On the trajectory method for the reconstruction of differential equations from time series

This work investigates the trajectory method [1] for the reconstruction of ordinary differential equations (ODEs) from time series. The potentials of the method are analyzed for dynamical systems described by second- and third-order ODEs, focusing in particular on the role of the parameters of the method and on the influence of the quality of the time series in terms of noise, length and sampling frequency. Typical models are investigated, such as the van der Pol, the linear mechanical, the Duffing and the Rossler equations, resulting in a robust and versatile method which is capable of allowing interesting applications to experimental cases. The method is then applied to the measured time series of a nonlinear mechanical oscillator, a typical velocity oscillation of the bursting phenomenon in near-wall turbulence and the averaged annual evolution of rainfall, temperature and streamflow over a hydrological basin.