Abstract:

Since the second part of the 16th Hilbert problem becomes so difficult Charles Pugh proposed the following related problem.

Problem 1. Let $a_0,a_1,ldots,a_n: mathbb{S}^1 to mathbb{R}$ be continuous $2pi-$periodic functions and consider the differential equation
begin{equation}
frac{dr}{dtheta}= a_0(theta)+a_1(theta)r+ ldots + a_n(theta) r^n,
end{equation}

on the cylinder $(theta,r) in mathbb{S}^1 times mathbb{R}$. Then the problem is to know the maximum number of isolated periodic solutions (i.e.limit cycles) of the above differential equation in function of $n$.

The solution of the Hilbert number for Problem 1 is known. In this talk we solve the Hilbert number for the next problem.

Problem 2. Let $a,a_0,a_1,ldots,a_n: mathbb{S}^1 to mathbb{R}$ be continuous $2pi-$periodic functions and consider the differential equation
begin{equation}
frac{dr}{dtheta}= frac{a(theta)}{a_0(theta)+a_1(theta)r+ ldots + a_n(theta) r^n},
end{equation}
on the region of the cylinder $(theta,r) in mathbb{S}^1 times mathbb{R}$ where the denominator of the above equation does not vanish. Then the problem is to know the number of limit cycles of the above differential equation in function of $n$.

The talk is based in the paper:

J. Llibre and A. Makhlouf, On the Hilbert number of a class of differential equation, J. Applied Analysis and Computation 5 (2015), 141-145.