Global dynamics of hybrid van der Pol–Rayleigh oscillators

We study the dynamics of a hybrid van der Pol–Rayleigh oscillator that has been used to model the self-sustained walking behaviors of humans or bipedal robots in literature. By using qualitative and bifurcation analysis, we discover new dynamics that are different from van der Pol and Rayleigh oscillators. This hybrid oscillator can exhibit two limit cycles and rich bifurcations phenomena. The global bifurcation diagram is shown in the parameter space, and the corresponding global phase portraits of this hybrid oscillator are sketched in the phase space. Furthermore, the locations or amplitudes of the periodic oscillations (limit cycles) are also characterized. •The hybrid Van der Pol–Rayleigh oscillator exhibits new dynamics.•There exist two limit cycles and a heteroclinic cycle.•Complete bifurcation diagram and the corresponding phase portraits are sketched.•The locations of these limit cycles are characterized.•These results are helpful to understand the motion of human or bipedal robot.