A new bifurcation in non-smooth continuous system inducing semi-limit cycle

In this paper, we presented a study on a non-smooth continuous system with emphasis on a special bifurcation. As the parameter varies, a series of concentric closed orbits appear near the equilibrium point. Moreover, the outermost closed orbit attracts all the trajectories outside. It is called as a semi-limit cycle as the trajectories at only one side of this orbit are attracted. By using the theory of generalized Jacobian matrix, it is revealed that this bifurcation can be featured by a pair of complex conjugate eigenvalues reaching exactly but not crossing the imaginary axis. The bifurcation can somewhat be considered to be a degenerate case of the Hopf bifurcation, in which the eigenvalues cross the imaginary axis totally. This study enriches the knowledge of bifurcation analysis for non-smooth dynamical systems.