×

Simple conditions for parametrically excited oscillations of generalized Mathieu equations. (English) Zbl 1367.34033

In this paper, the authors provide parametric conditions for oscillation and nonoscillation of generalized Mathieu equations. Parametric oscillation and nonoscillation regions are drawn for a better understanding of the obtained results.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: DOI

References:

[1] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1073.34002
[2] Benjamin, T. B.; Ursell, F., The stability of the plane free surface of a liquid in vertical periodic motion, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 225, 505-515 (1954) · Zbl 0057.18801
[3] Binney, J., Twisted and warped disks as consequences of heavy halos, Mon. Not. R. Astron. Soc., 183, 779-797 (1978)
[4] Binney, J., Resonant excitation of motion perpendicular to galactic planes, Mon. Not. R. Astron. Soc., 196, 445-467 (1981) · Zbl 0466.70012
[5] Çakmak, D., A note on M. K. Kwong and J. S. W. Wong’s paper, Dynam. Systems Appl., 15, 409-414 (2006)
[6] Cerda, E. A.; Tirapegui, E. L., Faraday’s instability in viscous fluid, J. Fluid Mech., 368, 195-228 (1998) · Zbl 0915.76037
[7] Coppel, W. A., Disconjugacy, Lecture Notes in Math., vol. 220 (1971), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0224.34003
[8] El-Sayed, M. A., An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc., 118, 813-817 (1993) · Zbl 0777.34023
[9] Faraday, M., On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Philos. Trans. R. Soc. Lond., 121, 299-340 (1831)
[10] Fite, W. B., Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc., 19, 341-352 (1918) · JFM 46.0702.02
[11] Hill, G. W., On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon, Acta Math., 8, 1-36 (1886) · JFM 18.1106.01
[12] Komkov, V., A technique for the detection of oscillation of second order ordinary differential equation, Pacific J. Math., 42, 105-115 (1972) · Zbl 0241.34038
[13] Kong, Q.; Pašić, M., Second-order differential equations some significant results due to James S. W. Wong, Differ. Equ. Appl., 6, 99-163 (2014) · Zbl 1294.34038
[14] Kwong, M. K.; Wong, J. S.W., Oscillation and nonoscillation of Hill’s equation with periodic damping, J. Math. Anal. Appl., 288, 15-19 (2003) · Zbl 1039.34026
[15] Kwong, M. K.; Wong, J. S.W., On the oscillation of Hill’s equations under periodic forcing, J. Math. Anal. Appl., 320, 37-55 (2006) · Zbl 1102.34021
[16] Leighton, W., Hill’s equation revisited, J. Math. Anal. Appl., 114, 497-502 (1986) · Zbl 0594.34030
[17] Magnus, W.; Winkler, S., Hill’s Equation (1979), Dover: Dover New York · Zbl 0158.09604
[18] Mathieu, É., Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique, J. Math. Pures Appl., 13, 137-203 (1868) · JFM 01.0354.02
[19] McLachan, N. W., Theory and Application of Mathieu Functions (1964), Dover: Dover New York · Zbl 0128.29603
[20] Nayfeh, A. H.; Mook, D. T., Nonlinear Oscillations (1995), John Wiley & Sons: John Wiley & Sons New York · Zbl 0919.73156
[21] Rajchenbach, J.; Clamond, D., Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited, J. Fluid Mech., 777, Article R2 pp. (2015) · Zbl 1381.76070
[22] Sugie, J., Geometrical conditions for oscillation of second-order half-linear differential equations, Acta Math. Hungar., 118, 369-394 (2008) · Zbl 1164.34016
[23] Sugie, J.; Matsumura, K., A nonoscillation theorem for half-linear differential equations with periodic coefficients, Appl. Math. Comput., 199, 447-455 (2008) · Zbl 1217.34056
[24] Sun, Y. G.; Ou, C. H.; Wong, J. S.W., Interval oscillation theorems for a second-order linear differential equation, Comput. Math. Appl., 48, 1693-1699 (2004) · Zbl 1069.34049
[25] Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations, Math. Sci. Eng., vol. 48 (1968), Academic Press: Academic Press New York/London · Zbl 0191.09904
[26] Wong, J. S.W., Second order nonlinear forced oscillations, SIAM J. Math. Anal., 19, 667-675 (1988) · Zbl 0655.34023
[27] Wong, J. S.W., On a theorem of Sobol, Bull. Inst. Math. Acad. Sin. (N.S.), 27, 253-264 (1999) · Zbl 0960.34023
[28] Wong, J. S.W., On Kamenev-type oscillation theorems for second-order differential equations with damping, J. Math. Anal. Appl., 258, 244-257 (2001) · Zbl 0987.34024
[29] Yan, J., A note on an oscillation criterion for an equation with damped term, Proc. Amer. Math. Soc., 90, 277-280 (1984) · Zbl 0542.34028
[30] Yan, J., Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc., 98, 276-282 (1986) · Zbl 0622.34027
[31] Zheng, Z., Note on Wong’s paper, J. Math. Anal. Appl., 274, 466-473 (2002) · Zbl 1025.34030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.