Dynamics of the delay-coupled bubble system combined with the stochastic term

The purpose of this paper is to study the mechanisms of the delay-coupled bubble system combined with the stochastic term from a mathematical point of view. Firstly, the dynamic behaviors of the bubble system were studied with the normal form theory, center manifold and stochastic averaging method. The stochastic bifurcation behaviors (P-bifurcation and D-bifurcation) of the bubble system were analyzed. Besides, the relationship between the dynamic behaviors of the interaction between the time delay term and the stochastic term and the stability of the bubble was numerically simulated. Finally, we compared the joint probability density curves of the bubble system and It$\hat{o}$ equation. The numerical simulation results are consistent, which indicated that it is feasible to study the stochastic bifurcation behaviors of the bubble system by using the It$\hat{o}$ stochastic differential equation.