Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 98, pp. 1-28.
Title: Structural stability of polynomial second order differential
equations with periodic coefficients
Author: Adolfo W. Guzman (Univ. Federal de Vicosa, Brazil)
Abstract:
This work characterizes the structurally stable second order
differential equations of the form
$x''= \sum_{i=0}^{n}a_{i}(x)(x')^{i}$ where
$a_{i}:\Re \to \Re$ are $C^r$ periodic functions.
These equations have naturally the cylinder $M= S^1\times \Re$
as the phase space and are associated to the vector fields
$X(f) = y \frac{\partial}{\partial x}
+ f(x,y) \frac{\partial}{\partial y}$, where
$f(x,y)=\sum_{i=0}^n a_i(x) y^i \frac{\partial}{\partial y}$.
We apply a compactification to $M$ as well as to $X(f)$ to
study the behavior at infinity. For $n\geq 1$, we define a set
$\Sigma^{n}$ of $X(f)$ that is open and dense and characterizes
the class of structural differential equations as above.
Submitted April 29, 2004. Published August 9, 2004.
Math Subject Classifications: 37C20.
Key Words: Singularity at infinity; compactification; structural stability;
second order differential equation