Structural stability of polynomial second order differential equations with periodic coefficients

Guzman, Adolfo W.

Electronic journal of differential equations, Tome 2004 (2004)

Résumé

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.

Détail

Classification : 37C20
Keywords: singularity at infinity, compactification, structural stability, second order differential equation