Structural stability of polynomial second order differential equations with periodic coefficients

Guzman, Adolfo W.

Electronic Journal of Differential Equations, Tome 2004 (2004).

Résumé

Summary: 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