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1. CJM 1998 (vol 50 pp. 134)

Médan, Christine
 On critical level sets of some two degrees of freedom integrable Hamiltonian systems We prove that all Liouville's tori generic bifurcations of a large class of two degrees of freedom integrable Hamiltonian systems (the so called Jacobi-Moser-Mumford systems) are nondegenerate in the sense of Bott. Thus, for such systems, Fomenko's theory~\cite{fom} can be applied (we give the example of Gel'fand-Dikii's system). We also check the Bott property for two interesting systems: the Lagrange top and the geodesic flow on an ellipsoid. Categories:70H05, 70H10, 58F14, 58F07

2. CJM 1997 (vol 49 pp. 338)

Rousseau, C.; Toni, B.
 Local bifurcations of critical periods in the reduced Kukles system In this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gr\"obner bases, theory of the resultant, Sturm's algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves. Categories:34C25, 58F14

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