  Based on our studies done on two-dimensional autonomous systems, forced
non-autonomous systems and time-delayed systems, we propose a unified
methodology - that uses renormalization group theory - for finding out
existence of periodic solutions in a plethora of nonlinear dynamical systems
appearing across disciplines. The technique will be shown to have a non-trivial
ability of classifying the solutions into limit cycles and periodic orbits
surrounding a center. Moreover, the methodology has a definite advantage over
linear stability analysis in analyzing centers.
