Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The damped Mathieu equation

Author: Lawrence Turyn
Journal: Quart. Appl. Math. 51 (1993), 389-398
MSC: Primary 34C15; Secondary 34B30, 34C25, 34D10, 70J40
DOI: https://doi.org/10.1090/qam/1218375
MathSciNet review: MR1218375
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish an asymptotic lower bound for the minimum excitation needed to cause instability for the damped Mathieu equation. The methods used are Floquet theory and Liapunov-Schmidt, and we use a fact about the width of the instability interval for the undamped Mathieu equation. Our results are compared with published numerical data.

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  • [1] M. Bell, A note on Mathieu functions, Proc. Glasgow Math. Assn. 3, 132-134 (1957) MR 0120408
  • [2] T. B. Benjamin and F. Ursell, The stability of the plane free surface of a liquid in vertical periodic motion, Proc. Roy. Soc. London Ser. A 225, 505-515 (1954) MR 0065315
  • [3] A. Cañada and P. Martínez-Amores, Bifurcation in the Mathieu equation with three independent parameters, Quart. Appl. Math. 37, 431-441 (1980) MR 564734
  • [4] Jack K. Hale, Oscillations in Nonlinear Systems, McGraw-Hill, New York, 1963 MR 0150402
  • [5] Jack K. Hale, On the behavior of the solutions of linear periodic differential systems near resonance points, Contributions to the Theory of Nonlinear Oscillations, Ann. of Math. Stud., vol. 5, Princeton Univ. Press, Princeton, NJ, 1960, pp. 55-89 MR 0141827
  • [6] Jack K. Hale, Ordinary Differential Equations, 2nd ed., Robert E. Kreiger Publ. Co., Huntington, New York, 1980 MR 587488
  • [7] Chihiro Hayashi, Nonlinear Oscillations in Physical Systems, Princeton Univ. Press, Princeton, NJ, 1985
  • [8] Th. Lieber and H. Risken, Stability of parametricallv excited dissipative systems, Phys. Lett. A 129, 214-218 (1988) MR 944248
  • [9] N. W. McLachlan, Theory and Applications of Mathieu Functions, Oxford Univ. Press, London, 1947 MR 0021158
  • [10] Wilhelm Magnus and Stanley Winkler, Hill's Equation, Interscience Publ., John Wiley & Sons, New York, 1966 MR 0197830
  • [11] C. Pierre and E. H. Dowell, A study of dynamic instability of plates by an extended incremental harmonic balance method, Trans. ASME Ser. E J. Appl. Mech. 52, 693-697 (1985)

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Additional Information

DOI: https://doi.org/10.1090/qam/1218375
Article copyright: © Copyright 1993 American Mathematical Society

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