Stochastic bifurcation and safety basin study of nonlinear vibration systems in Li-doped graphene nanoplates with time delays

From the perspective of nonlinear vibration systems in graphene nanoplates, chaos, safe basins, and stochastic bifurcations are three crucial indicators for assessing stability. This study analyzes the Li-doped graphene nanoplates' chaos in the Smale sense and its feasibility based on stochastic Melnikov theory, and verifies the control of noise, dual time delays, and time-delay feedback intensity. By combining stochastic dynamic theory with the simple cell mapping method, we conduct an in-depth analysis of the system's dynamic response and the evolution of stochastic safe basins, revealing the erosion mechanisms of safe basins under different parameters. Finally, topological data analysis is introduced to analyze stochastic bifurcations in the nonlinear vibration system of Li-doped graphene nanoplates, capturing more comprehensive stochastic P-bifurcation characteristics under diverse parameters. This provides theoretical support and strategic recommendations for chaos control and vibration stability optimization in Li-doped graphene nanoplate systems.