Advances in Differential Equations

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

Marta Calanchi and Bernhard Ruf

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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


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