Analysis of dissipative structures in the Gierer-Mainhardt model

Abstract

To analyze a manifold that approximates an attractor of a dissipative system we propose a Gaussian variational principle.For a Gierer-Mainhardt system we obtain closed-form expressions for the approximating stationary solution and study its stability.The evolution of distributed dissipative systems can be described by the following nonlinear equations:Ot "'where i = l, N, N eN, and {gti(r, t)} arc fields that describe the state of the system and are regarded as realvalued functions of the time t and the spatial coordinates r; (F~} are the components of the nonlinear evolution operator F, which depends on the fields {tg~} and the external parameters A I ..... A~t, M e N. Because of the dissipation, as t -~ oo the system tends to a ccrtain state in the space T of functions {~i}. In what follows we shall denote this state by f~" and call it an attractor of the dissipative system. The problem is to study the geometry of the attractor of the corresponding system and the behavior of the system in a neighborhood of it. For isolated dissipative systems numerical analysis can bc used to determine a priori the general form of an attractor, that is the form of the fields {gti(r, t)} that are approximate solutions of Eqs. (I) as t--~ co. Using such a method we can construct a certain manifold f2 in the space W that characterizes these solutions and is an approximating manifold of the attractor ~'. Such a manifold may have finite dimension p, and in that case it can bc described as follows: f~ = {v,(r) = (I),(r, u, ..... u,)},where ( The main idea of the approximation in the present work is to reduce the system of equations (1) to evolution equations that contain only the variables (u~, ..., up), and whose time dependence characterizes the motion of the system along the attractor.To analyze dissipative systems we consider the well-known variational principle of Gauss, which is connected with the following functional, called the minimal constraint functional:We regard the fields {~, ~u, } as pairwise independent variables, and the conditionleads to Eq. (1). To obtain the evolution equations in the variables (u~ ..... up), we substitute the functions %(r, t) = dO~ ( r, u~ (t) ..... up(t) ) into the functional (3) and minimize it with respect to ti~, zi 2 .... , zip. It should be remarked that this procedure requires justification. Indeed, these variational principles come from classical mechanics, where reducing the dimension of a system requires the existence of additional ideal (holonomic) constraints on the variables of the system. These constraints cause additional forces to arise and act on the