. A. A. EL-Ganaini, M. M. Kamel and Y. S. Hamed, "Vibration reduction in ultrasonic machine to external and tuned excitation forces", Applied Mathematical Modeling, Vol. 33, Issue 6, pp. 2853-2863, 2009. [12] H. A.EL-Gohary, W. A. A. EL-Ganaini, “Vibration suppression of a dynamical system to multi-parametric excitations via time-delay absorber", Applied Mathematical Modeling Vol. 36, pp. 35-45, 2012. [13] A. Y. T. Leung, Z. J. Guo, A. Myers, " Steady state bifurcation of a periodically excited system under delayed feedback controls", Communications in Nonlinear Science and Numerical Simulation Vol. 17, pp. 5256-5272, 2012. [14] N. A. Saeed, W. A. A. El-Ganini, M. Eissa, "Nonlinear time delay saturation-based controller for suppression of nonlinear beam vibrations", Applied Mathematical Modeling, Vol. 37, Issue 20, pp. 8846-8864, 2013. [15] Y. A. Amer and M. N. Abdelslam, "Stability and control of dynamical system subjected to multi external forces", International Journal of Mathematics and Computer (IJMCAR), Vol. 3, Issue 4, 41-52, 2013. [16] Y. A. Amer and M. N. Abdelslam, "Vibration reduction of nonlinear dynamical system at combined resonance subject to tuned excitation", Journal of Advances in Mathematics, Vol. 9, 1774-1786, 2014. [17] J. J. Thomsen, Vibration suppression by using self-arranging mass: effects of adding restoring force, Journal of Sound and Vibration 197 (1996) 403-425 [18] H. Rong, X. Wang, W. Xu, and T. Fang, Saturation and resonance of non-linear system under bounded noise excitation absorber, Journal of Sound Vibration 291 (2006) 48-59. [19] J. Warminski, M. P. Cartmell, A. Mitura, and M. Bochenski, Active vibration control of a non-linear beam with self-and external excitations, Journal of Shock and Vibration 20 (2013) 1033-1047. [20] O. Orhan and J. A. Peter, Nonlinear response of flapping beam to resonant excitations under nonlinear damping, ActaMech (2015) 1-27. [21] B. Elena, G. Dimitri, B. Philippe and F. Orla, Steady-State Oscillations in Resonant Electrostatic Vibration Energy Harvesters, IEEE Transactions on Circuits and Systems I, 60 (2015) 875-884.




How to Cite

Amer, Y. A., Ahmed, S. M., Dahshan, M., & Al, N. M. (2016). DYNAMICAL CHAOS IN 6  -RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN AN AMPLITUDE MODULATED FORCE. JOURNAL OF ADVANCES IN MATHEMATICS, 12(8), 6545–6552. https://doi.org/10.24297/jam.v12i8.5074