# Phase Portraits of a Class of Continuous Piecewise Linear Differential Systems

## TL;DR

This paper classifies phase portraits for a specific class of continuous piecewise linear differential systems and proves the existence and uniqueness of limit cycles.

## Contribution

The paper provides a classification of phase portraits for a specific class of continuous piecewise linear differential systems with a|x|+by+c and α|x|+βy+γ structure.

## Key findings

- The phase portraits of the given differential systems are classified in the Poincaré disc.
- Existence and uniqueness of limit cycles are proven for the systems under the condition aβ - bα ≠ 0.

## Abstract

The phase portraits of the planar linear differential systems are very well known. This is not the case for the phase portraits of the planar continuous piecewise linear differential systems. In this paper we classify the phase portraits of the class of planar continuous piecewise linear differential systems of the form \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\dot{x}}= a|x|+by+c,\qquad {\dot{y}}= \alpha |x|+\beta y+\gamma , \end{aligned}$$\end{document}x˙=a|x|+by+c,y˙=α|x|+βy+γ,in the Poincaré disc when \documentclass[12pt]{minimal}
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\begin{document}$$a\beta -b\alpha \ne 0$$\end{document}aβ-bα≠0, and prove the existence and uniqueness of limit cycles. Note that on the straight line \documentclass[12pt]{minimal}
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\begin{document}$$x=0$$\end{document}x=0 these differential systems are only continuous.