Structural stability of polynomial second order differential equations with periodic coefficients

• Main Author: Adolfo W. Guzman
• Format: Article
• Language: English
• Published: Texas State University 2004-08-01
• Series: Electronic Journal of Differential Equations
• Subjects: Singularity at infinity; compactification; structural stability; second order differential equation.
• Online Access: http://ejde.math.txstate.edu/Volumes/2004/98/abstr.html

This work characterizes the structurally stable second order differential equations of the form $x''= sum_{i=0}^{n}a_{i}(x)(x')^{i}$ where $a_{i}:Re o Re$ are $C^r$ periodic functions. These equations have naturally the cylinder $M= S^1imes Re$ 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 $ngeq 1$, we define a set $Sigma^{n}$ of $X(f)$ that is open and dense and characterizes the class of structural differential equations as above.