The double-exponential asymptotic sliding mode control is applied to the single-phase three-level inverter system, which exhibits a multitude of operating modes and intricate nonlinear dynamic behavior. The operational principles of the system across various modes are thoroughly analyzed, and a discrete model of the system is established utilizing the flash mapping method. The impact of system parameters on its nonlinear behavior is investigated through bifurcation diagrams and folding diagrams, leading to the identification of a two-dimensional stable operation domain for both control parameters and main circuit parameters. The stability of the system under double-exponential sliding mode control is examined using the fast-varying stability theorem, with comparisons made against bifurcation and folding diagrams for validation, as well as against other control strategies. Finally, the nonlinear behavior of the system is corroborated through time-domain waveform diagrams and their corresponding spectra under varying control parameters. The study reveals that the three-level inverter topology demonstrates more complex nonlinear dynamics; furthermore, it establishes that double-exponential asymptotic rate sliding mode control offers an expanded parameter stable domain. Specifically, the stable operational range for control parameters has been extended from 0.15~0.95 to 0.05~1.65, while shifting the unstable starting point from 1.3 (under improved exponential asymptotic rate sliding mode control) to 1.65. These research findings provide theoretical support for parameter design in implementing double-exponential asymptotic rate sliding mode control within three-level inverters.