Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations

Some precise descriptions of the structure and behavior of bounded solutions for a class of nonautonomous second-order differential equations are obtained. The nonautonomous differential equations here are in the form x″=f(θth,x,x′), where the driving system θ is defined in a compact base space H and the vector field f(h,x,p) is strictly increasing in state variable x on some interval (a,b). The stability of the structure of bounded solutions with respect to parameter perturbations in f is also addressed.