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26.—Singularities of Solutions of Certain Plane Autonomous Systems

Published online by Cambridge University Press:  14 February 2012

Russell A. Smith
Russell A. Smith
Affiliation:
Department of Mathematics, University of Durham.
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Extract

The paper discusses the nature of the singularities at any finite point t = t0 of real solutions x(t), y(t) of the system of differential equations x′ = P(x, y), y′ = Q(x, y) in which P, Q are polynomials in both x and y. Its main interest is in cases when the leading terms of P, Q are of the same degree n. Conditions are given under which the only possible singularities are algebraic poles and pseudo-poles. Conditions are also given under which the only possible singularities are of wildly oscillatory type.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 72 , Issue 4 , 1975 , pp. 307 - 315
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

References to Literature

Dulac, H. 1912. Solutions d'un système d'équations différentielles dans le voisinage de valeurs singulièrs. Bull. Soc. Math. Fr., 40, 324–392.Google Scholar
Hille, E., 1974. A note on quadratic systems. Proc. Roy. Soc. Edinb. A, 72, 17–37.Google Scholar
Malmquist, J., 1921. Sur les points singuliers des équations différentielles, 1e memoire. Ark. Mat. Astr. Fys., 15 (3).Google Scholar
Painlevé, P., 1897. Lecons sur la théorie analytique des équations différentielles professées à Stockholm. Paris: Herman.Google Scholar
Sasone, G. and Conti, R., 1964. Nonlinear Differential Equations (English translation by Diamond, A. H.). Oxford: Pergamon.Google Scholar
Valiron, G., 1950. Équations Fonctionelles Applications. 2e Édn. Paris: Masson.Google Scholar