The Hilbert's number for some classes of differential equations [ Back ]

Date:
16.02.15   
Times:
15:30 to 16:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Jaume Llibre
University:
UAB

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 $2\pi-$periodic functions and consider the differential equation
\begin{equation}
\frac{dr}{d\theta}= 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 $2\pi-$periodic functions and consider the differential equation
\begin{equation}\
\frac{dr}{d\theta}= \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.