Abstract: Bifurcation of periodic solutions widely exists in nonlinear dynamical systems. In this paper, categories of bifurcations of systems with two state variables with different types of constraints are discussed, where some new types of transition sets are added. Additionally, the bifurcation properties of two-dimensional systems without constraints are compared with the ones with constraints. The results obtained in this paper can be used by engineers for the choice of the structural parameters of the systems.