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

Published online by Cambridge University Press:  24 October 2008

G. R. Morris
G. R. Morris
Affiliation:
University of Queensland, Brisbane
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Extract

The most general differential equation to which the dynamical description of the title applies is

where dots denote differentiation with respect to t. The essential problem for this equation is to determine the behaviour of solutions as t → ∞. When we attack this problem, the most obvious question is whether, under reasonable conditions on p(t), every solution is bounded as t → ∞ this question is open except when g(x) is linear. In the special case when p(t) is periodic, (1·1) may have periodic solutions; it is clear that any such solution is bounded, and it is worth mentioning that finding periodic solutions is the easiest way of finding particular bounded ones. So long as the bounded-ness problem is unsolved, there is a special interest in finding a large class of particular bounded solutions: if we know such a class then, although we cannot say whether the general solution is bounded or not, we can make the imprecise comment that either the general solution is in fact bounded or the structure of the whole set of solutions is quite complicated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 133 - 155
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1)Harvey, C. A.Periodic solutions differential equation ẍ + g(x) = p(t). Contributions to differential equations 1 (1963), 425–451.Google Scholar
(2)Morris, G. R.A differential equation for undamped forced non-linear oscillations. I. Proc. Cambridge Philos. Soc. 51 (1955), 297–312.CrossRefGoogle Scholar
(3)Morris, G. R.A differential equation for undamped forced non-linear oscillations. II. Proc. Cambridge Philos. Soc. 54 (1958), 426–438.CrossRefGoogle Scholar
(4)Morris, G. R.An infinite class of periodic solutions of ẍ + 2x 3 = p(t). Proc. Cambridge Philos. Soc. 61 (1965), 157–164.CrossRefGoogle Scholar
(5)Moser, J. Perturbation theory for almost periodic solutions for undamped nonlinear differential equations, from International symposium on non-linear differential equations and nonlinear mechanics (ed. LaSalle, J. P. and Lefschetz, S.; New York, 1963), pp. 71–79.CrossRefGoogle Scholar
(6)Moser, J.On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ila (1962), pp. 1–20.Google Scholar