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On the recurrent solutions of a class of non-linear differential equations

Published online by Cambridge University Press:  24 October 2008

F. G. Friedlander
F. G. Friedlander
Affiliation:
Department of MathematicsThe UniversityManchester
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Extract

1. This paper is concerned with certain properties of the solutions of differential equations of the type

where f, g have continuous partial derivatives up to the second order satisfying Lipschitz conditions in some bounded domain and are periodic in t with period 2π, ω is a positive constant and k is a small parameter. If f = 0, (1·1) is equivalent to

an equation representing the forced vibrations of a quasi-linear system of one degree of freedom.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 2 , April 1951 , pp. 315 - 330
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Birkhoff, G. D.Surface transformations and their dynamical applications, Acta Math. 43 (1922), 1–119.CrossRefGoogle Scholar
(2)Levinson, N.Transformation theory of non-linear differential equations of the second order, Ann. Math. 45 (1944), 723–37.CrossRefGoogle Scholar
(3)Bohl, P.Über die hinsichtlich der unabhängigen und abhängigen Variabeln periodische Differentialgleichung erster Ordnung, Acta Math. 40 (1916), 321–36.CrossRefGoogle Scholar
(4)Denjoy, A.Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. (9) 11 (1932), 333–75.Google Scholar
(5)Kryloff, N. and Bogoliuboff, N. Introduction to non-linear mechanics, trans. by Lefschetz, S., Ann. Math. Stud. no. 11 (Princeton, 1943).Google Scholar
(6)Kryloff, N. and Bogoliuboff, N.Méthodes de la mécanique non-linéaire appliquées à l'étude des oscillations stationnaires (Kiev, 1934).Google Scholar
(7)Friedlander, F. G.On the forced vibrations of quasi-linear systems, Quart. J. Mech. Appl. Math. 3 (1950) 364–76.CrossRefGoogle Scholar
(8)Zygmund, A.Trigonometrical series (Warsaw, 1935), p. 136.Google Scholar
(9)Cartwright, M. L.On non-linear differential equations of the second order, III. Proc. Cambridge Phil. Soc. 45 (1949), 495–501.CrossRefGoogle Scholar