Phase space reconstruction of semiconductor laser dynamics using reservoir computing

The geometry of the phase space of a dynamical system contains information about the dynamics of the system. The Takens embedding theorem shows that the full dynamical evolution of the system can be extracted from the structure of the phase space and it can be reconstructed just by measuring one of the variables of the dynamical system. This result has many applications such as recovering lost time series data, testing data encryption security in chaotic synchronization cryptography or data forecasting. This can also be used in control engineering to create a state observer. There are real-world systems that have some variables that can be measured easily but it might be unfeasible to measure the others. In this work we implement reservoir computing techniques to reconstruct and forecast the dynamics of a 3-dimensional dynamical system that describes the evolution of an optically injected semiconductor laser. This system has 3 variables, the amplitude, phase and carrier density but usually only the first one is measured. A reservoir computing state observer is utilized to infer the evolution of unmeasured variables provided that time series are available for measurements of one of the dynamical variables. Later on, an autonomous reservoir computing algorithm is used to predict the evolution of the system dynamics.