Elliptical functions and ordinary differential equations

In this paper, we detail some results of a previous note concerning a trigonometric expansion of the Weierstrass elliptic function {p(z); 2omega, 2omega'}. In particular, this implies its classical Fourier expansion. We use a direct integration method of the ODE: (E) is d(2)u/dt(2) = P(u,lambda), u(0) = sigma, du/dt(0) = tau 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.