Constructing differential equations using only a scalar time-series about continuous time chaotic dynamics

We propose a simple method of constructing a system of differential equations of chaotic behavior based on the regression only from a scalar observable time-series data. The estimated system enables us to reconstruct invariant sets and statistical properties as well as to infer short time-series. Our successful modeling relies on the introduction of a set of Gaussian radial basis functions to capture local structure. The proposed method is used to construct a system of ordinary differential equations whose orbit reconstructs a time-series of a variable of the well-known Lorenz system as a simple but typical example. A system for a macroscopic fluid variable is also constructed.