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A differential equation for undamped forced non-linear oscillations. I

Published online by Cambridge University Press:  24 October 2008

G. R. Morris
G. R. Morris
Affiliation:
Mathematics DepartmentThe UniversityHull
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Extract

In this paper and its sequels I consider the differential equation

in which e(t) is an even periodic function of t and dots denote differentiation with respect to t. We can regard (1·1) as a special case of the equation

(the ‘differential equation of non-linear oscillations’), which describes the motion on the x–axis of a particle of unit mass subject to a restoring force g(x), a variable damping force f(x, ẋ)ẋ and an external force p(t); the dynamical description used in the title appeals to this interpretation. The most interesting problems on (1·2) and its specializations concern the behaviour of solutions as t → ∞ for example, we may seek conditions on f, g and p which allow us to assert that some or all solutions are bounded or, having assumed that p(t) is periodic, seek conditions on f and g which allow us to assert the existence of periodic solutions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 2 , April 1955 , pp. 297 - 312
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Cartwright, M. L. Forced oscillations in nonlinear systems. In Contributions to the theory of nonlinear oscillations (Ann. Math. Stud. no. 20, Princeton, 1950, pp. 149–241; see p. 174.Google Scholar
(2)Cartwright, M. L. and Littlewood, J. E.On non-linear differential equations of the second order. II. Ann. Math., Princeton, (2), 48 (1947), 472–94.CrossRefGoogle Scholar
(3)Duffing, G.Erzwungene Schwingungen bei veränderlicher Eigenfrequenz (Brunswick, 1918).Google Scholar
(4)Hamel, G.Über erzwungene Schwingungen bei endlichen Amplituden. Math. Ann. 86 (1922), 1–13.CrossRefGoogle Scholar
(5)Iglisch, R.Über die Lösungen des Duffingschen Schwingungsproblems bei groBen Parameterwerten. Math. Ann. 111 (1935), 568–81.Google Scholar
(6)John, F.On harmonic vibrations out of phase with the exciting force. Commun. appl. Math. 1 (1948), 341–59.Google Scholar
(7)Levinson, N.Transformation theory of non-linear differential equations of the second order. Ann. Math., Princeton, (2), 45 (1944), 723–37.Google Scholar
(8)Newman, M. H. A.On the ultimate boundedness of the solutions of certain differential equations. Compos. math. 8 (1950), 142–56.Google Scholar
(9)Reuter, G. E. H.A boundedness theorem for non-linear differential equations of the second order. Proc. Camb. phil. Soc. 47 (1951), 49–54.Google Scholar
(10)Stoker, J. J.Nonlinear vibrations (New York, 1950).Google Scholar
(11)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th ed. (Cambridge, 1935), see p. 524.Google Scholar