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The interplay among the topological bifurcation diagram, integrability and geometry for the family QSH(D)

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Abstract

Consider the class QSH of all non-degenerate planar quadratic differential systems possessing an invariant algebraic hyperbola. In this paper we consider a specific two-parametric family of systems in QSH possessing three pairs of infinite singularities, namely family QSH(D). For this family we have generically a presence of one couple of parallel invariant straight lines and one invariant algebraic hyperbola. Our goal is to explore the relationship among the topological bifurcation diagram, geometry of configurations of invariant algebraic curves and its type of integrability. For this study we construct the topological bifurcation diagram of configurations and phase portraits of the family QSH(D) altogether. We also study the integrability, we obtain all the distinct configurations of invariant algebraic curves, and we get all the topologically distinct phase portraits in the Poincaré disc. More precisely, we prove that the family under consideration is Liouvillian integrable, there are 53 distinct configurations of invariant algebraic curves, and there exist 18 topologically distinct phase portraits.

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Acknowledgements

The first author is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Grant Number 166449/2020-2. The second author is partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Grant “Projeto Temático” 2019/21181-0 and CNPq Grant Number 304766/2019-4. This paper was developed during the Postdoctoral Program of the first and third authors at ICMC-USP.

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Authors and Affiliations

  1. Departamento de Matemática, ICMC-Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, São Carlos, São Paulo, 13566-590, Brazil

    Marcos C. Mota, Regilene Oliveira & Ana Maria Travaglini

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  1. Marcos C. Mota
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  2. Regilene Oliveira
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  3. Ana Maria Travaglini
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All authors contributed equally in all stages of the manuscript elaboration, including on the revision.

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Correspondence to Marcos C. Mota.

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Mota, M.C., Oliveira, R. & Travaglini, A.M. The interplay among the topological bifurcation diagram, integrability and geometry for the family QSH(D). Geom Dedicata 217, 95 (2023). https://doi.org/10.1007/s10711-023-00827-6

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