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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.06203 |
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| _version_ | 1866908524461686784 |
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| author | Braun, Francisco da Cruz, Leonardo Torregrosa, Joan |
| author_facet | Braun, Francisco da Cruz, Leonardo Torregrosa, Joan |
| contents | In this paper, we introduce an alternative method for applying averaging theory of orders $1$ and $2$ in the plane. This is done by combining Taylor expansions of the displacement map with the integral form of the Poincaré--Poyntriagin--Melnikov function. It is known that, to obtain results of order $2$ with averaging theory, the first-order averaging function should be identically zero. However, when working with Taylor expansions of the $i$th-order averaging function, we usually cannot guarantee it is identically zero. We prove that the vanishing of certain coefficients of the Taylor series of the first-order averaging function ensures it is identically zero. We present our reasoning in several concrete examples: a quadratic Lotka--Volterra system, a quadratic Hamiltonian system, the entire family of quadratic isochronous differential systems, and a cubic system. For the latter, we also show that a previous analysis contained in the literature is not correct. In none of the examples is it necessary to precisely calculate the averaging functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06203 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local and Global Analysis of the Displacement Map Some Near Integrable Systems Braun, Francisco da Cruz, Leonardo Torregrosa, Joan Dynamical Systems In this paper, we introduce an alternative method for applying averaging theory of orders $1$ and $2$ in the plane. This is done by combining Taylor expansions of the displacement map with the integral form of the Poincaré--Poyntriagin--Melnikov function. It is known that, to obtain results of order $2$ with averaging theory, the first-order averaging function should be identically zero. However, when working with Taylor expansions of the $i$th-order averaging function, we usually cannot guarantee it is identically zero. We prove that the vanishing of certain coefficients of the Taylor series of the first-order averaging function ensures it is identically zero. We present our reasoning in several concrete examples: a quadratic Lotka--Volterra system, a quadratic Hamiltonian system, the entire family of quadratic isochronous differential systems, and a cubic system. For the latter, we also show that a previous analysis contained in the literature is not correct. In none of the examples is it necessary to precisely calculate the averaging functions. |
| title | Local and Global Analysis of the Displacement Map Some Near Integrable Systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.06203 |