Global classification of oscillatory dynamics in symmetric zero-divergence 3D piecewise-linear Filippov systems with a visible--visible two-fold
Abstract
A global classification of the asymptotic oscillatory dynamics is established for a symmetric zero-divergence class of three-dimensional piecewise-linear Filippov systems, up to a set of initial conditions of zero Lebesgue measure. The affine fields are related by an involution, and the switching plane contains a visible--visible two-fold. In canonical coordinates, the eigenvalues are μ± i and −2μ, while H measures the focal-line inclination. For every μ>0, a simple-period crossing cycle exists if and only if H∈ℐμ, and is unique, symmetric, hyperbolic, and orbitally asymptotically stable. Its half-period parametrizes ℐμ and determines the crossing points, period, and Floquet multipliers. Global dissipation excludes crossing cycles with any higher number of crossings and makes the classified cycle the ω-limit set of every crossing-only trajectory. If attractive sliding has no interior pseudo-equilibria, sliding is transient unless the trajectory reaches the two-fold. The initial conditions leading to the two-fold lie in a countable union of analytic surfaces and curves. Under sufficiently small perturbations in the Whitney C1 topology, the cycle persists as the unique simple-period crossing limit cycle and remains hyperbolic and orbitally asymptotically stable.
Related
- Piecewise Symmetric Tensors
- Dynamics of piecewise linear Li\'enard systems near infinity
- A simple classification of the special matchings in the symmetric group
- Classification of compatible left-symmetric conformal algebras on solvable Lie conformal algebras of rank two
- Classification of Symmetric Hadamard Matrices Up to Order 32
BibTeX
@misc{ferreira2026global,
title = {{Global classification of oscillatory dynamics in symmetric zero-divergence 3D piecewise-linear Filippov systems with a visible--visible two-fold}},
author = {Samuel Carlos S. Ferreira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33034},
eprint = {2609.33034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33034},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Global classification of oscillatory dynamics in symmetric zero-divergence 3D piecewise-linear Filippov systems with a visible--visible two-fold
AU - Samuel Carlos S. Ferreira
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.33034
UR - https://arxiv.org/abs/2609.33034
AB - A global classification of the asymptotic oscillatory dynamics is established for a symmetric zero-divergence class of three-dimensional piecewise-linear Filippov systems, up to a set of initial conditions of zero Lebesgue measure. The affine fields are related by an involution, and the switching plane contains a visible--visible two-fold. In canonical coordinates, the eigenvalues are μ± i and −2μ, while H measures the focal-line inclination. For every μ>0, a simple-period crossing cycle exists if and only if H∈ℐ_μ, and is unique, symmetric, hyperbolic, and orbitally asymptotically stable. Its half-period parametrizes ℐ_μ and determines the crossing points, period, and Floquet multipliers. Global dissipation excludes crossing cycles with any higher number of crossings and makes the classified cycle the ω-limit set of every crossing-only trajectory. If attractive sliding has no interior pseudo-equilibria, sliding is transient unless the trajectory reaches the two-fold. The initial conditions leading to the two-fold lie in a countable union of analytic surfaces and curves. Under sufficiently small perturbations in the Whitney C^1 topology, the cycle persists as the unique simple-period crossing limit cycle and remains hyperbolic and orbitally asymptotically stable.
ER -