PHYSICAL REVIEW A VOLUME 30, NUMBER 3 SEPTEMBER 1984 Conditions for the validity of Cxinzburg-Landau equations in far-from-equilibrium kinetics Ariel Fernandez and Oktay Sinanoglu Department of Chemistry, Yale University, 225 Prospect Street, New Haven, Connecticut 06511 (Received 16 November 1983; revised manuscript received 23 February 1984) We consider open reactive systems driven far from equilibrium sustaining a hard-mode instability. We derive the analytic form of the coefficients in the Ginzburg-Landau potential functional by first reducing the problem to Poincare normal form and then restricting the equations to the locally attractive manifold of states emerging beyond the instability. We conclude that the Hopf transversality conditions for bifurcation are not only sufficient but also necessary for the application of the Ginzburg-Landau equation. The Poin- care normal form reduction yields additional restrictions besides the Hopf hypotheses. I. INTRODUCTION The application of the Ginzburg-Landau (GL) theory to describe the critical dynamics of the attractive mode in chemical kinetics can yield the same results that are ob- tained from the Hopf-Birkhoff theorem approach (see Refs. 1 and 2). We can pose the following natural questions. (1) Are there bifurcation conditions other than the ones given in the Hopf theorem such that the time-dependent chemical oscillations are governed by the mean-field GL ap- proximation? The answer to this question is no if we as- surne analytic dependence of the bifurcation parameter (representing the external constraint) with respect to the amplitude of the rotating wave. (2) Are there restrictions in a physical system besides the Hopf hypothesis for the application of the GL equations? We shall demonstrate the existence of further restrictions. (3) Which is the analytic, rigorous form of the coeffi- cients of the GL potential functional? J(F) ~ x is the Jacobian matrix of F at = [BN(0)/tlX~] =0, j = 1, ... , N. Consider that J (F„) ~ x has two complex conjugated "c p A. +, X with zero as a real part: Ht(v, ) = 0, )i+ = + I H2(v, ) Xp', N(0) v= v, such eigen values dm H)(v, ) &0 dvm (3) for certain m ~ 1 (Ref. 5 ). In the particular case when the first integer m for which (3) is fulfilled is m = 1, we get the classical Hopf tranversal- ity condition. For m =1, v is analytic in the amplitude e of the time periodicity. That is a consequence of the Hopf theorem. Since we consider the general case m ~ 1 this condition has to be imposed as an additional restriction: and the remaining eigenvalues all lie in the left-half com- plex plane. The hard-mode instabilities leading to time periodicities occur if II. RESTRICTIONS IN APPLYING THE MEAN-FIELD THEORY TO TEMPORAL ORGANIZATIONS Consider an open system operating far from equilibrium in which a set of chemical reactions is occurring. The sys- tem is assumed in mechanical equilibrium and subject to isothermal and isobaric conditions. The equations of rate for the concentration vector X C R take the usual form: X= =F„(X) . dt Y= J(F) i-x Y+N(Y) 0 (2) The vector field F representing the rate of change due to chemical reactions is assumed analytic in the control param- eter v. This parameter is the bifurcation parameter and ac- counts for the external constraints that create and sustain a hard-mode instability in the system. If X0 represents the stationary state of the equation of motion (1) after the translation Y = X — Xp, problem (1) reads V=Pg+ QPiE (4) J~] There exists a linear transformation to convert problems (1) and (2) into Poincare normal form. 6 The change of vari- ables is Y=Q Z Q= (Revt, — Imvt, v3, ... , vtv) (5) z= @(z), where 0 0 Z& = — H2(Vg)Z2+ p(Z, , Z2) Z2=Hg(v, )Zi 't — H2(v, ) 0 0 v & is the eigenvector of J(F„) ~ -„corresponding to c 0 +i H2(v, ). (vij, j = 3, . .. , Ã span the union of eigen- spaces corresponding to the eigenvalues different from +i H2(v, ). Under this transformation, problems (1) and (2) read 30 1522 Q~1984 The American Physical Society