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 thir d-order ODEs, focusing in particular on the role of the parameters of the m ethod and on the influence of the quality of the time series in terms of no ise, length and sampling frequency. Typical models are investigated, such a s the van der Pol, the linear mechanical, the Duffing and the Rossler equat ions, resulting in a robust and versatile method which is capable of allowi ng interesting applications to experimental cases. The method is then appli ed to the measured time series of a nonlinear mechanical oscillator, a typi cal velocity oscillation of the bursting phenomenon in near-wall turbulence and the averaged annual evolution of rainfall, temperature and streamflow over a hydrological basin.