Orbites periodiques dans un potentiel a trois dimensions II. Bifurcations

In a previous paper, Hayliet al. (1983), two families of periodic orbits in the three-dimensional potential $$U = \frac{1}{2}(Ax^2 + By^2 + Cz^2 ) - \varepsilon xz^2 - nyz^2 $$ with \(\sqrt A :\sqrt B :\sqrt C = 6:4:3\) and ?=0.5 were described. It was found empirically that the characteristic curves of the two families intersect in the space (x₀, y₀, η) for |η|?0.2. This property is demonstrated in the present paper by writing explicitely the Poincaré mapping and by giving an approximation directly comparable with the numerical results obtained in Hayliet al. (1983). It is thus shown that one family bifurcates off the other.