Bifurcation Analysis Of A Neutral Heat Conduction Equation

The heat conduction process,as an important research area of mathematics and physics,is a diffusion phenomenon that exists widely in nature.However,the classical heat conduction equation has some limitations in describing heat propagation in medium.For this reason,some scholars have established the modified heat conduction equation by introducing time delays to avoid these limitations.In this thesis,the main work is to study the bifurcation problem of a neutral heat conduction equation.Firstly,considering the actual heat conduction process,the time-lag effect of heat flux and internal energy with the rise in temperature is described by two different delays respectively.Thus,a neutral heat conduction equation with double delays is derived.Secondly,the root distribution of the characteristic equation is analyzed using a geometric method.With the help of the eigenvalue problem of Laplacian operator,the corresponding characteristic equation is transformed into a series of equivalent transcendental equations.Then the geometric method is utilized to determine the range of purely imaginary roots and the expression of time delays.As a result,the crossing curves are obtained,which describing the critical delays and stability switch.The implicit function theorem is utilized to derive explicit expression of transverse direction on crossing curves.Further,the parameter range for deciding stability of the equilibrium is given based on the distribution of the crossing curve.Finally,Hopf bifurcation of neutral heat conduction equation is analyzed.The transversal condition is verified regarding the neutral delay as bifurcation parameter,then combining with the corresponding pure imaginary roots on the crossing curves,it is shown that the equation undergoes Hopf bifurcation at the critical parameter.The normal form of neutral heat conduction equation is calculated,and then the key coefficients are obtained which explain bifurcation properties.In addition,we also consider the neutral functional differential equation associated with the equation,and then demonstrate the influence of diffusion term on the Hopf bifurcation of the equation by proving the equality of the two normal forms.At last,appropriate parameters are selected for numerical simulation to illustrate the theoretical results of this thesis.