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# Fonctions elliptiques et équations différentielles ordinaires

Published:1997-09-01
Printed: Sep 1997
• Raouf Chouikha
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## Abstract

In this paper, we detail some results of a previous note concerning a trigonometric expansion of the Weierstrass elliptic function $\{\wp(z);\, 2\omega, 2\omega'\}$. In particular, this implies its classical Fourier expansion. We use a direct integration method of the ODE $$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right.$$ where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case, the bifurcations of $(E)$ depend on one parameter only. Moreover, this global method seems not to apply to the cases $n > 3$.
 MSC Classifications: 33E05 - Elliptic functions and integrals 34A05 - Explicit solutions and reductions 33E20 - Other functions defined by series and integrals 33E30 - Other functions coming from differential, difference and integral equations 34A20 - unknown classification 34A2034C23 - Bifurcation [See also 37Gxx]

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