On the existence of limit cycles and invariant surfaces of sewing piecewise linear differential systems on $\mathbb{R}^3$

TL;DR

This paper investigates the existence of various invariant structures such as limit cycles, periodic orbits, scrolls, and cylinders in a class of discontinuous piecewise linear differential systems in three-dimensional space.

Contribution

It demonstrates the existence of complex invariant structures, including limit cycles and invariant cylinders, in piecewise linear systems with discontinuities in three dimensions.

Findings

Existence of a unique limit cycle.

Presence of a one-parameter family of periodic orbits.

Identification of scrolls and invariant cylinders foliated by orbits.

Abstract

We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique one-parameter family of periodic orbits, scrolls, invariant cylinders foliated by orbits which can be periodic or no.