# A Bayesian nonparametric approach to the approximation of the global stable manifold

## TL;DR

This paper introduces a Bayesian nonparametric method using MCMC to reconstruct the global stable manifold from noisy time-series data, applicable to noninvertible maps, demonstrated on polynomial maps with synthetic data.

## Contribution

It presents the SM-GSBR model, a novel Bayesian approach for estimating the global stable manifold and underlying dynamics from observed data, including noninvertible maps.

## Key findings

- Successfully reconstructs stable manifolds from synthetic data.
- Works for noninvertible maps without modifications.
- Provides a new tool for analyzing noisy dynamical systems.

## Abstract

We propose a Bayesian nonparametric model based on Markov Chain Monte Carlo (MCMC) methods for unveiling the structure of the invariant global stable manifold from observed time-series data. The underlying unknown dynamical process is possibly contaminated by additive noise. We introduce the Stable Manifold Geometric Stick Breaking Reconstruction (SM-GSBR) model with which we reconstruct the unknown dynamic equations and in parallel 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.