Abstract: The respiratory control system precisely regulates the generation and maintenance of the breathing rhythm to adapt to the body's requirements in different physiological and pathological conditions. Pathological dyspnea can occur when oxygen levels in the arterial blood drop below the standard threshold. The pre-Bötzinger complex serves as a crucial site for the generation of the respiratory rhythm, contains expiratory neurons, inspiratory neurons, and post-inspiratory neurons. These neurons work in collaboration to regulate the respiratory rhythm through a variety of mechanisms, including neural modulation and chemical modulation. The breathing process involves interaction of multiple time scales, including the firing activity of neuronal networks, expansion and contraction of alveoli, and transport of oxygen in the blood. Understanding the mechanisms underlying the regulation of respiratory rhythm has long been a central focus of research. Under certain hypoxic conditions, the closed-loop respiratory control system exhibits self-recovery capabilities, as demonstrated in this study which investigates the system's self-recovery capability following sustained hypoxia interference under different initial conditions. The study demonstrates that the closed-loop respiratory control system can recover completely to normoxic level, partially recover to mild hypoxic level or completely fail to recover. Based on the simplicity of square wave currents, the study simplifies the closed-loop respiratory model and applies dynamic analysis methods to investigate the dynamic mechanisms of the system's different responses. Bifurcation analysis results indicate that changes of bifurcation structures at different stages during sustained hypoxia are key factors affecting the recovery capability. Specifically, the closed-loop respiratory control system can fully recover to normoxic level only when the bifurcation structure remain unchanged before and after sustained hypoxia, This research enchances our understanding of how sustained hypoxic perturbs respiratory rhythms, investigates the relationship between external factors and physiological conditions related to rhythm recovery, as well as the intrinsic dynamical mechanisms governing rhythm recovery.