 ## Dynamical analysis of fractional-order Mathieu equation

Shaofang Wen1 , Yongjun Shen2 , Xianghong Li3 , Shaopu Yang4 , Haijun Xing5

1Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang, China

2, 4, 5Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, China

3Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China

2Corresponding author

Journal of Vibroengineering, Vol. 17, Issue 5, 2015, p. 2696-2709.
Received 1 February 2015; received in revised form 25 March 2015; accepted 11 April 2015; published 15 August 2015

Copyright © 2015 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract.

The dynamical characteristics of Mathieu equation with fractional-order derivative is analytically studied by the Lindstedt-Poincare method and the multiple-scale method. The stability boundaries and the corresponding periodic solutions on these boundaries for the constant stiffness ${\delta }_{0}={n}^{2}$ ($n$ = 0, 1, 2, …), are analytically obtained. The effects of the fractional-order parameters on the stability boundaries and the corresponding periodic solutions, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient (ELDC) and the equivalent linear stiffness coefficient (ELSC). The comparisons between the transition curves on the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. The following analysis is focused on the effects of the fractional parameters on the stability boundaries located in the $\delta -\epsilon$ plane. It is found that the increase of the fractional order $p$ could make the ELDC larger and ELSC smaller, which could result into the rightwards and upwards moving of the stability boundaries simultaneously. It could also be concluded the increase of the fractional coefficient ${K}_{1}$ would make the ELDC and ELSC larger, which could move the transition curves to the left and upwards at the same time. These results are very helpful to design, analyze or control this kind of system, and could present beneficial reference to the similar fractional-order system.

Keywords: fractional-order derivative, Mathieu equation, Lindstedt-Poincare method, multiple-scale method, stability boundaries.

#### Acknowledgements

The authors are grateful to the support by National Natural Science Foundation of China (No. 11372198), the Program for New Century Excellent Talents in University (NCET-11-0936), the Cultivation Plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province (LJRC018), the Program for Advanced Talent in the Universities of Hebei Province (GCC2014053), and the Program for Advanced Talent in Hebei (A201401001).

1. Miller K. S., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993. [CrossRef]
2. Podlubny I. Fractional Differential Equations. Academic, London, 1998. [CrossRef]
3. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006. [CrossRef]
4. Das S. Functional Fractional Calculus for System Identification and Controls. Springer-Verlag, Berlin, 2008. [CrossRef]
5. Caponetto R., Dongola G., Fortuna L., Petras I. Fractional Order Systems: Modeling and Control Applications. World Scientific, New Jersey, 2010. [CrossRef]
6. Monje C. A., Chen Y. Q., Vinagre B. M., Xue D. Y., Feliu V. Fractional-order Systems and Controls: Fundamentals and Applications. Springer-Verlag, London, 2010. [CrossRef]
7. Petras I. Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press, Beijing, 2011. [CrossRef]
8. Shen Y. J., Yang S. P., Xing H. J., Gao G. S. Primary resonance of Duffing oscillator with fractional-order derivative. Communications in Nonlinear Science and Numerical Simulation, Vol. 17, Issue 7, 2012, p. 3092-3100. [CrossRef]
9. Shen Y. J., Yang S. P., Xing H. J., et al. Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives. International Journal of Non-Linear Mechanics, Vol. 47, Issue 9, 2012, p. 975-983. [CrossRef]
10. Shen Y. J., Wei P., Yang S. P. Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dynamics, Vol. 77, Issue 4, 2014, p. 1629-1642. [CrossRef]
11. Shen Y. J., Yang S. P., Sui C. Y. Analysis on limit cycle of fractional-order van der Pol oscillator. Chaos, Solitons & Fractals, Vol. 67, 2014, p. 94-102. [CrossRef]
12. Gorenflo R., Abdel-Rehim E. A. Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. Journal of Computational and Applied Mathematics, Vol. 205, Issue 2, 2007, p. 871-881. [CrossRef]
13. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results. Computers and Mathematics with Applications, Vol. 51, Issue 9-10, 2006, p. 1367-1376. [CrossRef]
14. Ishteva M., Scherer R., Boyadjiev L. On the Caputo operator of fractional calculus and C-Laguerre functions. Mathematical Sciences Research Journal, Vol. 9, Issue 6, 2005, p. 161-170. [CrossRef]
15. Agnieszka B. Malinowska, Delfim F. M. Torres Fractional calculus of variations for a combined Caputo derivative. Fractional Calculus and Applied Analysis, Vol. 14, Issue 4, 2011, p. 523-537. [CrossRef]
16. Chen L. C., Zhu W. Q. Stochastic dynamics and fractional optimal control of quasi-integrable Hamiltonian systems with fractional derivative damping. Fractional Calculus and Applied Analysis, Vol. 16, Issue 1, 2013, p. 189-225. [CrossRef]
17. Chen L. C., Zhu W. Q. The first passage failure of SDOF strongly nonlinear stochastic system with fractional derivative damping. Journal of Vibration Control, Vol. 15, Issue 8, 2009, p. 1247-1266. [CrossRef]
18. Chen L. C., Zhu W. Q. Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations. International Journal of Non-linear Mechanics, Vol. 46, Issue 12, 2011, p. 1324-1329. [CrossRef]
19. Chen L. C., Li H. F., Li Z. S., Zhu W. Q. First passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative. Journal of Vibration and Control, Vol. 19, Issue 14, 2013, p. 2154-2163. [CrossRef]
20. Chen L. C., Wang W. H., Li Z. S., Zhu W. Q. Stationary response of Duffing oscillator with hardening stiffness and fractional derivative. International Journal of Non-linear Mechanics, Vol. 48, Issue 1, 2013, p. 44-50. [CrossRef]
21. Wang Z. H., Hu H. Y. Stability of a linear oscillator with damping force of fractional-order derivative. Science in China G: Physics, Mechanics and Astronomy, Vol. 39, Issue 10, 2009, p. 1495-1502. [CrossRef]
22. Wang Z. H., Du M. L. Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system. Shock and Vibration, Vol. 18, Issues 1-2, 2011, p. 257-268. [CrossRef]
23. Li C. P., Deng W. H. Remarks on fractional derivatives. Applied Mathematics and Computation, Vol. 187, Issue 2, 2007, p. 777-784. [CrossRef]
24. Cao J. X., Ding H. F., Li C. P. Implicit difference schemes for fractional diffusion equations. Communication on Applied Mathematics and Computation, Vol. 27, Issue 1, 2013, p. 61-74. [CrossRef]
25. Zeng F. H., Li C. P. High-order finite difference methods for time-fractional subdiffusion equation. Chinese Journal of Computational Physics, Vol. 30, Issue 4, 2013, p. 491-500. [CrossRef]
26. Chen A., Li C. P. Numerical algorithm for fractional calculus based on Chebyshev polynomial approximation. Journal of Shanghai Jiaotong University, Vol. 18, Issue 1, 2012, p. 48-53. [CrossRef]
27. Wahi P., Chatterjee A. Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dynamics, Vol. 38, Issue 1-4, 2004, p. 3-22. [CrossRef]
28. Xu Y., Li Y. G., Liu D., Jia W. T., Huang H. Responses of Duffing oscillator with fractional damping and random phase. Nonlinear Dynamics, Vol. 74, Issue 3, 2013, p. 745-753. [CrossRef]
29. Mclachlan N. W. Theory and Application of Mathieu Functions. Oxford University Press, London, 1951. [CrossRef]
30. Ge Z. M., Yi C. X. Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems. Chaos, Solitons & Fractals, Vol. 32, Issue 1, 2007, p. 42-61. [CrossRef]
31. Ebaid A., ElSayed D. M. M., Aljoufi M. D. Fractional calculus model for damped Mathieu equation: approximate analytical solution. Applied Mathematical Sciences, Vol. 6, Issue 82, 2012, p. 4075-4080. [CrossRef]
32. Rand R. H., Sah S. M., Suchorsky M. K. Fractional Mathieu equation. Communications in Nonlinear Science and Numerical Simulation, Vol. 15, Issue 11, 2010, p. 3254-3262. [CrossRef]
33. Leung A. Y. T., Guo Z. J., Yang H. X. Transition curves and bifurcations of a class of fractional Mathieu-type equations. International Journal of Bifurcation and Chaos, Vol. 22, 2012, p. 1-13. [CrossRef]
34. Nayfeh A. H. Nonlinear Oscillations. Wiley, New York, 1979. [CrossRef]
35. Nayfeh A. H. Introduction to Perturbation Techniques. Wiley, New York, 1981. [CrossRef]
36. Rossikhin Y. A., Shitikova M. V. On fallacies in the decision between the Caputo and Riemann-Liouville fractional derivative for the analysis of the dynamic response of a nonlinear viscoelastic oscillator. Mechanics Research Communications, Vol. 45, 2012, p. 22-27. [CrossRef]