Mathematical Modeling and Analysis of Lissajous Curves Based on Non-harmonic and Nonstationary Dynamics

Lissajous is a complex pattern composed of two or more simple harmonic vibrations, which still have important applications in many fields. The frequency ratio and phase difference of the two signals can be measured by using the Lissajous pattern. The traditional representation of the Lissajous curves is only based on two simple harmonic vibrations, which is too idealistic a model and considers only one single frequency component. However, signals in the real world are all made up of different frequency components and share non-stationary characteristics, which are not considered by the traditional Lissajous model. Therefore, this study aims to provide an analysis of Lissajous curves based on non-harmonic and nonstationary dynamics, enhancing the utility of the Lissajous Curves model in real-world scenarios. In this study, we first investigate the influence of frequency component parameters on the shape of the Lissajous graph. Then, to assess the effects of white noise on the Lissajous curves, Lissajous curves with various noise levels were also computed. Finally, we studied the non-stationarity of Lissajous curves based on two chirp signals with different frequency-changing rates. The results show that the overall characteristic of the Lissajous curve remains unchanged for noisy and non-stationary signals with the proportional frequency change rate. Our results validated the feasibility and robustness of the Lissajous Curves model for non-harmonic and nonstationary dynamics, we expected this study could contribute to the broader understanding of Lissajous curves, bridging the gap between traditional linear Lissajous approaches and the complexities introduced by nonsinusoidal and nonstationary dynamics.