The contact and wear between seal faces were caused by the angular wobble of a dry gas seal system and then will influence the stability, therefore, both the theoretical calculation and experiment were applied to study the stability. According to the theory of nonlinear vibration in consideration of the heat dissipation effect on the gas film thickness and stiffness, an angular wobble model of the gas filmsealing ring system was set up. In specific cases, the gas film stiffness and damping were expressed as multiple variables involving the wobby angle  in order to obtain the differential equation of nonlinear vibration. Without external incentives, the system bifurcation was found through the Floquet index, and the influence of spiral angle of dry gas seal on the system stability was analysed. The hopf bifurcation will occur when the spiral angle reaches 75°48′32″ in consideration of  thermal dissipation deformation and the spiral angle 75°42′55″ without consideration of thermal dissipation deformation. The spiral angle stable ranges were acquired for the cases of under thermal dissipation or nonthermal dissipation. The spiral angle range to stabilize the system under heat dissipation and deformation is larger than that of no heat dissipation, which indicates that changes will happen for the bifurcation when considering the heat dissipation and deformation. Finally, the spiral angle 75°50′and 75° were chosen to process the specimen and an experiment was designed to observe the effect of angular wobble on the sealing faces and leakage. The dry gas seal with 75°50′ spiral angle presents a bad serious abrasion face and continuall leakage. The experimental results were compared with those of the theoretical calculation to show that the spiral angle has certain interference on the sealing stability under thermal dissipation deformation, which provides a theoretic guidance for optimizing the dynamic design of dry gas seals.