  We have probed the condition of periodic oscillation in a class of two
variable nonlinear dynamical open systems modeled with
Lienard-Levinson-Smith(LLS) equation which can be a limit cycle, center or a
very slowly decaying center type oscillation. Using a variety of examples of
open systems like Glycolytic oscillator, Lotka-Volterra(L-V) model, a
generalised van der Pol oscillator and a time delayed nonlinear feedback
oscillation as a non-autonomous system, each of which contains a family of
periodic orbits, we have solved LLS systems in terms of a multi-scale
perturbation theory using Krylov-Bogoliubov(K-B) method and it is utilised to
characterise the size and shape of the limit cycle and center as well as the
approach to their steady state dynamics. We have shown the condition when the
average scaled radius of a center undergoes a power law decay with exponent
1/2.
