Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

[摘要] 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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-05

[关键词] Filippov vector fields;Monodromic tangential singularities;Center-focus problem;Lyapunov coefficients;Limit cycles