Homogeneous bifurcations, nonequilibrium

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

Here X is the nonequilibrium constraint. If the system under consideration is a homogeneous chemical system, then Zk is specified by the rates of chemical reactions. For an inhomogeneous system, Zk may contain partial derivatives to account for diffusion and other transport processes. It is remarkable that, whatever the complexity of Zk, the loss of stability of a solution of (19.2.4) at a particular value of A, and bifurcation of new solutions at this point are similar to those of (19.2.1). As in the case of (19.2.1), the symmetries of (19.2.4) are related to the multiplicity of solutions. For example, in an isotropic system, the equations should be invariant under the inversion r —r. In this case, if Xk[r,t) is a solution then Xk -r,t) will also be a solution if Xk(r,t) Xk —r,t) then there are two distinct solutions which are mirror images of each other. 

When a nonlinear system ewolwes under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [l,2]. As far as we are concerned with the stochastic approach, the rrLa te.n. equation, has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system s attractors, turns out to display marked analogies with the theory of normal forms. 

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