Isochronicity of bi-centers for symmetric quartic differential systems

Wilker Fernandes; Viviane Pardini Valério; Patricia Tempesta

doi:10.3934/dcdsb.2021215

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2022, Volume 27, Issue 7: 3991-4006. Doi: 10.3934/dcdsb.2021215

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Isochronicity of bi-centers for symmetric quartic differential systems

Departamento de Matemática e Estatística, Universidade Federal de São João del-Rei, São João del-Rei, 36307-352, Brazil

^*Corresponding author: Wilker Fernandes (wilker@ufsj.edu.br)
^*Corresponding author: Wilker Fernandes (wilker@ufsj.edu.br)

Received: February 2021

Revised: June 2021

Published: July 2022

The authors thank the reviewer for careful reading and valuable suggestions which helped to improve the manuscript. W. Fernandes and P. Tempesta are partially supported by FAPESP Grand number 2019/07316-0.

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Abstract

In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.

Addendum: "W. Fernandes and P. Tempesta are partially supported by FAPESP Grand number 2019/07316-0." is added under Fund Project. We apologize for any inconvenience this may cause.

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Mathematics Subject Classification: Primary: 34C05, 34C14; Secondary: 37C10, 37C15.

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Figure 1. Global phase portraits of systems (14), (16), (18), (19), (20), (21) and (23), respectively

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