Dynamics for a Rayleigh–Liénard–Duffing oscillator in wing rock

During the past few decades there has been significant scholarly interest in the investigation of piecewise differential systems, primarily because these systems facilitate the modeling of various complex phenomena. To effectively characterize the dynamics of differential systems, it is essential to analyze their local and global dynamical behaviors, with particular emphasis on their limit cycles. In this paper we study the dynamics for a classical type of Rayleigh–Liénard–Duffing oscillators, that is utilized to simulate the wing rock motion with a single degree of freedom. This hybrid oscillator can exhibit rich bifurcations phenomena, including pitchfork bifurcation, Hopf bifurcation and heteroclinic bifurcation. By applying the qualitative theory of differential equations and the averaging theory, we prove that a lower bound for the maximum number of small-amplitude limit cycles (isolate periodic oscillations) for the corresponding piecewise differential system is 2. Furthermore, we classify for this oscillator its phase portraits in the Poincaré disc.