# American Institute of Mathematical Sciences

September  2019, 24(9): 4851-4861. doi: 10.3934/dcdsb.2019034

## Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields

 1 Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Av. Bandeirantes 3900, CEP 14040-901, Ribeirão Preto, SP, Brazil 2 Universidade Federal de Goiás, IME, CEP 74001-970, Caixa Postal 131, Goiânia, Goiás, Brazil

* Corresponding author: Tiago de Carvalho

Received  March 2018 Revised  July 2018 Published  February 2019

Fund Project: The first author is partially supported by the CAPES grant number 1576689 (from the program PNPD) and also is grateful to the FAPESP/Brazil grants numbers 2013/34541-0 and 2017/00883-0, the CNPq-Brazil grant number 443302/2014-6 and the CAPES grant number 88881.030454/2013-01 (from the program CSF-PVE).

The T-singularity (invisible two-fold singularity) is one of the most intriguing objects in the study of 3D piecewise smooth vector fields. The occurrence of just one T-singularity already arouses the curiosity of experts in the area due to the wealth of behaviors that may arise in its neighborhood. In this work we show the birth of an arbitrary number, including infinite, of such singularities. Moreover, we are able to show the existence of an arbitrary number of limit cycles, hyperbolic or not, surrounding each one of these singularities.

Citation: Tiago de Carvalho, Bruno Freitas. Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4851-4861. doi: 10.3934/dcdsb.2019034
##### References:

show all references

##### References:
Return map of $Z = (X,Y)$
$\Sigma-$centers at the T-singularities
Topological cylinders
Behavior of the orbits outside and inside of the planes
Behavior at Theorems B and C
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