Electron. J. Diff. Eqns., Vol. 2004(2004), No. 98, pp. 1-28.

Structural stability of polynomial second order differential equations with periodic coefficients

Adolfo W. Guzman

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.

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Adolfo W. Guzman
Departamento de Matematica
Universidade Federal de Vicosa
Campus Universitario CEP 36571-000. Vicosa - MG. Brasil
email: guzman@ime.usp.br   guzman@ufv.br

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