Reconstruction of the parameter spaces of dynamical systems

Parameter variations in the equations of motion of dynamical systems are identified by time series analysis. The information contained in time series data is transformed and compressed to feature vectors. The space of feature vectors is an embedding for the unobserved parameters of the system. We show that the smooth variation of d system parameters can lead to paths of feature vectors on smooth d-dimensional manifolds in feature space, provided the latter is high-dimensional enough. The number of varying parameters and the nature of their variation can thus be identified. The method is illustrated using numerically generated data and experimental data from electromotors. Complications arising from bifurcations in deterministic dynamical systems are shown to disappear for slightly noisy systems.