In this paper, a new chaotic system is proposed. Through the analysis of the bifurcation diagram, Lyapunov exponent and phase diagram of the system, it is proved that the system is a chaotic system under certain parameters. When the parameters are changed, the system exhibits the alternation behavior of period--doubling period--chaos--period--doubling period--chaos, which indicates that the system has more complex dynamic characteristics and sensitivity to the initial value. In addition, the Multisim circuit simulation software is applied to verify the physical realizability of the system. Moreover, based on the new chaotic system, a mismatch projection synchronization controller is designed, which theoretically confirms that the system is synchronized with the Chen chaotic system under the action of the controller, the effectiveness of the synchronous controller is validated by Matlab simulation. Finally, the digital encrypted transmission is realized under the action of the synchronous controller.