Choice Of Educational Innovation And Its Reaction-diffusion Based On Dynamic System

With the increasing emphasis on education innovation of the whole society，the academiccommunity springs up a great deal of academic achievements concerning with educationinnovation and its reaction-diffusion problem recently．However，most of scholars’ researchesnow are specific to the conceptions，ideas and studies of technological products on education，which means that the researches of dynamic change rule belonging to the educationinnovation are still in the initial stage．The education innovation would appear the effects ofspread，reaction and diffusion with the change of environment，so we can establish thecorresponding reaction-diffusion equation model to study the dynamic changes byconsidering the environment change，the interaction and others factors．By qualitative analysisof the model，the educational innovation promoters can understand the evolution of marketstructure better and take appropriate countermeasures．In "the education innovation modelbased on dynamic system"，the author describes the reaction-diffusion process of educationalinnovation by the differential equation model，and has done a preliminary analysis of themodel．Some theoretical properties of the model need further research，and some further workalso should be done about the model．This paper studies the stability of educational innovation diffusion system，the existenceof limit cycles and Hopf bifurcation problems by applying stability theory，limit cycles andbifurcation theory in the ordinary differential equations and Routh-Hurwitz criterion，Liapunov stability theorem， Dulac function， Dulac criteria and other theoreticalmethods．Some work is done as follows：(1) This paper further analyses competition，complementary and alternative basic typesof education innovation reaction-diffusion models，and the global asymptotic stability ofequilibrium points and existence of limit cycles in each model is discussed．By constructing afunctionV，we get a sufficient condition for the existence and uniqueness of positiveequilibrium to satisfy global uniform asymptotic stability is obtained．And there is no limitcycle in the first quadrant to the three models by the construction of eligible Dulac functionand Dulac criterion．(2) The original competition education innovation diffusion model has been improved toestablish a competition education innovation model based on the impact of media and do adetailed analysis the stability of the new system equilibrium and the existence of limit cycles．According with the difference of taking the decision variable values，the paperdiscusses the trends of the track in4conditions to analyze the impact of the media toreaction-diffusion under different conditions．These works provide a theoretical basis ofnumerical simulations to determine the depth and time of reaction-diffusion．And theeducational innovation promoters can understand the evolution of market structure better andtake appropriate countermeasures．(3) More factors should be considered to establish a more realistic model，so we establisha new competitive-type delay differential equation model of educational innovation．Byqualitative analysis of the new system，a condition of the existence of Hopf bifurcation isobtained．...