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arXiv · 2010.00497

Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

Douglas D. Novaes·Leandro A. Silva
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Abstract

In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map defined around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields $Z=(Z^+ ,Z^-)$. First, we prove that the first-return map, defined in a neighborhood of a monodromic tangential singularity, is analytic, which allows the definition of the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a general recursive formula together with a Mathematica algorithm for computing the Lyapunov coefficients is obtained. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analyzed.

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Douglas D. Novaes, Leandro A. Silva. 2020-10-01. Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields. https://doi.org/10.1016/j.jde.2021.08.008

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