Hilbert number for a family of piecewise nonautonomous equations

TL;DR

This paper investigates a family of piecewise nonautonomous equations, solving key problems related to their centers, limit cycles, and providing an upper bound for the number of limit cycles, contributing to the understanding of their complex dynamics.

Contribution

It characterizes conditions for centers, proves finiteness of limit cycles, and establishes a uniform upper bound for their number in the given family.

Findings

Characterization of center conditions

Finite number of limit cycles for each equation

A uniform upper bound for the number of limit cycles

Abstract

For family $x'=(a_0+a_1\cos t+a_2 \sin t)|x|+b_0+b_1 \cos t+b_2 \sin t$, we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincar\'e center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert's 16th problem).