Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

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.

References in corpus (3)

• New lower bound for the Hilbert number in piecewise quadratic differential systems
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• A note on Vishik's normal form

Cited by in corpus (7)

• Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region
• Sliding mode on tangential sets of Filippov systems
• A succinct characterization of period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness
• On the non-existence of isochronous tangential centers in Filippov vector fields
• Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds
• On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set
• On partial maps derived from flows