Stable Takens' Embedding for linear dynamical systems

Abstract

Takens' Embedding Theorem gives theoretical justification for the use of delay coordinate maps in characterizing and predicting nonlinear dynamical systems. However, in practice imperfections such as system and measurement noise may render these results unusable. In this paper, we consider conditions allowing for a stable version of Takens' Embedding Theorem in the restricted case of linear dynamical systems. Our work is inspired from results from the field of Compressive Sensing, where signals from a low-dimensional signal family residing in a high-dimensional space can be robustly recovered from compressive measurements only if the measurement form a stable embedding of the signal family. In particular, we show that a stable embedding of the attractor of the dynamical system is indeed possible and give sufficient conditions on the number of delays and the observation function for the delay coordinate maps to be stabilized. In addition, we also show that when the attractor is an ellipse, the conditioning of the embedding is lower bounded by a positive constant dependent only on the dynamical system and not within control of the experimentalist. We illustrate our results with an example linear dynamical system converging to an elliptical attractor. Our analysis in this paper will give insights into stable Takens' Embedding of general dynamical systems.