Bifurcation analysis of a self-excited oscillator with quasi-zero-stiffness properties

This paper presents a theoretical study of both conventional and non-smooth bifurcations in a friction-driven self-excited oscillator. A distinctive feature of the model is its Van der Pol damping and nonlinear restoring force, which includes a cubic term and is designed to function as an equivalent quasi-zero-stiffness device. Firstly, theoretical conditions for the occurrence of various sliding bifurcations are established. Secondly, when equilibrium points are located in some region, their stability conditions and the existence criteria for both supercritical and subcritical Hopf bifurcations are derived. It is well known that there is little theoretical research on bifurcations in non-smooth systems and these research findings bridge the theoretical gaps in the analysis of non-smooth dynamical systems. The theoretical analysis is validated through numerical simulations, which confirms the presence of sliding bifurcations (crossing–sliding bifurcation, grazing–sliding bifurcation, switching–sliding bifurcation), as well as supercritical and subcritical Hopf bifurcations. Furthermore, we observe that a periodic orbit emerging from a supercritical Hopf bifurcation, upon further parameter variation, may begin to intersect the switching boundary, leading to various sliding bifurcations, which is different from the phenomenon described in previous references. A novel coexistence of subcritical Hopf bifurcation and crossing–sliding bifurcation, or switching–sliding bifurcation is found, which is rarely reported in prior studies. In a word, this study provides a refined theoretical framework for analyzing both sliding bifurcations and Hopf bifurcations in non-smooth dynamical systems.