Invariant manifolds for piecewise smooth differential systems in $\mathbb{R}^{3}$

TL;DR

This paper studies how small perturbations in three-dimensional piecewise smooth systems can create invariant manifolds filled with periodic orbits, using averaging theory to analyze their emergence from cylindrical structures.

Contribution

It demonstrates the existence of invariant manifolds filled with periodic orbits in 3D piecewise smooth systems via averaging theory and geometric transformations.

Findings

Invariant manifolds emerge from cylinders in D systems.

Small perturbations induce periodic orbit-filled manifolds.

Averaging theory effectively analyzes these manifolds.

Abstract

In this paper some piecewise smooth perturbations of a three-dimensional differential system are considered. The existence of invariant manifolds filled by periodic orbits is obtained after suitable small perturbations of the original differential system. These manifolds emerge from a continuum of cylinders of $\mathbb{R}^3$ which does exist for the piecewise smooth differential systems after a rotation of some planar algebraic polynomial curves. The main tool used in order to obtain the results is the averaging theory for piecewise smooth differential systems.