This paper constructs a new four-dimensional dissipative chaotic system with coexisting attractors. The system's dynamic behaviors are analyzed through the numerical simulation of the phase portrait, Lyapunov exponent graph, bifurcation diagram , etc. Numerical simulation results indicate that the system has abundant dynamic characteristics, and the polarity of the chaotic signal can be flexibly changed by introducing offset boosting control. By studying the spectral entropy (SE) complexity of the two different initial conditions, an initial condition with higher complexity is selected as the initial state when the system is synchronized. Finally, a backstepping controller is designed to implement the synchronization of the chaotic system. Multisim is used to simulate the system's circuit and field programmable gate arrays (FPGA) are used to implement the system on actual hardware.