Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

TL;DR

This paper extends the concept of Lyapunov coefficients to monodromic tangential singularities in piecewise analytic vector fields, providing theoretical results, recursive formulas, and computational tools for analyzing stability and bifurcations.

Contribution

It proves the analyticity of the first-return map for these singularities, establishes that the first non-vanishing Lyapunov coefficient's index is even, and offers a recursive formula and Mathematica implementation.

Findings

First-return map is analytic near monodromic tangential singularities.

The index of the first non-zero Lyapunov coefficient is always even.

Provides a recursive formula and Mathematica code for computing Lyapunov coefficients.

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.