On the Trajectory Method for the Reconstruction of Differential Equations from Time Series

This work investigates the trajectory method [1] for thereconstruction of ordinary differential equations (ODEs) from timeseries. The potentials of the method are analyzed for dynamical systemsdescribed by second- and third-order ODEs, focusing in particular on therole of the parameters of the method and on the influence of the qualityof the time series in terms of noise, length and sampling frequency.Typical models are investigated, such as the van der Pol, the linearmechanical, the Duffing and the Rössler equations, resulting in arobust and versatile method which is capable of allowing interestingapplications to experimental cases. The method is then applied to themeasured time series of a nonlinear mechanical oscillator, a typicalvelocity oscillation of the bursting phenomenon in near-wall turbulenceand the averaged annual evolution of rainfall, temperature andstreamflow over a hydrological basin.