We propose a Bayesian nonparametric model based on Markov Chain Monte Carlo methods for unveiling the structure of the invariant global stable manifold from observed time-series data. The underlying unknown dynamical process could have been contaminated by additive noise. We introduce the Stable Manifold Geometric Stick Breaking Reconstruction model with which we reconstruct the unknown dynamic equations, while at the same time, we estimate the global structure of the perturbed stable manifold. Our method works for noninvertible maps without modifications. The stable manifold estimation procedure is demonstrated specifically in the case of polynomial maps. Simulations based on synthetic time-series are presented.