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Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

Part of: Qualitative theory

Published online by Cambridge University Press:  01 December 2021

D. Marín ,
M. Saavedra  and
J. Villadelprat
D. Marín
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193, Cerdanyola del Vallès, Barcelona, Spain (davidmp@mat.uab.cat) Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193, Cerdanyola del Vallès, Barcelona, Spain
M. Saavedra
Affiliation:
Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Barrio Universitario, Concepción, Casilla 160-C, Chile (mariansa@udec.cl)
J. Villadelprat
Affiliation:
Departament d'Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain (jordi.villadelprat@urv.cat)
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Abstract

In this paper we consider the unfolding of saddle-node

\[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \]
parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.

Keywords

Period functionsaddle-node unfoldingDulac timeasymptotic expansions

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C20: Transformation and reduction of equations and systems, normal forms 34C23: Bifurcation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 1 , February 2023 , pp. 104 - 114
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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