Bifurcation of critical periods from the rigid quadratic isochronous vector field |
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Authors: | A Gasull Y Zhao |
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Institution: | a Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain b Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P.R. China |
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Abstract: | This paper is concerned with the study of the number of critical periods of perturbed isochronous centers. More concretely, if X0 is a vector field having an isochronous center of period T0 at the point p and X? is an analytic perturbation of X0 such that the point p is a center for X? then, for a suitable parameterization ξ of the periodic orbits surrounding p, their periods can be written as T(ξ,?)=T0+T1(ξ)?+T2(ξ)?2+?. Firstly we give formulas for the first functions Tl(ξ) that can be used for quite general vector fields. Afterwards we apply them to study how many critical periods appear when we perturb the rigid quadratic isochronous center , inside the class of centers of the quadratic systems or of polynomial vector fields of a fixed degree. |
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Keywords: | 34C23 37C10 37C27 |
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