[en] Highlights: • 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. 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.

[en] Highlights: • The exact number of limit cycles of a piecewise linear system with three zones and asymmetry. • The hyperbolicity of limit cycles of the piecewise linear system. • The development and generalization of Coppel’s method in the piecewise linear system. -- Abstract: We study a planar piecewise linear differential system with three zones and asymmetry $\dot{x}=F\left(x\right)-y,\dot{y}=g\left(x\right)$, where $xg\left(x\right)>0$ for $x\ne 0$ and the graph of $F\left(x\right)$ is a U-shaped curve. The system was researched in Llibre et al. (2015) for its limit cycles. However, the number and hyperbolicity of limit cycles were still unclear in some parameter regions. We give the exact number of limit cycles and obtain the hyperbolicity of limit cycles in these parameter regions, where the maximum number is two.