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Bifurcation of nonsymmetric solutions for some duffing equations

Published online by Cambridge University Press:  17 April 2009

Fumio Nakajima
Fumio Nakajima
Affiliation:
Department of MathematicsFaculty of EducationIwate UniversityMorioka 020–0066Japan
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Abstract

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For some symmetric Duffing equation, the existence of bifurcation of nonsymmetric, periodic solutions from symmetric periodic solutions is proved by using the change of index of symmetric, periodic solutions for variation of parameters.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 119 - 128
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bowman, T.T., ‘Periodic solutions of Liénard systems with symmetries’, Nonlinear Anal. 2 (1978), 457–464.CrossRefGoogle Scholar
[2]Coddington, E.A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, New York, Toronto, London, 1955).Google Scholar
[3]Hayashi, C., Ueda, Y. and Kawakami, H., ‘Transformation theory as applied to the solutions of nonlinear differential equations of the second order’, Internat. J. Non-Linear Mech. 4 (1969), 235–255.CrossRefGoogle Scholar
[4]Levinson, N., ‘Transformation theory of non-linear differential equations of the second order’, Ann. of Math. 45 (1944), 723–737.CrossRefGoogle Scholar
[5]Loud, W.S., ‘Some growth theorems for linear ordinary differential equations’, Trans. Amer. Math. Soc. 85 (1957), 257–264.CrossRefGoogle Scholar
[6]Loud, W. S., ‘Periodic solutions of nonlinear differential equations of Duffing type’, in Proceedings of U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn 1967) (Benjamin, New York (1968)), pp. 199–224.Google Scholar
[7]Loud, W.S., ‘Nonsymmetric periodic solutions of certain second order nonlinear differentia equations’, J. Differential Equations 5 (1969), 352–368.CrossRefGoogle Scholar
[8]Minorsky, N., Nonlinear oscillations (D. Van Nostrand, Princeton, NJ, 1962).Google Scholar
[9]Murthy, P., ‘Periodic solutions of two-dimensional forced systems: the Massera theorem and its extension’, J. Dynam. Differential Equations 10 (1998), 275–302.CrossRefGoogle Scholar
[10]Nakajima, F. and Seifert, G., ‘The number of periodic solutions of 2-dimensional periodic systems’, J. Differential Equations 49 (1983), 430–440.CrossRefGoogle Scholar
[11]Nakajima, F., ‘Index theorems and bifurcations in Duffing's equations’, in Recent Topics in Nonlinear PDE II (Sendai, 1985), North-Holland Math. Studies 128 (North Holland Amsterdam, New York, 1985), pp. 133–161.CrossRefGoogle Scholar
[12]Nakajima, F., ‘Nonlinear Mathieu equations I’, in Nonlinear Waves, International Series 10 (Gakuto, Tokyo, 1997), pp. 353–359.Google Scholar
[13]Sansone, G. and Conti, R., Nonlinear differential equations (Pergamon Press, New York, 1964).Google Scholar