Structural stability of polynomial second order differential equations with periodic coefficients

Abstract

This work characterizes the structurally stable second order differential equations of the form x'' = ni=0 αi(x)(x')i where ai : ℜ → ℜ are C^r periodic functions. These equations have naturally the cylander M = S1 x ℜ as the phase space and are associated to the vector fields X(ƒ) = y ∂/∂x + ƒ(x, y) ∂/∂y, where ƒ(x, y) = ni=0αi(x)yi ∂/∂y. We apply a compactification to M as well as to X(ƒ) to study the behavior at infinity. For n ≥ 1, we define a set ∑n of X(ƒ) that is open and dense and characterizes the class of structural differential equations as above.