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On the Stability of Duffing Type Fractional Differential Equation with Cubic Nonlinearity

DOI: 10.4236/oalib.1106184, PP. 1-14

Subject Areas: Ordinary Differential Equation, Integral Equation

Keywords: Duffing Equation, Eigenvalues, Fractional Derivative, Hard Spring, Jacobian Matrix, Soft Spring, Stability

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Abstract

The purpose of this paper is to study the stability of nonlinear fractional Duffing equation where , by analysing the eigenvalues generated from the system of the given differential equation. Graphical results furthermore show the effect of the damping and nonlinear parameter on the system. Our contribution relies on its application to the choice of hard/soft spring in the mechanism of shock absorbers.

Cite this paper

Ugochukwu, N. D. , Oguagbaka, M. F. , Samuel, E. S. , Aguegboh, N. S. and Ebube, O. N. (2020). On the Stability of Duffing Type Fractional Differential Equation with Cubic Nonlinearity. Open Access Library Journal, 7, e6184. doi: http://dx.doi.org/10.4236/oalib.1106184.

References

[1]  Hassan, A. (1973) Perturbation Method. John Wiley & Sons Inc., Hoboken.
[2]  Amazigo, J.C. (2011) Postgraduate Lecture Notes on Perturbation Methods. Department of Mathematics, University of Nigeria, Nsukka.
[3]  Rand, R.H. (2003) Lecture Notes on Nonlinear Vibration. http://www.tam.cornell.edu/randdocs/nlvibe54.pdf
[4]  Correig, A.M. and Urquizu, M. (2002) Some Dynamical Aspects of Microseism Time Series. Geophysical Journal International, 149, 589-598. https://doi.org/10.1046/j.1365-246X.2002.01602.x
[5]  Oyesanya, M.O. (2008) Duffing Oscillator as a Model for Predicting Earthquake Occurrence. Journal of Nigerian Association of Mathematical Physics, 12, 133-142. https://doi.org/10.4314/jonamp.v12i1.45495
[6]  Oyesanya, M.O. and Nwamba, J.I. (2013) Stability Analysis of Damped Cubic-Quintic Duffing Oscillator. World Journal of Mechanics (Scientific Research), 3, 43-57. https://doi.org/10.4236/wjm.2013.31003
[7]  Sedighi, H.M., Shirazi, K.H. and Zare, J. (2012) An Analytic Solution of Transversal Oscillation of Quintic Nonlinear Beam with Homotopy Analysis Method. International Journal of Nonlinear Mechanics, 47, 777-784. https://doi.org/10.1016/j.ijnonlinmec.2012.04.008
[8]  Eze, E.O., Obasi, U.E. and Agwu, E.U. (2019) Stability Analysis of Periodic Solution of Some Duffing’s Equations. Open Journal of Applied Sciences, 9, 198-214. https://doi.org/10.4236/ojapps.2019.94017
[9]  Zeeman, E. (1976) Duffing Equation in Brain Modelling. Bulletin Institute of Mathematics and Its Applications, 12, 207-214.
[10]  Zeeman, E. (2008) Duffing Oscillator as Model for Predicting Earthquake Occurrence I. Journal of Nigerian Association of Mathematical Physics, 12, 133-142. https://doi.org/10.4314/jonamp.v12i1.45495
[11]  Wang, G., Zheng, W. and He, S. (2002) Estimation of Amplitude and Phase of a Weak Signal by Using the Property of Sensitive Dependence on Initial Condition of a Non-Linear Oscillator. Signal Processing, 82, 103-115. https://doi.org/10.1016/S0165-1684(01)00166-9
[12]  Yang, L. and Li, Y.K. (2014) Existence and Global Exponential Stability of Almost Periodic Solution for a Class of Delay Duffing Equation on Time Scale. Abstract and Applied Analysis, 2014, Article ID: 857161. https://doi.org/10.1155/2014/857161
[13]  Hadi, D., Dumitru, B. and Jalil, S. (2012) Stability Analysis of Caputo Fractional-Order Nonlinear Systems Revisited. Nonlinear Dynamics, 67, 2433-2439. https://doi.org/10.1007/s11071-011-0157-5
[14]  Li, C.P. and Zhang, F.R. (2011) A Survey on the Stability of Fractional Differential Equations. The European Physical Journal Special Topics, 193, 27-47. https://doi.org/10.1140/epjst/e2011-01379-1
[15]  Bagley, R.L. and Calico, R.A. (1991) Fractional Order State Equations for the Control of Viscoelastically Damped Structures. Journal of Guidance, Control, and Dynamics, 14, 304. https://doi.org/10.2514/3.20641
[16]  Sun, H.H., Abdelwahab, A.A. and Onaral, B. (1984) Linear Approximation of Transfer Function with a Pole of Fractional Power. IEEE Transactions on Automatic Control, 29, 441-444. https://doi.org/10.1109/TAC.1984.1103551
[17]  Ichisea, M., Nagayanagia, Y. and Kojima, T.J. (1971) An Analog Simulation of Non-Integer Order Transfer Functions for Analysis of Electrode Processes. Journal of Electroanalytical Chemistry, 33, 253.
[18]  Laskin, N. (2000) Fractional Market Dynamics. Physica A: Statistical Mechanics and Its Applications, 287, 482-492. https://doi.org/10.1016/S0378-4371(00)00387-3
[19]  Kusnezov, D., Bulgac, A. and Dang, G.D. (1999) Quantum Lévy Processes and Fractional Kinetics. Physical Review Letters, 82, 1136. https://doi.org/10.1103/PhysRevLett.82.1136
[20]  Alidousti, J., Ghaziani, R.K. and Bayati, A.E. (2017) Stability Analysis of Nonlinear Fractional Differential Order Systems with Caputo and Riemann-Liouville Derivatives. Turkish Journal of Mathematics, 41, 1260-1278. https://doi.org/10.3906/mat-1510-5
[21]  Matignon, D. (1996) Stability Results for Fractional Differential Equations with Applications to Control Processing. Computational Engineering in Systems and Application Multiconference, Vol. 2, 963-968.
[22]  Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R. and Abdeljawad, T. (2010) Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay. Abstract and Applied Analysis, 2010, Article ID: 108651. https://doi.org/10.1155/2011/213485
[23]  Li, Y., Chen, Y.Q. and Podlubny, I. (2009) Mittag-Leffler Stability of Fractional Order Nonlinear Dynamics Systems. Automatica, 45, 1965-1969. https://doi.org/10.1016/j.automatica.2009.04.003
[24]  Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.
[25]  Qian, D., Li, C., Agarwal, R.P. and Wong, P.J. (2010) Stability Analysis of Fractional Differential System with Riemann-Liouville Derivative. Mathematical and Computer Modelling, 52, 862-874. https://doi.org/10.1016/j.mcm.2010.05.016
[26]  Diethelm, K. (2010) The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin. https://doi.org/10.1007/978-3-642-14574-2
[27]  Garrappa, R. (2010) On Linear Stability of Predictor-Corrector Algorithms for Fractional Differential Equations. International Journal of Computer Mathematics, 87, 2281-2290. https://doi.org/10.1080/00207160802624331

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