Fractional Mathieu equation

Communications in Nonlinear Science and Numerical Simulation

Volumn 15, Issue 11, 2010, Pages 3254-3262

  Rand, R H ^a,b   Sah, S M ^c   Suchorsky, M K ^a  

^a Cornell University   (United States)

^b Department of Obstetrics Gynecology   (United States)

^c HASSAN II UNIVERSITY   (Morocco)

Author keywords

Fractional calculus; Fractional derivative; Parametric excitation

Indexed keywords

FRACTIONAL CALCULUS; FRACTIONAL DERIVATIVES; HIGH-ORDER APPROXIMATION; HIGHER-ORDER APPROXIMATION; LOW ORDER APPROXIMATION; MATHIEU'S EQUATION; METHOD OF HARMONIC BALANCES; PARAMETRIC EXCITATIONS; REGION OF STABILITIES; TRANSITION CURVES;

CALCULATIONS;

EID: 77952239495     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2009.12.009     Document Type: Article

References (30)

1. Ahmad W.M., and Sprott J.C. Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fract 16 (2003) 339-351
2. Barbosa R.S., Machado J.A.T., Vinagre B.M., and Calderon A.J. Analysis of the van der pol oscillator containing derivatives of fractional order. J Vib Control 13 (2007) 1291-1301
3. Bridge J., Rand R., and Sah S.M. Slow passage through multiple parametric resonance tongues. J Vib Control 15 (2009) 1581-1600
4. Cao J, Ma C, Xie H, Jiang Z. Nonlinear dynamics of duffing system with fractional order damping. In: DETC2009-86401, Proceedings of ASME IDETC/CIE 2009 conference, San Diego, CA, August 30-September 2; 2009.
5. Chen J.H., and Chen W.C. Chaotic dynamics of the fractionally damped van der pol equation. Chaos Solitons Fract 35 (2008) 188-198
6. Galucio A.C., Deu J.F., and Ohayon R. Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput Mech 33 (2004) 282-291
7. Ge Z.M., and Yi C.X. Chaos in a nonlinear damped Mathieu system. Chaos Solitons Fract 32 (2007) 42-61
8. Jesus I.S., and Tenreiro Machado J.A. Implementation of fractional-order electromagnetic potential through a genetic algorithm. Commun Nonlinear Sci Numer Simul 14 (2009) 838-843
9. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and applications of fractional differential equations (2006), Elsevier, Amsterdam
10. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fract 7 (1996) 1461-1477
11. Meral F.C., Royston T.J., and Magin R. Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15 (2010) 939-945
12. Miller K., and Ross B. An introduction to the fractional calculus and fractional differential equations (1993), Wiley, New York
13. Morrison T.M., and Rand R.H. 2:1 resonance in the delayed nonlinear Mathieu equation. Nonlinear Dyn 50 (2007) 341-352
14. Naber M. Linear fractionally damped oscillator. Int J Differ. Equ. 2010 (2010) 10.1155/2010/197020 Article ID 197020, 12 p.
15. Oldham K.B., and Spanier J. The fractional calculus (1974), Academic Press, New York
16. Petras I. A note on the fractional-order Volta's system. Commun Nonlinear Sci Numer Simul 15 (2010) 384-393
17. Podlubny I. Fractional differential equations (1990), Academic Press, San Diego
18. Rand R.H. Dynamics of a nonlinear parametrically-excited PDE: 2-term truncation. Mech Res Commun 23 (1996) 283-289
19. Rand RH. Lecture notes on nonlinear vibrations (version 52). Available from http://audiophile.tam.cornell.edu/randdocs/; 2007.
20. Ross B. A brief history and exposition of the fundamental theory of fractional calculus. In: Fractional calculus and its applications. Springer lecture notes in mathematics, vol. 57; 1975. p. 1-36.
21. Rudinger F. Tuned mass damper with fractional derivative damping. Eng Struct 28 (2006) 1774-1779
22. Ryabov Y.E., and Puzenko A. Damped oscillation in view of the fractional oscillator equation. Phys Rev B 66 (2002) 184201
23. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional integrals and derivatives: theory and applications (1993), Gordon and Breach, Amsterdam
24. Stoker J.J. Nonlinear vibrations in mechanical and electrical systems (1950), Wiley, New York
25. Tavazoei M.S., Haeri M., Attari M., Bolouki S., and Siami M. More details on analysis of fractional-order van der pol oscillator. J Vib Control 15 (2009) 803-819
26. Torvik P.J., and Bagley R.L. On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech 51 (1984) 294-298
27. Wahl P., and Chatterjee A. Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dyn 38 (2004) 3-22
28. Xie F., and Lin X. Asymptotic solution of the van der pol oscillator with small fractional damping. Phys Scripta T136 (2009) 014033
29. Yonggang K, Zhang Xiue. Some comparison of two fractional oscillators. Physica B. doi:10.1016/j.physb.2009.08.092.
30. Zounes R.S., and Rand R.H. Transition curves in the quasiperiodic Mathieu equation. SIAM J Appl Math 58 (1998) 1094-1115