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Local Bifurcations of Critical Periods in the Reduced Kukles System

Published online by Cambridge University Press:  20 November 2018

C. Rousseau  and
B. Toni
C. Rousseau
Affiliation:
Département de mathématiques et de statistique and CRM, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, H3C 3J7
B. Toni
Affiliation:
Départment of Mathematics, Faculdad de Ciencias, UAEM, Cuernavaca 62210, Mexico
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Abstract

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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 (Gröbner bases, theory of the resultant, Sturm’s algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

Keywords

34C2558F14
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 338 - 358
Copyright
Copyright © Canadian Mathematical Society 1997

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