Bifurcation And Turing Patterns In Two Kinds Of Autocatalytic Reaction Diffusion System

Since 1970s, people began to pay attention to much resemble behaviour between dynamical instability and equilibrium phase changing on steady state of system when they are far away thermodynamics steady state. For example, they emerge tremendous fluctuation near the critical point, that is to say, connecting distance is divergent; Their kinetics behaviour are critical slowly at the critical point; When phases are changed. systems may give rise to critical transition to pattern formation and the transition to such stationary symmetry of temporal-spatial structures, that is to say, symmetry of temporal-spatial structures before phase changing is better than that after phase changing. There exists critical exponential regularity in order parameter of systems near critical points. These phenomenon enlighten people to give kinetics classify on critical behaviour of system by bifurcation theory in mathmatics.Generally, most dynamics in chemical reaction are nonlinear. However, when systems approach thermodynamics steady state, their kinetics behaviour are studied approximatly by linear non-equilibrium thermodynamics. When systems are far away thermodynamics steady state, sometimes, nonlinear effacts become principal factor of dynamics behaviour, the nonlinear behaviour couple by diffusive linear behaviour. The coupling initially may give rise to the spontaneous appearance of the order and chaotic pattern. Since 1970s, as dissipative theory is accepted generally, people begin to studied phenomenen of chemical osillation. To study the chemical reaction diffusion systems, it is important to master the kinetics generality of systems near critical points . These are principal content of pattern formation which is an important branch of nonlinear science.In this paper, we investigate two kinds of autocatalytic chemical reaction. One kind of reaction-diffusion system based on the cubic autocatalator, with the reaction taking place within a closed two-dimensional region, is considered. The linear stability of the steady state (a, b) = (1/μ,μ), where a and b are the dimensionless concentrations of reactant A and autocatalyst B and μ, is a parameter representing the initial concentration of the precursor p, is discussed firstly by the linearized theory. It is shown that a necessory condition for the existence of Turing instabilities is that the parameter D < 3 -22 where D = λb/λa(λa, λbare the diffusion coefficients of chemical species A and B respectively). Secondly, we investigate the possibility of occurance of two dimensional Turing patterns consisting of rhombic arrays of rectangles and hexagonal, respectively, by performing the appropriate weakly nonlinear stability analyses of the homogeneous solution and amplitude functions to our model 1. We are primarily concerned with evaluating those Landau constants and determiningthe stability of the equlibrium points of the amplitude equation. The other kind of the spatiotemporal structures coupled via the diffusive interchange of the autocatalyst. are discussed. We find that the kinetics have a unique homogeneous steady state and discuss stability of the steady state by the linearized theory. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, solutions in the uncoupled system and coupled system is 0 < δ <(n-1)2/n-1, with 0 <β > 1.