Hopf And Turing Bifurcations In Fluctuational Models Of Self-Organization

The uctuation models of self-organization are proposed with uc-tuation Hamiltonians taken into account. Conditions of the Hopf and Turing bifurcations are examined, special attention is paid for the analysis of limit-cycle and focus solutions. The principal possibility of the formation of ordered structures in the uctuation models of self-organization are discussed for open systems in a stationary states.


Introduction
The problem of self-organization and related problem of the formation of spatial and temporal structures, i.e. the Turing and Hopf bifurcations, seems to be important up to now (see [1][2][3]).The development of the general theory of self-organization has been stimulated by the latest advance connecting with the universal behavior of the systems having di erent physical natures near the continuous phase transition point and critical (bifurcation) points.The modern theory of self-organization (synergetics) is based on such powerful and well-developed methods as those of nonlinear mechanics, kinetic models and physics of phase transition.It is obvious that the phase-transition methods using fundamental ideas of scaling invariance 4], renormalization group 5], collective variables 6] are preferable because various systems (not only physical but chemical, biological, sociological, etc.) exhibit an anomalous high susceptibility to external in uences.
The main idea of our approach to study the development of the space, time and spatio-temporal structures in open systems is based on the assumption that many processes in chemically (biochemically) reacting systems can be treated as those isomorphous to the second-order phase transitions.It means that uctuational e ects must be taken into consideration near the critical (bifurcation) points.Thus, the corresponding uctuation models of the processes of self-organization have to be proposed on the background of the phase transition theory.20 A.V.Chalyi, D.A.Khiznyak, Ya.V.Tsehmister

E ective uctuation Hamiltonian for chemically reacting systems
For a system with chemical reactions the main governing mode (in terms of synergetics 2]) or the order parameter (in terms of physics of phase transitions 4-7]) is the degree of completeness (coordinate) de ned by the formula d = d n i = i ; ( where d n i is the variation of the number of moles, i are the stoichiometric coe cients.The quantity conjugate to of (in the thermodynamic sense) is the reaction a nity A = @G @ T;P; represented by a linear combination of the chemical potentials i .
Near the boundary of stability the chemical reaction susceptibility = ?@ @A T;P;f g ; ?1 = X i;j i j (@ i =@n i ) P;T;n k 6 =nj (3) goes to in nity.This principal circumstance allows to use the basic assumptions of the phase transition theory, namely to expand the uctuation part of the Gibbs potential into a functional Taylor series It is natural to choose this uctuational part in the generalized well-known Landau-Ginzburg form.For the case of two chemical reactions with order parameters x = 1 ?h 1 i; y = 2 ?h 2 i (5) the e ective uctuational Hamiltonian has the following form + C (0) xxxx x 4 + C (0) yyyy y 4 + 4C (0) xxxy x 3 y + 4C (0) xyyy xy 3   + 6C (0) xxyy x 2 y 2 + C (2)  xx ( rx) 2 + C (2)  yy ( ry) 2 + 2C (2)  xy ( rx ry) + ; Here C (0) and C (2) are the zeroth and second spatial moments of the direct correlation function @ @ P;T = @A @ P;T; ; Thus, the value of ?1 , i.e. the zeroth moments C (0) , speci es the coefcients in front of the squares of order parameters in the expansion (6) for the uctuation part of the Gibbs potential.

Kinetic equations and theorems on bifurcation
According to (6), the equations of motion have the form of the known kinetic equations of the Landau-Ginzburg time-dependent theory _ x(r; t) = ?qx G fl x(r; t) ; _ y(r; t) = ?qy G fl y(r; t) ; (8) or _ x = (x; y) + D 1 r 2 x; _ y = (x; y) + D 2 r 2 y; where q i are the Onsager coe cients, D i are the di usion coe cients, and and are nonlinear functions of the order parameters.One can easily see that these kinetic equations coincide in their form with the kinetic equations of the reaction-di usion models widely used in synergetics 2].

Singular points in uctuation models of self-organization
We shall now investigate the concrete uctuation models with two order parameters.
The rst model is the uctuation model similar to well-known kinetic \brusselator" model 1,2]: _ x = A ? Cx + By ?xy 2 + D x @ 2 x @ 2 ; _ y = Bx ?x 2 y + D y @ 2 y @ 2 : (17) The di erence between the \brusselator" model and model ( 17) is in the form of the nonlinear cubic term: the \brusselator" model produces the term x 2 y on the motion equation for x i.e. the same term as in the motion equation for y.
One can obtain the following stationary states for the uctuation model ( 17 Then by taking into account the characteristic equation det( M ?Dk 2 ?Î] = 0; 2 ?(Sp M) + = 0; Sp M = T 1 + T 2 ; = T 1 T 2 ?B 2 ; (21) one can obtain the necessary conditions of the Turing bifurcation The second uctuation model is the model similar to the Landau-Ginzburg model with two order parameters and the forth-order interaction H int x; y] = a 4 Z x 2 y 2 dr: x @ 2 ; _ y = By + C 2 y 3 + Q 2 x 2 y + D y @ 2 y @ 2 ; (24) have a number of stationary states It follows from a characteristic equation 2 ?(Sp M) + = 0; Sp M = 1 + 2 ; = j Mj = 1 2 ? 3 ; (26) that conditions of the Hopf bifurcation Q 1 Q 2 < 0; q 1 q 2 < 0 (27) realize only on the non-thermodynamic branch of states.Figure 1.These results are con rmed by the numerical calculations performed in 8].Namely, singular points exist under the following conditions q 1 q 2 < 0, i.e. on the non-thermodynamic branch of states in which the change of the coe cients A i and (or) D i takes place.Computer simulations of the ordered structure formation give three di erent types of structures: (a) so called \contrast" structures with many sharp peaks ( gure 1), (b) quasiperiodic structures slowly decaying in space and time ( gure 2), and (c) soliton-like structures exhibiting a self-organization process within the range of parameters predicted theoretically ( gure 3).

Fluctuation e ects in stationary states of open systems
According to the Prigogine's de nition of the stationary states of open systems the total entropy remains constant due to the complete compensation of the entropy increase S i connecting with the dissipative processes inside a system and the entropy ow S e from this system outside: S = S i + S e = 0:  1) positive entropy changes S (d)   i caused by of dissipative processes inside a system, 2) negative entropy changes S (j)   i connecting with the ordered structures formation, i.e. S i = S (d)  i + S (s) i : In these cases stationary states can be guaranteed as follows: a) if S (d)  i > S (s)  i , then S i > 0 and in order to ful l the condition of the entropy conservation S = 0 the following inequality similar to the Prigogine condition of stationarity has to take place: S e = ?jS (d)  i ?S (s) i j < 0; (32) 2) if S (d)  i < j S (s) i j , then S i < 0 and in this case an entropy ow from outside S e > 0 becomes necessary, i.e.
S e = j S (s) i ?S (d) i j > 0; (33) It is important to stress that uctuation e ects must be taken into account in treating the formation of the spatial, temporal, and spatio-temporal structures near the bifurcation points.Therefore, the formula for the full entropy change reads as follows S = S (d)  i + S (s) i + S e + S fl : 21 Figure 2.
Let us generalize a de nition of stationary states given by Prigogine on the cases of arbitrary open systems with ordered structures.The entropy change consists of two contributions it the processes of the ordered structures formation are taking into account:

Figure 3 .
Figure 3.The last inequality gives a generalization of the Prigogine's de nition of the stationary states of open systems.It is important to stress that uctuation e ects must be taken into account in treating the formation of the spatial, temporal, and spatio-temporal structures near the bifurcation points.Therefore, the formula for the full entropy change reads as follows from (34) that under the conditions of stationarity (32) or (33) the last term describing uctuation e ects gives the only contribution to S fl .For the uctuational part of the free energy one has respectively: F fl = ?T S fl : (35) It gives a necessary grounding to use the uctuation Hamiltonians of the Landau-Ginzburg form for investigation of the ordered structures formation in stationary states of open systems.