P-分岔是随机分岔现象的重要形式之一，得到研究人员的广泛关注。由于对分岔条件缺乏严格的数学描述，不论在理论和数值分析方面都缺少系统性工作。本项目拟在总结现有成果的基础上，借助确定系统约束分岔分析的奇异性理论和方法，给出P-分岔条件的数学描述，推导P-分岔参数条件的通用表达式，揭示P-分岔的分类形式，以期为后续P-分岔的理论和数值分析奠定基础。然后，通过对随机激励下多稳态van der Pol-Duffing系统的分析，揭示系统结构参数、激励参数对随机P-分岔的影响规律。通过对眼颤系统的随机P-分岔分析，为治病机理和临床治疗方案的制定提供了直接指导。为进一步探索随机P-分岔的工程应用奠定基础。

P-bifurcation is one typical type of bifurcation among stochastic bifurcations, which has attracted wide attention of many researchers. However, due to inadequate rigorous mathematical description of bifurcation condition, there are few theoretical and numerical analysis of this problem. This project proposes a mathematical description of the stochastic P-bifurcation based on existed results. General formula of the explicit parameter conditions is deduced by using the singularity method of constrained bifurcations. Meanwhile, the current project reveals possible classification of the P-bifurcation. Theoretical achivements would serve as the foundation for both theoretical and numerical researches in the future. Two typical systems, i.e., the van der Pol-Duffing system and the nystagmus system are studied to explore the mechanism and engineering application of P-bifurcation. The influence of structural and excitation parameters over stohcastic P-bifurcation are discussed through the analysis of multi-stable van der Pol-Duffin system. The study on nystagmus system is able to provide direct guidance for the clinical treatment plan.