Research On The Dynamics And Regularization Of Several Types Of Filippov Systems

A Filippov system is a discontinuous dynamical system composed of two or more smooth vector fields that are separated by discontinuity boundaries.It plays an important role in the study of ecology,such as controlling the species values,controlling the economics costs,controlling the diseases spreading,and so on.Therefore,the study of the Filippov system is of great significance.This paper studies the dynamics of several types of Filippov systems: competitive system with linear relationship,competitive and symbiotic system with Holling type II or III response function,and Filippov systems with two different thresholds.This paper can be divided into six parts.The first part is the introduction,which presents the research background and significance,research status and the main results of this paper.The second part is the preliminary,which gives the concepts and theory that will be used in this work.The third part studies the dynamics of several types of Filippov systems with competitive relationship.We will prove that a persistence bifurcation or a non-smooth fold bifurcation will occur in such systems.After that,we show the numerical simulation of these bifurcations and analyze their ecological significance.The forth part studies the dynamics of symbiotic Filippov system with Holling type II or type III response function.The results shows that a persistence bifurcation will occur.All the numerical simulation results are given and the ecological meaning are derived.The fifth part studies the regularization dynamics of two special types of Filippov systems with two thresholds.According to our analysis,after regularization,the persistence bifurcation will disappear,while the non-smooth fold bifurcation becomes a saddle-nodelike bifurcation.The last part is the conclusion,which summarizes the main results of this paper and present the potential future research direction.