Abstract
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.
Highlights
It is well known that vector fields can be used in order to model a wide range of phenomena and applications
In this paper we consider a PSVF composed by two 3D smooth vector fields
The boundary Σ separating the limit of operation of each smooth vector field is called the switching manifold
Summary
It is well known that vector fields can be used in order to model a wide range of phenomena and applications. The dynamic of the model is given by a Piecewise Smooth Vector Field (PSVF for short). In this paper we consider a PSVF composed by two 3D smooth vector fields. In order to exemplify the richness of behavior around each T-singularity, we prove the existence of PSVFs presenting a center around each T-singularity These centers can be destroyed giving rise to an arbitrary number of limit cycles, hyperbolic or not, surrounding each one of the T-singularities considered. The idea here is perturb the model of Theorem A in order to preserve the n T-singularities and obtain limit cycles. In the plane πxl , the first return φ2Yερ (−y0) is the first coordinate of the point in Σ given by the intersection of the graph of the function z = G(y) =. Stable) manifold it is enough consider the first component of vector fields Xδλ and Yερ (see [1], Proposition 8).
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