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Local bifurcations of critical periods in the reduced Kukles system

 Printed: Apr 1997
  • C. Rousseau
  • B. Toni
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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.
MSC Classifications: 34C25, 58F14 show english descriptions Periodic solutions
unknown classification 58F14
34C25 - Periodic solutions
58F14 - unknown classification 58F14

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