Finitude of Limit Cycles of Linear Piecewise ODEs in the Cylinder

TL;DR

This paper establishes an upper bound on the number of limit cycles for linear piecewise ODEs in a cylindrical domain, extending Hilbert's 16th problem to this specific setting.

Contribution

It provides a novel upper bound on limit cycles based on the number of regions and coefficient degree for piecewise linear ODEs in a cylinder.

Findings

Derived an explicit upper bound for limit cycles

Extended classical Hilbert's 16th problem to piecewise linear systems

Applicable to systems with trigonometric coefficients in time

Abstract

Let $x'=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the piecewise equation and the degree of the coefficients, that is, an analogue of Hilbert's 16th problem in this context.