Time-reversibility and integrability of $ p:-q $ resonant vector fields

Jaume Giné; Valery G. Romanovski; Joan Torregrosa; Jaume Giné; Valery G. Romanovski; Joan Torregrosa

doi:10.3934/math.2024005

AIMS Mathematics

2024, Volume 9, Issue 1: 73-88. doi: 10.3934/math.2024005

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Time-reversibility and integrability of $ p:-q $ resonant vector fields

1.
Departament de Matemàtica, Universitat de Lleida, Av. Jaume Ⅱ, 69, Lleida 25001, Catalonia, Spain
2.
Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, SI-2000 Maribor, Slovenia
3.
Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia
4.
Faculty of Natural Science and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
5.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra 08193, Barcelona, Catalonia, Spain
6.
Centre de Recerca Matemàtica, Campus de Bellaterra, Bellaterra 08193, Barcelona, Catalonia, Spain

Received: 26 August 2023 Revised: 05 October 2023 Accepted: 12 October 2023 Published: 27 November 2023
MSC : 34C14, 37C79

We study the local analytical integrability in a neighborhood of $ p:-q $ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $ \varphi(x, y) = (y^{p/q}, x^{q/p}). $ Some generalizations of the theory developed by Sibirsky for the $ 1:-1 $ resonant case to the $ p:-q $ resonant case are presented.
- planar systems of ODEs,
- time-reversibility,
- integrability,
- resonant singularity
Citation: Jaume Giné, Valery G. Romanovski, Joan Torregrosa. Time-reversibility and integrability of $ p:-q $ resonant vector fields[J]. AIMS Mathematics, 2024, 9(1): 73-88. doi: 10.3934/math.2024005

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Abstract

We study the local analytical integrability in a neighborhood of $ p:-q $ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $ \varphi(x, y) = (y^{p/q}, x^{q/p}). $ Some generalizations of the theory developed by Sibirsky for the $ 1:-1 $ resonant case to the $ p:-q $ resonant case are presented.

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