The behaviors of a system that alternates between the R¨ossler oscillator and Chua’s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems. The behaviors of a system that alternates between the ¨ossler oscillator and Chua’s circuit is investigated to explore the influence of the switches on the dynamical evolution. Switch-related to the state variables are introduced, upon which a typical switching dynamical model is established. sets of the subsystems are derived via analysis of the related equilibrium points, which divide the parameters into several regions corresponding to different types of attractors. the dynamics behave typically in period orbits with the variation of the parameters. focus / cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement. period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches. Morerther, period-decreasing sequences have been obtained, which can be explained by the variation of the eigenvalues ​​associated with the equilibrium points of the subsystems.