Automatic identification of dynamical system excited by time-dependent factor without prior information

A moving object is often excited by time-dependent factors. If the equation of motion and time-dependent factor can be distilled from data, a foundation for further analysis can be established. This paper proposes a method to discover the equation of motion and time-dependent factors without prior information and customized preparation. We first construct statistics to determine whether the system is subject to time-dependent factors. For the motivated system, we introduce the time-dependent Fourier series to approximate the excitation. Finally, we achieve the identification by simple steps. The new method does not require the prior knowledge of the system. Since the new method determines whether the time-dependent factor is present and which state variable is motivated, it avoids the confusion caused by inappropriately adding time-dependent terms to the identification. As the designed Fourier series can approximate the unknown model accurately, the new method does not require developing customized identification for each problem. We apply the new method to discover dynamical systems from data collected from Lienard equation and SD oscillator driven by various time-dependent factors. The new method can accurately discover dynamical systems from the collected short-time data and predict many dynamical behaviors, including the long-time evolution of the system, the multi-stable dynamics, and qualitative changes in the dynamics as the time-dependent factor varies. The results show that the new identification method can discover nonautonomous dynamical systems effectively.