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

  • * Corresponding author: Tiago de Carvalho

    * Corresponding author: Tiago de Carvalho 

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)

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  • 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.

    Mathematics Subject Classification: Primary 34A36, 34A26, 37G15, 37G35.


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  • Figure 1.  Return map of $ Z = (X,Y) $

    Figure 3.  $ \Sigma- $centers at the T-singularities

    Figure 2.  Topological cylinders

    Figure 4.  Behavior of the orbits outside and inside of the planes

    Figure 5.  Behavior at Theorems B and C

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