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

David Marín Pérez; M. Saavedra; Jordi Villadelprat Yagüe

doi:10.1017/prm.2021.72

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

David Marín Pérez, M. Saavedra, Jordi Villadelprat Yagüe

Research output: Contribution to journal › Article › Research › peer-review

Abstract

In this paper we consider the unfolding of saddle-node X=1xUa(x,y)(x(xμ−ε)∂x−Va(x)y∂y), parametrized by (ε,a) with ε≈0 and a in an open subset A of Rα, and we study the Dulac time T(s;ε,a) of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative ∂sT(s;ε,a) tends to −∞ as (s,ε)→(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.

Original language	English
Pages (from-to)	104-114
Number of pages	11
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume	153
Issue number	1
DOIs	https://doi.org/10.1017/prm.2021.72
Publication status	Published - 1 Dec 2021

Keywords

asymptotic expansions
Dulac time
Period function
saddle-node unfolding
Asymptotic expansions
Saddle-node unfolding

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https://ddd.uab.cat/record/257112Licence: CC BY-NC-ND

Cite this

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title = "Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres",

abstract = "In this paper we consider the unfolding of saddle-node X=1xUa(x,y)(x(xμ−ε)∂x−Va(x)y∂y), parametrized by (ε,a) with ε≈0 and a in an open subset A of Rα, and we study the Dulac time T(s;ε,a) of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative ∂sT(s;ε,a) tends to −∞ as (s,ε)→(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 = "asymptotic expansions, Dulac time, Period function, saddle-node unfolding, Asymptotic expansions, Saddle-node unfolding",

author = "\{Mar{\'i}n P{\'e}rez\}, David and M. Saavedra and \{Villadelprat Yag{\"u}e\}, Jordi",

note = "Funding Information: This work has been partially funded by the Ministry of Science, Innovation and Universities of Spain through the grants PGC2018-095998-B-I00 and MTM2017-86795-C3-2-P, and by the Agency for Management of University and Research Grants of Catalonia through the grants 2017SGR1725 and 2017SGR1617. Publisher Copyright: Copyright {\textcopyright} The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.",

year = "2021",

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language = "English",

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pages = "104--114",

journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",

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AU - Marín Pérez, David

AU - Saavedra, M.

AU - Villadelprat Yagüe, Jordi

N1 - Funding Information: This work has been partially funded by the Ministry of Science, Innovation and Universities of Spain through the grants PGC2018-095998-B-I00 and MTM2017-86795-C3-2-P, and by the Agency for Management of University and Research Grants of Catalonia through the grants 2017SGR1725 and 2017SGR1617. Publisher Copyright: Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

PY - 2021/12/1

Y1 - 2021/12/1

N2 - In this paper we consider the unfolding of saddle-node X=1xUa(x,y)(x(xμ−ε)∂x−Va(x)y∂y), parametrized by (ε,a) with ε≈0 and a in an open subset A of Rα, and we study the Dulac time T(s;ε,a) of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative ∂sT(s;ε,a) tends to −∞ as (s,ε)→(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.

AB - In this paper we consider the unfolding of saddle-node X=1xUa(x,y)(x(xμ−ε)∂x−Va(x)y∂y), parametrized by (ε,a) with ε≈0 and a in an open subset A of Rα, and we study the Dulac time T(s;ε,a) of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative ∂sT(s;ε,a) tends to −∞ as (s,ε)→(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.

KW - asymptotic expansions

KW - Dulac time

KW - Period function

KW - saddle-node unfolding

KW - Asymptotic expansions

KW - Saddle-node unfolding

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

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Keywords

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Invariantes locales y globales en geometria

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

Abstract

Keywords

Access to Document

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Invariantes locales y globales en geometria

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