Stability Of Steady State And Turing Patterns In The Three Kinds Of Chemical Reaction Diffusion Systems

Generally, most dynamics in chemical reaction are nonlinear. However, when systems approach thermodynamics steady state, their kinetics behaviour are studied approximately by linear non-equilibrium thermodynamics, when systems are far away from thermodynamics steady state, sometimes, nonlinear effects become principal factor of dynamics behaviour. The coupling initially may give rise to the spontaneous appearance of the ordinal and chaotic pattern.This thesis is made up of three chapters to investigate the stability of steady state and the properties of pattern formations solutions for three kinds of chemical reaction diffusion systems, respectively.The autocatalytic reaction is a classical kind of reaction which is used to study pattern formations. One kind of reaction-diffusion system is cubic autocatalytic reaction, model 1 is such kind of autocatalytic one, i.e.Here a and b are the dimensionless concentration of A and autolyst B and μ is a parameter representing the initial concentration of the precursor P. A is commen diffusional coefficient of reactants of A and B. The boundary condition satisfy Dirichlet boundary condition, i.e.With the usage of linearized theory and multiple scale methods, Predholm alternative theory, and the analysis the local stability of non-uniform state, we get a necessary condition for the existence of Turing instability. We describe the properties of the bifurcation solutions carefully.Model 2 is a kind of classical activator-inhibitor modelHere , . a and b represent the concentrations of activator and inhibitor respectively. are the diffusive coefficients of the reactant A and B. μis bifurcation parameter. Its boundary condition isModel 2 is more sophisticated than model 1 at some aspects. We study the stability of the unique homogeneous stead state. The pattern solution can be found only when the parameter is large enough. By performing the weakly nonlinear theory, we get the amplitude equation of square pattern in two dimensional region. We calculate its Landau constant and analysize the stability of the four kinds of the equlibrium points of the amplitude equation.Model 3 also is of activator-inhibitor reaction diffusive system, i.e.Its boundary condition satisfy the below,When the parameters m,n,p,q satisfy , we determine homogeneous stead state and discuss the property of Turing bifurcation solution via nonlinear theory also, we obtain the amplitude equation and analyse the amplitude equilibrium state.