## Advances in Differential Equations

- Adv. Differential Equations
- Volume 7, Number 2 (2002), 197-216.

### On the number of closed solutions for polynomial ODE's and a special case of Hilbert's 16th problem

Marta Calanchi and Bernhard Ruf

#### Abstract

In this paper we prove that the equation ${du \over dt} + \sum_{i=0}^n a_i(t)u^i = f(t),$ $t \in [0,1]$, $u(0) = u(1)$, has for every continuous $f$ at most $n$ solutions provided that $n$ is odd, and the continuous coefficients $a_i$ satisfy $|a_n(t)| \ge \alpha > 0$ and $|a_i(t)| \le \beta,$ $i = 1,\dots,n-1$, with $\beta > 0$ sufficiently small. Furthermore, we show that this result implies that for a restricted subclass of polynomial vector fields of order $n$ in $\mathbb R^2$ the maximal number of limit cycles is $n$. This constitutes a special case of Hilbert's 16th problem.

#### Article information

**Source**

Adv. Differential Equations Volume 7, Number 2 (2002), 197-216.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651851

**Mathematical Reviews number (MathSciNet)**

MR1869561

**Zentralblatt MATH identifier**

1087.34012

**Subjects**

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)

Secondary: 34C25: Periodic solutions

#### Citation

Calanchi, Marta; Ruf, Bernhard. On the number of closed solutions for polynomial ODE's and a special case of Hilbert's 16th problem. Adv. Differential Equations 7 (2002), no. 2, 197--216. https://projecteuclid.org/euclid.ade/1356651851