Dynamical Behavior Of Several Kinds Of Stochastic Fast-slow Systems With Bifurcation Points

Fast-slow systems have found broad applications in various domains such as finance,ecology,chemistry,and medicine in recent years.Stochastic fast-slow systems have garnered significant attention due to the growing abundance of research findings on deterministic fastslow systems.Stochastic perturbation exists objectively in the real world,so the research on stochastic fast-slow systems is of great practical significance.In this paper,three models of stochastic fast-slow systems with bifurcation points are studied.One is a predator-prey model with a fold point and a transcritical bifurcation point,one is a typical model of stochastic fast-slow systems with a transcritical bifurcation point accompanied by bifurcation delay,the other is a typical neuronal model with fold/homoclinic bifurcations.For the first two models,by using the sample paths approach,we estimate the probability that the paths of the perturbed solutions remain in some region or that the paths of the perturbed solutions escape from some region by precisely controlling the whole paths of the process,in order to investigate the effect of additive noise with different intensities on the dynamic behavior near the bifurcation points.For the third model,we explored the effect of uniformly bounded real noise on the system from the perspective of random dynamical systems.The paper is structured into six chapters.The first two chapters are introduction and preliminary knowledge.In Chapter 3,we study a predator-prey model with Holling-II type functional response function,which is represented by two-dimensional fast-slow differential equations with a fold point and a transcritical bifurcation point.In the neighborhoods of two bifurcation points,we investigate the effect of small additive noise varying with slow variable on the dynamics of fast variables by the sample-paths approach.In this paper,by giving accurate probability estimates for the behavior of individual paths of the stochastic solutions,we show that the sufficiently small but non-exponentially small additive noise destroys the bifurcation delay phenomenon near the transcritical bifurcation point.Meanwhile,we show that,with high probability,the critical transition phenomenon near the fold point remains unaffected as long as the additive noise is sufficiently small,while the locations where the critical transitions of the paths occur are altered.In Chapter 4,on the basis of the study in Chapter 3,we continue to investigate the effect of additive noise with reasonable large intensity on the dynamical behavior near the transcritical bifurcation point with bifurcation delay.We also provide estimations of the locations where the stochastic solutions perturbed by additive noise with different intensities experience critical transitions with high probability,and summarize the results of the effect of additive noise with different intensities on the dynamics near the transcritical bifurcation point with bifurcation delay.In addition,during the analysis,we demonstrate that for nonautonomous fast-slow systems,the effect of small additive noise on trajectories with large shifts in the fast variables is negligible by the sample-paths approach.In Chapter 5,we investigate the effect of uniformly bounded real noise on the discharge and oscillation behavior due to fold/homoclinic bifurcations,using the invariant manifold theory of random dynamical systems to explain a mechanism of chaotic oscillation generated by uniformly bounded random external forces and present numerical simulations to support it.In Chapter 6,we summarize the main work and results of this paper,and point out the problems that can be further studied.