The location of limit cycles and the associated mechanism in the state switched nonlinear system

By introducing switching law associated with the values of the state variables, a switched mathematical model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point corresponding to the limit cycle of the switched system is calculated by the shooting method. To investigate switching behavior of this system, the equilibrium points and their bifurcations of the subsystems are derived. An interesting switching behavior, i.e., the so-called 4T-focus/focus/focus periodic switching is explored in detail to present the mechanism of the movement. With the increase of the parameter, the turning points on the switching surface may be attracted by different attractors of the subsystem, causing the turning points decrease from four to two. Then the system forms other types of periodic solutions. Furthermore, period-decreasing and period-adding sequences have been obtained, which can be explained by the changes of the duration time in the subsystems.