Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity
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| Format: | Preprint |
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2026
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| author | Ferreira, Samuel Carlos S. Freitas, Bruno R. Medrado, João Carlos R. |
| author_facet | Ferreira, Samuel Carlos S. Freitas, Bruno R. Medrado, João Carlos R. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25773 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity Ferreira, Samuel Carlos S. Freitas, Bruno R. Medrado, João Carlos R. Dynamical Systems 34A36, 34C25, 34C14, 34C20 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. |
| title | Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity |
| topic | Dynamical Systems 34A36, 34C25, 34C14, 34C20 |
| url | https://arxiv.org/abs/2604.25773 |