STRUCTURAL STABILITY OF POLYNOMIAL SECOND ORDER DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

Author(s)

Guzman, Adolfo W.

• Document type: Journal article
• Source: Electronic Journal of Differential Equations; v. N/A, p. 28-pg., 2004-01-01.

Abstract

This work characterizes the structurally stable second order differential equations of the form x '' = i over(n) a(i)(x)(x')(i) where a(i) : R -> R are C-r periodic functions. These equations have naturally the cylinder M = S-1 x R as the phase space and are associated to the vector fields X(f) = y partial derivative/partial derivative x + f(x, y) partial derivative/partial derivative y, where f(x, y) = i over(n) a(i)(x) y over(i) partial derivative/partial derivative y. We apply a compactification to M as well as to X(f) to study the behavior at infinity. For n >= 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. (AU)