Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres
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...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2023 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:257112 |
| Online Access: | https://ddd.uab.cat/record/257112 https://dx.doi.org/urn:doi:10.1017/prm.2021.72 |
| Access Level: | Open access |
| Keyword: | Period function Saddle-node unfolding Dulac time Asymptotic expansions |
| Summary: | 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. |
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