Hopf Bifurcations On A Class Of Partial Neutral Functional Differential Equations

Delay differential equations are widely used in various different fields such as ecology,medicine,and control.Among them,hysteresis exists in the highest order derivative in some equations,which is called the “neutral functional differential equation”.This paper,by using the central manifold theorem and normal formed theory,tends to transform a class of neutral functional differential equations into abstract ordinary differential equations,solve the expression of the corresponding first Lyapunov coefficient,and then analyze its Hopf bifurcation.And finally,it selects a specific parameter value for numerical simulation.Firstly,the study solves the characteristic equations of the neutral differential equations mentioned and analyzes the solution of the characteristic equations.At the same time,it solves the values of the parameters when Hopf bifurcation occurs,and verifies the transversal condition.Thus,it checks the existence of Hopf bifurcation phenomenon of selected parameters in the relevant system.In the continuing part,Riesz representation theorem is used to transform the neutral partial differential equation into an abstract ordinary differential equation,and decompose it in the finite-dimensional subspace and the infinite dimensional subspace of BC space.At the same time,the central manifold theorem and normal formed theory is applied.Then,the explicit expression of the first Lyapunov coefficient corresponding to the system is offered,which can be directly used to judge the bifurcation of the equation.In the last process,a numerical simulation is made.The correct value of the parameters in the equation is selected,and the Matlab is used for numerical simulation to verify the correctness of the conclusion.