# Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

## TL;DR

This paper investigates the coexistence of crossing and sliding limit cycles in planar piecewise linear vector fields with two zones, providing conditions for their existence, stability, and the maximum number of such cycles.

## Contribution

It introduces a canonical form for these systems, demonstrates the existence of multiple crossing limit cycles, and characterizes sliding cycles and their stability.

## Key findings

- Existence of three crossing limit cycles under perturbation.
- Detection of a sliding cycle after second order perturbation.
- Only one additional crossing limit cycle can appear with a sliding cycle.

## Abstract

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one additional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.