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

TL;DR

This paper studies symmetric limit cycles in 3D piecewise linear systems with specific singularities, using integrability and return map analysis to prove existence and stability of large-amplitude cycles.

Contribution

It introduces a canonical form reduction, uses Darboux integrability, and applies the Weierstrass Preparation Theorem to establish the existence and stability of symmetric limit cycles.

Findings

Existence of large-amplitude symmetric limit cycles proven.

Analytic expansions for return times near infinity derived.

Stability characterized via Floquet multipliers and Schur--Cohn inequalities.

Abstract

We analyze a three-dimensional discontinuous piecewise linear system \(Z=(X,Y)\) whose switching manifold \(\Sigma\) 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 \(\Sigma\) defines a hyperbola \(\Gamma\), 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.