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
Full-text PDF Free Access
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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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M. Bell, A note on Mathieu functions, Proc. Glasgow Math. Assn. 3, 132–134 (1957)
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)
A. Cañada and P. Martínez-Amores, Bifurcation in the Mathieu equation with three independent parameters, Quart. Appl. Math. 37, 431–441 (1980)
Jack K. Hale, Oscillations in Nonlinear Systems, McGraw-Hill, New York, 1963
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
Jack K. Hale, Ordinary Differential Equations, 2nd ed., Robert E. Kreiger Publ. Co., Huntington, New York, 1980
Chihiro Hayashi, Nonlinear Oscillations in Physical Systems, Princeton Univ. Press, Princeton, NJ, 1985
Th. Lieber and H. Risken, Stability of parametricallv excited dissipative systems, Phys. Lett. A 129, 214–218 (1988)
N. W. McLachlan, Theory and Applications of Mathieu Functions, Oxford Univ. Press, London, 1947
Wilhelm Magnus and Stanley Winkler, Hill’s Equation, Interscience Publ., John Wiley & Sons, New York, 1966
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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Article copyright:
© Copyright 1993
American Mathematical Society