From data to dynamical systems

This article is part of a series commemorating the 50th anniversary of Ed Lorenz's seminal paper 'Deterministic nonperiodic flow'. The system he studied has become the object of extensive mathematical investigations and serves as a paradigm of a low-dimensional chaotic dynamical system. Nonetheless, Lorenz maintained his focus on the relationship between dynamical models and empirical data. This paper reviews some of the difficulties encountered in fitting chaotic models to data and then pursues fits of models with stable periodic orbits to data in the presence of noise. Starting with Lorenz's model in a regime with a stable periodic orbit, it investigates how well properties of the deterministic system can be recovered from trajectories of stochastic perturbations. A numerical study suggests, surprisingly, that the stability properties of the periodic orbit cannot be fully recovered from a finite length trajectory as the magnitude of the stochastic perturbation converges to zero. A heuristic framework for the analysis is developed and shown empirically to yield good estimates for the return map of the periodic orbit and its eigenvalues as the length of a stochastic trajectory increases.