Z-source converter can have a high voltage transmission ratio, reduce the losses of switching devices, and improve the efficiency of the system, etc., because of embedding the Z-source network into the system, which makes it find wide applications in DC conversion, inverters, etc. Nonlinear dynamics of the peak-current-mode controlled synchronous switching Z-source converter is studied for the first time so far as we know. The discrete iterated mapping model under continuous current mode is established, while the effects of the reference current on the stability of the system are analyzed by using the trajectories of eigenvalues, and the steady state operation parameter domain is schemed. Period-doubling bifurcation, border-collision bifurcation, tangent bifurcation and intermittent chaos are found in this converter based on the bifurcation diagram and the Lyapunov exponent diagram, and the evolvement and mechanism of the border-collision bifurcation and chaos are analyzed. Finally, the circuit simulation and the experimental results show that the theoretical analysis is correct. Results obtained indicate that with the increase of the reference current, the peak-current-mode controlled synchronous switching Z-source converter goes from period 1 into period 2 and period 4 through the period-doubling bifurcation, and moves into the intermittent chaos due to the border-collision bifurcation. Then the system exhibits a period-3 behaviour because of the tangent bifurcation. Finally, the converter moves into chaos due to the border-collision bifurcation again.