Noise-induced tipping (N-tipping) emerges when random fluctuations prompt transitions from one (meta)stable state to another, potentially as a rare event. In this talk, we delineate new techniques for determining Most Probable Escapes Paths (MPEPs) in stochastic differential equations over periodic boundaries. We utilize a dynamical system approach to unravel MPEPs for the intermediate noise regime. We discuss the framework for computing the MPEPs by first looking at intersections of stable and unstable manifolds of invariant sets of a Hamiltonian system derived from the Euler-Lagrange equations of the Freidlin-Wentzell (FW) functional. The Maslov index helps identify which critical points of the FW functional are local minimizers and assists in explaining the effects of the interaction of noise and the deterministic flow. The Onsager-Machlup functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out the MPEP. We will illustrate our approach and compare our theoretical prediction with Monte Carlo simulations in the Inverted Van der Pol system and a carbon cycle model.