Chaotic Dynamics of an Extended Duffing Oscillator Under Periodic Excitation
World Journal of Applied Physics
Volume 3, Issue 2, June 2018, Pages: 34-50
Received: Jun. 29, 2018; Accepted: Jul. 12, 2018; Published: Aug. 6, 2018
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Hervé Lucas Koudahoun, Department of Physics, University of Abomey-Calavi, Abomey-Calavi, Benin
Yélomè Judicaël Fernando Kpomahou, Department of Industrial and Technical Sciences, University of Abomey, Abomey, Benin
Jean Akande, Department of Physics, University of Abomey-Calavi, Abomey-Calavi, Benin
Damien Kêgnidé Kolawolé Adjaï, Department of Physics, University of Abomey-Calavi, Abomey-Calavi, Benin
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In this paper, chaotic dynamics of a cubic-quintic-septic Duffing oscillator subjected to periodic excitation is investigated. The multiple scales method is used to determine the various resonance states of the model. It is found that the considered model posses thirteen resonance states whose seven are thoroughly studied. The steady-state solutions and theirs stabilities are determined. The frequency-amplitude curves show that the considered system presents mixed behavior, limit cycles, hysteresis, jump and bifurcation phenomena. It is also noticed that these phenomena are strongly influenced by quintic-septic nonlinearity and excitation amplitude. Bifurcation structures displayed by the model for each considered type of resonant states are investigated numerically using the fourth-order Runge-Kutta algorithm. As results, the quintic-septic nonlinearity, linear dissipation and excitation amplitude can be used to control the chaotic behavior of the system.
Extended Duffing Oscillator, Resonance States, Stability, Limit Cycles, Bifurcation and Jump Phenomena, Periodic and Quasi-periodic Oscillations, Chaos
To cite this article
Hervé Lucas Koudahoun, Yélomè Judicaël Fernando Kpomahou, Jean Akande, Damien Kêgnidé Kolawolé Adjaï, Chaotic Dynamics of an Extended Duffing Oscillator Under Periodic Excitation, World Journal of Applied Physics. Vol. 3, No. 2, 2018, pp. 34-50. doi: 10.11648/j.wjap.20180302.13
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Monwanou, A. V., Miwadinou, C. H., Aïnamon, C., and Chabi Orou, J. B. (2018). Hysteresis, Quasiperiodicity and Chaoticity in a Nonlinear Dissipative Hybrid Oscillator, International Journal of Basic and Applied Sciences, 7(1), 1–7.
Higgins, John P. (2002). Nonlinear Systems in Medicine, Yale Journal of biology and medicine, 75(5-6), 247-260.
Moon, Francis C. (2004). Chaotic Vibrations An Introduction for Applied Scientists and Engineers, John Wiley.
Weiss, James N., Garfinkel, Alain., Spano, Mark L., and Ditto, William L. (1994). Chaos and Chaos Control in Biology, The journal of Clinical Investigation, 93(4), 1355-1360.
Tamastel, and Gruiz, Marton (2006). Chaotic Dynamics an Introduction Based on Classical Mechanics, Cambridge University Press, New York.
Hayes, Scott., Grebogi, Celso., Ott, Edward and Mark, Andrea. (1994), Experimental Control of Chaos for Communication, Physical review letters, 73(13), 1781-1784.
Aihara, Kazuyuki. (2002). Chaos Engineering and Its Application to Parallel Distributed Processing With Chaotic Neural Networks using energy balance method, Proceeding of the IEEE, 90(5), 919-930.
Aihara, Kazuyuki. (2012). Chaos and its applications, Procedia IUTAM, 5, 199 -203.
Khan Ayub and Kumar, Sanjay. (2018). Study of chaos in chaotic satellite systems, Pramana-J. Phys, 90(13), 1-9.
Mallik, A. K. (2003). Response of A Hard Duffing Oscillator to Harmonic Excitation-An Overview, indian institute of technology, kharagpur 721302, 28-30, 1-5.
Luo, Albert C. J., and Huang Jianzhe. (2014). Period-3 Motions to Chaos in a Softening Duffing Oscillator, International Journal of Bifurcation and Chaos, 24(3), 1430010-1430010-26
Ghandchi-Tehrani Maryam., Wilmshurst, Lawrence I., and Stephen, J. Elliote. (2015). Bifurcation control of a Duffing oscillator using pole placement, Journal of Vibration and Control, 21(14), 2838-2851.
Kovacic, Ivanaand Brennan, Michael J. (2011). The Duffing Equation Nonlinear Oscillators and their Behaviour, John Wiley & Sons, Ltd.
Lou, Jing-jun., He, Qi-wei., and Zhu Shi-jian. (2004). Chaos In The Softening Duffing System Under Multi-Frequency Periodic Forces, Applied Mathematics and Mechanics, 25(12), 1421-1427.
Berger, J. E., and Nunes, G. Jr. (1997). A mechanical Duffing oscillator for the undergraduate laboratory, American Journal of Physics, 65(9), 841-846.
Nayfeh, A. H. and D. T. Mook. (1995). Nonlinear Oscillations, John Wiley & Sons, New York.
Donso, Guillermo and Celso L. Ladera. (2012). Nonlinear dynamics of a magnetically driven Duffing-type spring-magnet oscillator in the static magnetic field of a coil, European Journal of Physics, 33(6), 1473-1486.
Chua, Vivien. (2012). Cubic-Quintic Duffing Oscillators,, 1-19.
Oyesanya, Moses O., and Nwamba, J. I. (2013). Stability analysis of damped cubic-quintic Duffing oscillator, WorldJournal of Mechanics, 3(1), 43-57.
Kacem, N., Baguet, S., Dufour R., and Hentz, S. (2011). Stability control of nonlinear micromachanical resonators under simultaneous primary and superharmonic resonances, Applied Physics Letters, 98(19), 193507-193507-3.
Kacem, N., and Hentz, S. (2009) Bifurcation topology tuning of a mixed behavior in nonlinear micromechanical resonators, Applied PhysicsLetters, 95(18), 183104-183104-3.
Elshurafa, Amro M., Khirallah, Kareem, Tawfik, Hani H., Ahmed, Emira., Ahmed K. S. Abdel Aziz and Sedky, Sherif. M. (2011). Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators, Journal of Microelectromechanical Systems, 20(4), 943-958.
Oyesanya, M. O., and Nwamba, J. I. (2013). Duffing Oscillator with Heptic Nonlinearity under single Periodic Forcing, Int. J. of Mechanics and Applications, 3(2), 35-43.
Lesage, J. C., Liu, M. C. (2008). On the investigation of a restrained cargo system modeled as a Duffing oscillator of various orders, Proceeding of Early Career Technical Conference, ASME, Maimi, Florida, USA.
Koudahoun, L. H., Kpomahou, Y. J. F., Adjaï, D. K. K., Akande, J., Rath, B. Mallick, P. and Monsia, D. M. (2016). Periodic Solutions for nonlinear oscillations in Elastic Structures via Energy Balance Method, viXra 16110214v1, 1-12.
Deleanu, Dumitru. (2016). Transient and Steady-State Responses for the Ship Rolling Motion with Multiple scales Lindstedt Poincaré Method, ‘’Mircea Cel Batran’’ Naval Academy Scientific Bulletin, XIX(2), 208-215.
Luo, Albert C. J. (2010). Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, Springer-Verlag, New York.
Marinca, V. and Herisanu, N. (2005). Forced Duffing Oscillator with Slight Viscous Damping and Hardening Non-Linearity, Facta Universitatis, Series: Mechanics, Automatic Control and Robotics, 4(17), 245-255.
Nayfeh, Ali H. (2004). Perturbation Methods, WILEY-VCH.
Hayashi, Chihiro. (1986). Nonlinear Oscillations in Physical Systems, Princeton University Press, 41 William Sheet, Princeton, New Jersey 08540. Mc Graw-Hill, Inc.
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