Property Analysis And Application Of Discrete Competitive System Based On Fractal Theory

The development of natural processes is often accompanied by mult-participants competition,such as the trend of economic indicators,the price competition among multiple individuals in supply chains,the evolution law of ecological population and so on.Discrete coupled competition models provide effective theoretical tools for reasonably expounding the dynamic characteristics and laws of these competition phenomena.Based on the fractal M-J set theory in nonlinear science,this thesis takes several classical discrete coupled competition models as research objects,studies connectivity,boundedness and control optimization of the initial distribution fractal sets of competitive competition participators,and the preliminary application of relevant theoretical results in supply chain price competition model is realized.The specific research contents are as follows:Firstly,the classical Parrondo paradox in game theory is extended to the study of competition model.Aiming at a two-dimensional discrete competition model of two participants,considering the change laws of fractal Julia set connectivity under the alternating change of competition relationship.By analyzing the convergence characteristics of the critical point of the system,the paradox of "unconnected + unconnected = connected" in Julia set of the competition model under switching rules is proposed,and the relationship between the stable coexistence of participants and their competitive behavior is revealed.Secondly,in order to verify the different connectivity characteristics of Julia sets under the above switching rules,an improved escape time algorithm is proposed,which uses the 0-1normalization technique to map the connectivity of Julia sets into numerical indexes.Good connectivity of the initial distribution Julia sets of the competition model means the stability and anti-interference of the coexistence of competition relationship.Therefore,this study provides an effective algorithm tool for quantitative analysis of the stable coexistence of participants.Thirdly,for a high-dimensional logistic competition model,the boundedness of Julia sets is studied when the system gain is chain coupled.In addition,classical feedback control method and escape equation method are used to realize boundary control and optimization of the system Julia sets,and the variation law of the upper bound of the controlled Julia sets is given.Finally,relevant theoretical results are preliminarily applied to a kind of dual channel price competition model.By analyzing the relationship between consumer preferences(online and offline)and the connectivity of Julia sets,and using the gradient control optimization method,the boundary control of Julia sets is realized.The results show that excessive online preference leads to price instability and poor robustness in all channels,and reasonable active price intervention can make the channel price fluctuate steadily,thus ensuring the stability of the market.To sum up,based on the fractal M-J set theory,the fractal analysis and optimization of several types of discrete coupled competitive systems in this thesis can not only enrich the knowledge framework of discrete competitive model dynamics at the theoretical level,but also provide a new idea of cross research between fractal theory and management science at the application level.