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We have probed the condition of periodic oscillation in a class of twovariable nonlinear dynamical open systems modeled withLienard-Levinson-Smith(LLS) equation which can be a limit cycle, center or avery slowly decaying center type oscillation. Using a variety of examples ofopen systems like Glycolytic oscillator, Lotka-Volterra(L-V) model, ageneralised van der Pol oscillator and a time delayed nonlinear feedbackoscillation as a non-autonomous system, each of which contains a family ofperiodic orbits, we have solved LLS systems in terms of a multi-scaleperturbation theory using Krylov-Bogoliubov(K-B) method and it is utilised tocharacterise the size and shape of the limit cycle and center as well as theapproach to their steady state dynamics. We have shown the condition when theaverage scaled radius of a center undergoes a power law decay with exponent1/2.

When an oscillating center in an open system undergoes power law decay