A system, which alternates between autonomous and non-autonomous circuit systems observing the time periodic switched rules, is investigated in order to explore its complicated dynamical behaviors. By analyzing the equilibrium point, limiting cycles, and the stability of the autonomous subsystems, as well as deriving the Lyapunov exponents of the switching systems in theory and numerical calculation, we have studied the variation of periodic oscillation behaviors of the compound systems with different stable solutions to the two subsystems. By using the bifurcation diagram of the switched systems and their corresponding largest Lyapunov exponent diagrams, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations and chaotic oscillations with different parameters in the switched systems. Furthermore, dynamical evolutions of the switching system to chaos by period-doubling bifurcations, saddle-node bifurcations and torus bifurcations are observed.