Quasi-periodic Bursting in a Kind of Duffing–Van der Pol System with Two Excitation Terms

This paper mainly studies the quasi-periodic bursting of a kind of Duffing–Van der Pol system characterized by two disproportionate periodic frequencies. It aims to further improve the quasi-periodic bursting oscillation mechanism and provide theoretical guidance for practical application.

This paper mainly uses the frequency truncation method and fast-slow analysis method. First, by comparing the Lyapunov exponent spectra under different \({u}_{1}\) $u_1$ , the periodic, quasi-periodic and chaotic states of the system are analyzed. The frequency truncation method is employed to convert the system into a fast-slow system characterized by a single slow variable. Then, the dynamic behaviors of the system are analyzed when the frequency ratios are \(\pi \) $\pi$ and \(\sqrt{3}\) $\sqrt{3}$ . By means of the Lyapunov exponent, spectrum diagram, phase diagram, etc., the dynamic behavior is analyzed, and the quasi-periodic bursting phenomenon is discussed. Finally, by means of the Lyapunov exponential spectrum, the state change of the system from chaos to quasi-periodic and then to periodic is analyzed.

By comparing the Lyapunov exponent spectra under different \({u}_{1}\) $u_1$ , the periodic, quasi-periodic and chaotic states are obtained. Using the Lyapunov exponent, spectrum diagram, phase diagram and time history diagram, the dynamic behavior is determined, and the transition mechanism of quasi-periodic bursting oscillation is discussed. In virtue of the Lyapunov exponent spectrum, the state change of the system from chaos to quasi-periodic and then to periodic is analyzed.

Compared with the bursting oscillation with integer excitation ratio, when the excitation ratio is irrational, the system needs a certain method to transform and then analyze its dynamic behavior. In this paper, the frequency truncation method is used to transform the system, and then the quasi-periodic bursting phenomenon is studied, which will be helpful to the study of the bursting phenomenon of double-excitation coupled systems.