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Stability Analysis of Damped Cubic-Quintic Duffing Oscillator ()

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This paper presents a
comprehensive stability analysis of the dynamics of the damped cubic-quintic
Duffing oscillator. We employ the derivative expansion method to investigate
the slightly damped cubic-quintic Duffing oscillator obtaining a uniformly
valid solution. We obtain a uniformly valid solution of the un-damped
cubic-quintic Duffing oscillator as a special case of our solution. A phase
plane analysis of the damped cubic-quintic Duffing oscillator is undertaken
showing some chaotic dynamics which sends a signal that the oscillator may be
useful as model for prediction of earth- quake occurrence.

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*World Journal of Mechanics*, Vol. 3 No. 1, 2013, pp. 43-57. doi: 10.4236/wjm.2013.31003.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,” In: Applied Mathematical Sciences, Springer-Verlag, New York, 1983. |

[2] | V. Chua, “Cubic-Quintic Duffing Oscillators,” 2012. www.its.caltech.edu/mason/research/duf.pdf |

[3] | A. Hassan Nayfeh, “Perturbation Methods,” John Wiley & Sons, Inc., Hoboken, 1973. |

[4] | J. C. Amazigo, “Postgraduate Lecture Notes on Perturbation Methods,” Department of Mathematics, University of Nigeria, Nsukka, 2011 |

[5] | R. H. Rand, “Lecture Notes on Nonlinear Vibrations,” 2003. http://www.tam.cornell.edu/randdocs/nlvibe54.pdf |

[6] | A. M. Correig and M. Urquizu, “Some Dynamical Aspects of Microseism Time Series,” Geophysical Journal International, Vol. 149, No. 3, 2002, pp. 589-598. doi:10.1046/j.1365-246X.2002.01602.x |

[7] | M. O. Oyesanya, “Duffing Oscillator as a Model for Predicting Earthquake Occurrence 1,” Journal of Nigerian Association of Mathematical Physics, Vol. 12, 2008, pp. 133-142. |

[8] | H. M. Sedighi, K. H. Shirazi and J. Zare, “An Analytic Solution of Transversal Oscillation of Quintic Nonlinear Beam with Homotopy Analysis Method,” International Journal of Nonlinear Mechanics, Vol. 47, No. 10, 2012, pp. 777-784. doi:10.1016/j.ijnonlinmec.2012.04.008 |

[9] | J. C. Amazigo, B. Budiansky and G. F. Carrier, “Asymptotic Analyses of the Buckling of Imperfect Columns of Nonlinear Elastic Foundations,” International Journal of Solids and Structures, Vol. 6, No. 10, 1970, pp. 13411356. doi:10.1016/0020-7683(70)90067-3 |

[10] | J. D. Cole and J. Kevorkian, “Uniformly Valid Asymptotic Approximations for Certain Nonlinear Differential Equations,” In: J. P. Laselle and S. Lefschetz, Eds., Nonlinear Differential Equations and Nonlinear Mechanics, Academic, New York, 1963, pp. 113-120. doi:10.1016/B978-0-12-395651-4.50018-0 |

[11] | A. H. Nayfeh, “A Comparison of Three Perturbation Methods for the Earth-Moon-Spaceship Problem,” AIAA Journal: The American Institute of Aeronautics and Astronautics, Vol. 3, No. 9, 1965, pp. 1682-1687. doi:10.2514/3.3226 |

[12] | Y. Y. Shi and M. C. Eckstein, “Ascent or Descent from Satellite Orbit by Low Thrust,” AIAA Journal: The American Institute of Aeronautics and Astronautics, Vol. 4, No. 12, 1966, pp. 2203-2209. doi:10.2514/3.55302 |

[13] | Y. Y. Shi and M. C. Eckstein, “Application of Singular Perturbation Methods to Resonance Problems,” Astronomical Journal, Vol. 73, 1968, pp. 275-289. doi:10.1086/110629 |

[14] | H. Ashley, “Multiple Scaling in Flight Vehicle Dynamic Analysis—A Preliminary Look,” AIAA Paper No. 670560. |

[15] | L. A. Rubenfeld, “On a Derivative Expansion Technique and Some Comments on Multiple Scaling in the Asymptotic Approximation of Solutions of Certain Differential Equations,” Society for Industrial and Applied Mathematics Review, Vol. 20, No. 1, 1978, pp. 79-105. doi:10.1137/1020005 |

[16] | J. H. He, “The Homotopy Perturbation Method for Nonlinear Oscillators with Discontinuities,” Applied Mathematics and Computer, Vol. 151, No. 1, 2004, pp. 287-292. doi:10.1016/S0096-3003(03)00341-2 |

[17] | J. H. He, “Application of Homotopy Perturbation Method to Nonlinear Wave Equations,” Chaos, Solitons and Fractals, Vol. 26, No. 3, 2005, pp. 695-700. doi:10.1016/j.chaos.2005.03.006 |

[18] | J. H. He, “Homotopy Perturbation Methods for Solving Boundary Value Problems,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 87-88. doi:10.1016/j.physleta.2005.10.005 |

[19] | J. H. He, “New Interpretation of Homotopy Perturbation Method,” International Journal of Modern Physics B, Vol. 20, No. 18, 2006, pp. 2561-2568. doi:10.1142/S0217979206034819 |

[20] | J. H. He, “Non-Perturbative Methods for Strongly Nonlinear Problems,” Dissertation, De-Verlagim Internet GmbH, Berlin, 2006. |

[21] | J. H. He, “A New Perturbation Technique Which Is Also Valid for Large Parameters,” Journal of Sound and Vibrations, Vol. 229, No. 5, 2000, pp. 1257-1263. doi:10.1006/jsvi.1999.2509 |

[22] | Y. Farzaneh and A. A. Tootoonchi, “Global Error Minimization Method for Solving Strongly Nonlinear Oscillator Differential Equations,” Journal of Computers and Mathematics with Applications, Vol. 59, No. 8, 2010, pp. 2887-2895. doi:10.1016/j.camwa.2010.02.006 |

[23] | J. H. He, “Variational Approach for Nonlinear Oscillators,” Chaos, Solitons and Fractals, Vol. 34, No. 5, 2007, pp. 1430-1439. doi:10.1016/j.chaos.2006.10.026 |

[24] | J. H. He, “Variational Principles for Some Nonlinear Partial Differential Equations with Variable Coefficient,” Chaos, Solitons and Fractals, Vol. 19, No. 4, 2004, pp. 847-851. doi:10.1016/S0960-0779(03)00265-0 |

[25] | J. H. He, “Variational Iteration Method—A Kind of Nonlinear Analytical Technique: Some Examples,” International Journal of Nonlinear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1 |

[26] | J. H. He and X. H. Wu, “Construction of Solitary Solution and Compaction-Like Solution by Variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 29, No. 1, 2006, pp. 108-113. doi:10.1016/j.chaos.2005.10.100 |

[27] | T. Pirbodaghi, S. H. Hoseni, M. T. Ahmadian and G. H. Farrahi, “Duffing Equations with Cubic and Quintic Nonlinearities,” Journal of Computers and Mathematics with Applications, Vol. 57, No. 3, 2009, pp. 500-506. doi:10.1016/j.camwa.2008.10.082 |

[28] | D. D. Ganji, M. Gorji, S. Soleimani and M. Esmaeilpour, Solution of Nonlinear Cubic-Quintic Duffing Oscillator Using He’s Energy Balancing Method,” Journal of Zhejiang University Science A, Vol. 10, No. 9, 2009, pp. 1263-1268. doi:10.1631/jzus.A0820651 |

[29] | A. Belandez, G. Bernabeu, J. Frances, D. I. Mendez and S. Marini, “An Accurate Approximate Solution for the Quintic Duffing Oscillator Equation,” Journal of Mathematical and Computer Modelling, Vol. 52, No. 3-4, 2010, pp. 637-641. doi:10.1016/j.mcm.2010.04.010 |

[30] | S. K. Lai, C. W. Lim, B. S. Wu, C. Wang, Q. C. Zeng and X. F. He, “Newton-Harmonic Balancing Approach for Accurate Solutions to Nonlinear Cubic-Quintic Duffing Oscillators,” Journal of Applied Mathematical Modelling, Vol. 33, No. 2, 2009, pp. 852-866. doi:10.1016/j.apm.2007.12.012 |

[31] | R. H. Rand, “Topics in Nonlinear Dynamics with Computer Algebra,” In: Computation in Education, Gordon and Breach Publishers, 1994. |

[32] | W. E. Boyce and R. C. Diprima, “Elementary Differential Equations and Boundary Value Problems,” John Wiley & Sons, Inc., Hoboken, 2000. |

[33] | E. I. Rivin, “Dynamic Overload and Negative Damping in Mechanical Linkage: Case Study of Catastrophic Failure of Extrusion Press,” Engineering Failure Analysis, Vol. 14, No. 7, 2002, pp. 1301-1312. |

[34] | R. D. Peters, “Beyond the Linear Damping Model for Mechanical Harmonic Oscillators,” 2004. http://physics.mercer.edu/petepag/debunk.html. |

[35] | S. T. Wu, “Asymptotic Behavior of Solutions for Nonlinear Wave Equations of Kirchoff Type with a PositiveNegative Damping,” Journal of Applied Mathematics Letters, Vol. 25, No. 7, 2012, pp. 1082-1086. doi:10.1016/j.aml.2012.03.022 |

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