Polynomial differential equations with piecwise linear coefficients

TL;DR

This paper investigates the maximum multiplicity of periodic solutions in cubic and quartic non-autonomous differential equations with piecewise linear coefficients, comparing these with polynomial coefficients of the same degree.

Contribution

It demonstrates that for many classes, the multiplicity of periodic solutions remains the same whether coefficients are polynomial of degree n or piecewise linear with n segments.

Findings

Maximum multiplicity is the same for polynomial and piecewise linear coefficients in many classes.

The study extends understanding of periodic solutions in non-autonomous differential equations.

Results provide insights into the structure of solutions for piecewise linear systems.

Abstract

Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the mutiplicity is the same when the coefficients are polynomial functions of degree n or piecewise linear functions with n segments.