Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity

Samuel Carlos S. Ferreira; Bruno R. Freitas; João Carlos R. Medrado

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arXiv · 2604.25773

Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity

Samuel Carlos S. Ferreira·Bruno R. Freitas·João Carlos R. Medrado

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Abstract

We analyze a three-dimensional discontinuous piecewise linear system $Z=(X,Y)$ whose switching manifold $Σ$ contains visible-visible two-fold intersection lines. Assuming that the matrices $DX$ and $DY$ each have one nonzero real eigenvalue and one pair of complex conjugate eigenvalues, we reduce the system to a canonical form. Under a resonant condition, we use Darboux integrability theory to obtain a first integral common to $X$ and $Y$. Its restriction to $Σ$ defines a hyperbola $Γ$, which parametrizes the crossing points of symmetric periodic orbits. On this curve we construct the half-return maps, derive analytic expansions for the corresponding return times near infinity, and introduce a time-matching function given by their difference. By means of the Weierstrass Preparation Theorem, we prove the existence of a large-amplitude symmetric limit cycle for a suitable subfamily of systems. We then study stability through a saltation-corrected monodromy matrix and reduce the problem to Schur--Cohn inequalities for the two transverse Floquet multipliers.

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Samuel Carlos S. Ferreira, Bruno R. Freitas, João Carlos R. Medrado. 2026-04-28. Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity. https://arxiv.org/abs/2604.25773

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