A. Introductory and Inorganic Oscillations Thermodynamic Aspects and Bifurcation Analysis of Spatio-temporal Dissipative Structures BY G. NICOLIS*’ AND I. PRIGOGINEJ~ FacultC des Sciences UniversitC Libre de Bruxelles 1050 Brussels Belgium Keceiryed 6th December I 974 The thermodynamic prerequisites for the emergence of spatio-temporal patterns of organization in nonlinear chemical systems are reviewed. Analytic expressions for steady-state time-periodic and travelling wave like solution of reaction-diffusion equations are reported. A comparison with the results based on computer simulations is outlined. 1. INTRODUCTION-THE THERMODYNAMIC BACKGROUND It is well-established that certain types of chemical reactions subject to appropriate conditions organize themselves spontaneously in space and time to give rise to regular steady state spatial patterns or to time-periodic flashes of chemical activity.Obviously self-organization phenomena of this kind are cooperative in the sense that they require a strong coupling between the different subunits constituting the system. However in contrast to phase transitions and other familiar examples of cooperative processes which come from equilibrium thermodynamics self-organization in the context of chemical reactions seems always to involve large scale macroscopic elements. Thus the patterns which are eventually established in a reacting mixture reflect in many respects the global properties of the system such as the size the symmetry of the spatial domain and most importantly the nature of the constraints acting from the external world.Because of this thermodynamics of irreversible processes constitutes the natural framework for studying these phenomena. The principal goal of this discipline is to describe the properties of macroscopic systems in terms of the properties of certain state functionals like entropy or entropy production. Now in the neighbourhood of the state of thermodynamic equilibrium the behaviour of these functionals is deter-mined by the theorem of minimum entropy production ’:In a system subject to time- independent constraints evolution leads to a steady state where the entropy production per unit time P takes a minimum value compatible with the constraints acting on the system dP -< 0.dt At the same time by the second law P 2 0. (1-2) Inequalities (1.1) and (1.2) imply the asymptotic stability of the branch of states t Also Center for Statistical Mechanics and Thermodynamics The University of Texas at Austin Austin Texas 78712 U.S.A. 7 SPATIO-TEMPORAL DISSIPATIVE STRUCTURES constituting the continuation of the equilibrium states hereafter referred to as the thermodynamic branch. Thus for a single phase system where the equilibrium state itself is stable the emergence of new types of cooperative behaviour is ruled out. The extension of thermodynamics away from equilibrium to the so-called nonlinear range of irreversible processes has proved that inequality (1.1) cannot be extended to this range. Instead we have derived a stability condition for the thermo- dynamic branch of states.It is found that stability will be ensured whenever for all t > to. Here S2P is the second order excess entropy production and SJ and SX are respec- tively the excess flows (e.g. reaction rates) and forces (e.g. chemical affinities) due to the deviation of the system from the reference state i.e. from the thermodynamic branch. A close analysis shows that inequality (1.3) can be violated in open systems obeying nonlinear kinetics and driven beyond a critical distance from equilibrium In such systems an instability of the thermodynamics branch will develop and evolve subsequently to a new regime. The threshold for this instability will be determined by the equation PP((l,}) = 0 (1*4) where (Af} is a set of critical values of the parameters-such as the intensities of the external constraints the chemical rate constants or the diffusion coefficients-which influence the evolution of the system.In this case we can see that cooperative behaviour is allowed as the system can conceivably leave the thermodynamic branch and thus exhibit markedly non-equilibrium behaviour. We arrive therefore at the conclusion that spatio-temporal organization in chemical systems is a supercritical phenomenon accompanied by an instability of the thermodynamic branch. This phenomenon is created and sustained by the constraints that is by the dissipative processes inside the system. Dissipation becomes here an organizing factor contrary to what common intuition would suggest.To stress this point we have called these supercritical organized states dissipative structures. Ironically the first chemical model showing some form of ordering-the Volterra-Lotka oscillator-turns out to be a non-typical example of cooperative behaviour.2* It belongs to the class of conservative systems which do not exhibit an instability of the thermodynamic branch. As a result the oscillatory behaviour is not asymptoti-cally stable and can therefore be destroyed by the least external perturbation. As we shall see in detail in the subsequent sections the crossing of a critical point of instability (see eqn (1.4)) is a sufficient prerequisite to guarantee the stability of the subsequently emerging pattern. We close this section by a short remark concerning the origin of the deviations from the reference state causing the appearance of an excess entropy production in relations (1.3) and (1.4).A macroscopic system involving many degrees of freedom is always endowed with an internal mechanism permitting such deviations namely the fluctuations. Thus a system near the state determined by relation (1.4) will always have a non-vanishing probability of reaching the unstable region through fluctuations. An interesting and most unexpected result of our recent work on fluctuations in nonlinear systems shows that only those fluctuations whose range exceeds some critical value will be able to amplify and induce cooperative behaviour. This critical size is the result of a competition between the natural growth rate of a local fluctuation and the influence of the surroundings tending to damp this fluctuation.We expect therefore to have an interesting phenomenon of nucleation of a gew kind whic4 G. NICOLIS AND I. PRIGOGINE suggests surprising analogies between chemical instabilities and phase transitiom6 Molecular dynamics calculations developed by Portnow and aiming to substantiate these ideas are presently in progress. These methods have been applied recently to questions such as metastable and unstable states as described in a discussion remark by R. Lefever 1. Prigogine and J. Turner in this Symposium. 2. MATHEMATICAL FORMULATION Chemical reactions are the most important elements responsible for evolution beyond the thermodynamic branch.Because of their nonlinear character they are capable of generating instabilities by amplifying small effects like small inhonio- geneities induced by external disturbances or by internal fluctuations. The latter can then propagate into the medium through the long range coupling between spatial regions provided by transport processes. Thus we expect that diflusion and heat conduction as well as convection should be important factors in the understanding of cooperative behaviour in chemical systems. In this paper we focus attention on the phenomena originated by the chemical reactions themselves and by the spatial inhomogeneities. Hence we shall assume that the system is isothermal and at mechanical equilibrium. Moreover we shall ignore the influence of electric or of any other types of fields.Let N1,.. .,N be the concentrations of the n chemical substances involved. These variables which by our assumptions provide a complete macroscopic description of the system evolve according to the reaction-diffusion equations dNi (i = I,. . . n) __ = v~((N~])+D~v~N~ a’t where D1 are the diffusion coefficients of species i (assumed to be constant) and ui the rates of production of ifrom the chemical reactions. In general the presence of feedback processes of various kinds will cause 2ii to be nonlinear functions of the (N,)’s. Thus eqn (2.1) constitute a nonlinear partial differential system of the parabolic type. In order to have a well-posed problem we will have to supplement these equations with appropriate boundary conditions on the surface I:of the spatial domain of volume V.Two types of conditions will be considered (i) Dirichlet conditions (Nf) = {const.) (2.2a) (ii) Neumann conditions (n-ViVf> = {const.). (2.2b) The latter type of condition with (const.) = (0)applies in most of the experimental investigations on the Belousov-Zhabotinski reaction although quite recently Marek reported experiments on this reaction under open system conditions. Conditions (2.2a) and (2.2b)represent the constraints acting on the system which in the most general case will maintain it away from the state of thermodynamic equilibrium. Note that the entire formulation adopted in this section assumes bounded media. One reason for this is that in chemistry and to an even greater extent in biology the size and the surface of the system play a very important role owing to the long range nature of diffusion which establishes a means for communication inside the system as well as between system and environment.The other reason for dealing with bounded media is that the mathematics becomes much more trans- parent. In particular one can construct explicit expressions for the various types of SPATIO-TEMPORAL DISSIPATIVE STRUCTURES solution of system (2.1) and study their stability. The requirement of stability is of course extremely important as it will determine among the solutions available those which will actually be realized. Owing to the nonlinearity the mathematical theory of systems of equations of the type (2.1) is still at a rather primitive stage.* At best one has some qualitative information about the behaviour of the solutions based on bgurcation theory lo supplemented with stability considerations.The purpose of this theory is to study the possible branchings of solutions that may arise under certain conditions. This is linked of course to the point of view of thermodynamics developed in the previous section according to which the emergence of dissipative structures implies the instability of the thermodynamic branch. Under appropriate boundary conditions eqn (2.1) admit a ungorm steady sfate solution on the thermodynamic branch. In this paper we shall analyze the transitions from this state to new types of solutions such as (a) space-dependent steady states ; (b)time-periodic (and space-dependent) solutions and (c) travelling waves and more general types of time-dependent solutions.Part of our analysis will be performed on the general form of the eqn (2.1). However quite often we will find it useful to illustrate the ideas on the simple trimolecular model 2* A+X 2X+Y + 3x (2.3) B+X-*Y+D X+E which was analyzed recently in detail from the standpoint of bifurcation theory.’ The astonishing element of most of the transitions leading to patterns (a) to (c) is their symmetry-breaking character. Beyond a certain critical set of values of the parameters defining a bifurcation point of the differential system (2.1) the most symmetrical solution of this system ceases to be stable.One then obtains a macro-scopic quantization? of the various new solutions whose properties are determined by a set of a few “ quantum numbers ” expressing the influence of the rate constants and the diffusion coefficients the symmetry of the spatial domain and the boundary conditions. In the subsequent three sections we shall review briefly the most characteristic phenomena arising in self-organizing chemical systems. 3. STEADY-STATE DISSIPATIVE STRUCTURES 3(i). D E L o cA L I z ED sT R ucT uREs Consider first the case for which the solutions on the thermodynamic branch are spatially uniform. Let -k, 4mbe respectively the eigenvalues and eigenfunctions of the Laplace operator within the spatial domain of interest. Their specific properties will of course depend on the details of the geometry.We shall come back to this point later in this subsection. The stability properties of the thermodynamic branch will be determined by the characteristic equation detI(aui/aNj)o-6:J(co +Dikm)I= 0 (3.1) * In this respect we may note that Thorn’s theory of catastrophes refers to systems described by ordinary differential equations. A brief comparison between dissipative structures and catastrophe theory has been attempted in ref. (9). t This term is due to M. P. Hanson J. Chern. Pliys. 1975. G. NICOLIS AND I. PRIGOGINE where (i,j) = 1 . . . ti and the spectrum of k is determined by the Laplace operator and the spatial domain. Suppose eqn (3.1) predicts a critical set of bifurcation parameter values such that Re Wa = 0 Im 001= 0 Re On*a < 0.Then one will have bifurcation of a new steady state solution which will depend on space. The condition for this is A = det I(au,/aNj)o-dk,’D,k,l = 0. (3.2) The important point is that for n 2 2 eqn (3.2) may be fulfilled for norz-trivial values of k,, provided the bifurcation parameters (A) are within certain limits. Let (Ac] be the first values of (3.) compatible with these conditions k the corresponding values of k,. Then in the neighbourhood of {Ac) the emerging steady state structure will have a non-trivial space-dependence determined to first approximation by this ki and by the corresponding eigenfunction 4;. For any given model one can compute this as well as higher order approximations using bifurcation theory.The method is described in detail in ref. (9) and (10) and will not be reproduced here. As an illustration we consider a one-dimensional system of length L. Then : @r k = (m= 0 1,. . .) (3.3a) and 4mK sin rnrir 0 6 r < L (3.3b) L for the boundary condition (2.2a),whereas mnr ~,occos-L O<r<L (3.3c) for zero flux boundary condition. Thus k describes the wavelength of the emerging dissipative structure. The critical value of this wavelength is determined by eqn (3.2). I 0 1 space/arbitrary units FIG.1.-One-dimensional steady state dissipative structure for X in the model reaction (2.3). NA = 2, NB= 4.6 Dx = 0.0016 Dy = 0.0080 and L = 1. The boundary values for Nx,Ny are & = NA = 2 Ry =NB/NA= 2.3.The critical wave number mC= 8. SPATIO-TEMPORAL DISSIPATIVE STRUCTURES The important point is first that for two or more chemical substances m will be finite ; and second that the critical wavelength will depend on the size of the system. We find here a striking manifestation of the global character of dissipative structures. An additional important element is that the symmetry properties of m influence profoundly the form of the solutions. When m,is even the system exhibits a symmetry- breaking transition to two possible new states both of which are stable. Fig. 1 represents the corresponding spatial pattern for the model reaction (2.3). For the same model the analytic calculation based on bifurcation theory yields m nr sin -(3.4) X(r) = A & a(B-Bc)* sin + b(B-B,) C m 1 mnr L ,, odd (m2-m:)2 m2-4m where a b are well-defined functions of the chemical parameters and the diffusion coefficients and the concentration B of the initial substance B has been used as bifurcation parameter.When m is odd one observes more complex phenomena like hysteresis. The corresponding spatial patterns for model (2.3) are shown in fig. 2. In both cases the emerging structures are shown to be stable. U I spacelarbitrary units FIG.2.-Steady state dissipative structure in the case of an odd critical wave number m = 7. Dx = 0.0018 and all other parameters have the same values as in fig. 1. Consider now a two-dimensional medium for instance a circle of radius R. Then (3.5a) where (r 0) are polar coordinates and J, is the Bessel function of order n.The eigenvalue k of the Laplacian is determined by the boundary conditions. For instance for zero flux boundary conditions J;(k,R) = 0. (3.5b) Again for two or more chemicals both n and k can be different from zero at the point of the first bifurcation. Fig. 3 represents the spatial patterns resulting beyond bifurcation for the model system (2.3) in the case where n = 0 k; # 0. This pattern G. NICOLIS AND I. PRIGOGINE is .cylindrically symmetric. If on the other hand both k; and n are different from zero the resulting pattern exhibits apolarity in the sense that a macroscopic gradient sets up spontaneously across the system.ll Fig. 4 describes the spatial form of the solutions for the model reaction (2.3).FIG.3.-Cylindrically symmetric steady state dissipative structure on a circle for the trimolecular model (2.3). The radius of the circle is R = 0.20 and all other values are the same as in fig. 1. FIG.4.-Polar dissipative structure on a circle for model (2.3) in the case of zero flux boundary conditions. R = 0.10 Dx = 0.00325,Dy = 0.0162 A = 2 and B = 4.6. In the simplest cases of bifurcation the resulting pattern reflects to a good approximation the properties of the critical modes i.e. of the eigenfunctions of the Laplacian at the bifurcation point. In situations involving high symmetries like for example a square domain it is not uncommon to have as first bifurcation a state corresponding to a degelzerate eigenvalue of the Laplacian.The spatial structures are then a complex mixture of the degenerate eigenfunctions and the multiplicity of solutions can become very high. The situation is described in fig. 5. A closer study reveals that as the size of the system increases different dissipative structures can be realized successively. Thus below a critical size R,,,the only SPATIO-TEMPORAL DISSIPATIVE STRUCTURES possible solution is the uniform one on the thermodynamic branch. Between R, and R,,the system may exhibit a non-polar pattern like the one in fig. 3. Then between A, and R, the latter could be succeeded by a polar pattern such as that in fig. 4. Beyond R,,all steady state solutions may become unstable and a propagating wave can emerge (see section 4).Finally for still larger sizes the system may undergo homogeneous bulk oscillations with the same phase everywhere within the reaction space.12 Now growth is a very general phenomenon occurring in all living organisms. The fascinating point is that according to the previously developed ideas growth will engender a whole succession of different forms or patterns. FIG.5.-Steady state dissipative structure on a square for model (2.3) in the case of double degeneracy. The side L = 0.132 and the boundary conditions are zero flux ones. A = 2 B = 4.6 Dx = 0.0016 and Dy = 0.0080. To arrive at this structure initial perturbations along a diagonal of the square had to be considered. The intrinsic dependence on length described above is a particular case of a more general phenomenon occurring quite frequently in nonlinear chemical systems.Namely when a bifurcation parameter takes different values a whole succession of instabilities can take place leading to more and more complex spatial forms. These can be either bifurcations from the thermodynamic branch leading to a multiplicity of dissipative structures each one having its own domain of attraction of a set of initial conditions; or they can be the result of a secondary bifurcation from a previously established dissipative structure. Unfortunately in both cases bifurcation theory very rarely permits a rigorous study of the stability of the various states. An interesting thermodynamic property of these multiple transitions is that they may lead to an increase in dissipation as the wavelength of the structure becomes shorter.3(ii). LO cA L I zED STR uCTuREs In many multicomponent systems the diffusion of some of the chemicals may create an inhomogeneous environment for other chemical intermediates. In this G. NICOLIS AND 1. PRIGOGINE case the solutions on the thermodynamic branch are no longer space-independent. Computer simulations l4 have revealed the existence of spatial patterns which remain localized within a part of the reaction space. Recent theoretical analysis of these states 13? l5 has shown that these patterns are again of the dissipative structure type as they arise beyond the instability of the thermodynamic branch. Fig. 6 describes the spatial distribution of concentrations in a one-dimensional system for the model reaction (2.3).Xt 0 0.5 I space/arbitrary units FIG.6.-One-dimensional localized steady state dissipative structure for mode1-(2.3). DA = 0.197, Dx = 0.001 05 Dy = 0.005 26 B = 30 X = k = 14 = B/A = 2.14. It is important to realize that this localization phenomenon may be spontaneous. No spatial gradient needs to be preimposed. On the other hand localization may result in a weaker dependence of the properties of the dissipative structure on the system as a whole e.g. on the size. Still complete independence of the size is difficult to imagine or to justify mathematically. It appears tempting to suggest that localized structures are good candidates to explain the formation of leading centres in some systems like the Belousov-Zhabotinski reaction.In this respect however it is necessary to point out that the complete independence of those leading centres of the size of the system postulated by Zaikin and Zhabotinski l6 is never fulfilled rigorously in localized structures. 4. TIME-PERIODIC SOLUTIONS According to linear stability analysis a time-periodic solution will bifurcate in the neighbourhood of the thermodynamic branch provided the characteristic eqn (3.1) admits a pair of purely imaginary roots +ioG1 at the bifurcation point. In chemical systems involving only two variables the first bifurcation leads necessarily to a state where ki = 0.l’ If this state is allowed by the boundary conditions then the system will presumably exhibit this most symmetrical solution whose space dependence will be trivial.This actually happens in the case of zero flux boundary conditions where the system exhibits limit cycle oscillations throughout the reaction space. In contrast constant boundary conditions will in general rule out the state with ki = 0. The first allowed bifurcating solution will then exhibit both space structure as well as temporal periodicity. Fig. 7 describes this solution for the model reaction (2.3). Using bifurcation theory one can calculate explicitly these solutions in the neighbourhood of the bifurcation point. For contrast boundary conditions and for a SPATIO-TEMPORAL DISSIPATIVE STRUCTURES one-dimensional system involving two chemical intermediates one obtains the following form for the deviation X(Y t) from the thermodynamic branch sin -cos pt + x(r) = a(~-~,)f (3 00 cos 2pt) sin (Y) b(B-B,) (a i-d -+O((B-Bc)).(44 m=l odd The important point in this expression is the presence of a fundamental mode of period 2n/p and of its first harmonic. This is a general property of all systems in which the first nonlinearity around the thermodynamic branch is quadratic.” t = 7.97 spacelarbitrary units FIG.7.-Characteristic stages of evolution of the spatial distribution of X during one period of the wave for model (2.3) for a length L = 0.80and for constant boundary conditions. A = 2 B = 12.6, Dx = 0.0080 Dy = 0.0040. Relation (4.1) describes a superposition of standing waves. However the numerical evaluation of this formula for model (2.3) reveals a pattern which is strikingly similar to fig.7. It appears therefore that in a bounded medium propagating concentration fronts can appear during certain time intervals but not during the entire period of the phenomenon. Waves of this kind are capable of transporting matter during macro-scopic time intervals as the (Vx) at a certain point retains a given sign during these intervals. However they differ markedly from the waves encountered commonly G. NICOLIS AND I. PRIGOGINE in mechanics or in electromagnetic theory in the sense that it is meaningless to define for them a propagation velocity depending on x through the usual relationship In chemical systems involving more than two coupled variables an additional important element appears in that the first bifurcation can lead to states with k; # 0.’ Thus even for zero flux boundary conditions these systems will exhibit symmetry- breaking transitions leading to states with non-trivial space dependence. For instance in a circular domain the first bifurcating solution will in some cases be approximated by the function (4.3a) with (4.3b) At each instant of time the equal concentration curves +ml = const. will be symmetrical with respect to a diameter of the circle. As time proceeds this diameter will rotate at constant speed. Thus the time-periodic solutions of this form represent rotating waves although their form is not spiral in the approximation provided by relations (4.3). It is quite possible however that a bifurcation theoretical calculation leading to an expression such as eqn (4.1) will produce spiral waveforms arising from the influence of higher order terms.The latter contain a series having J,J(kmr)as coefficients which could break the symmetry of the equal concentration curves with respect to the diameter of the circle. The numerical evaluation of the bifurcation theoretical expression for a circle for the model reaction (2.3) is in progress. Let us recall that the appearance of angle-dependent solutions beyond the first bifurcation from the thermodynamic branch occurs typically in systems with three or more variables. In this respect we may note that the Noyes-Field model for the Belousov-Zhabotinski reaction involves three intermediates and therefore fulfils these conditions.Additional variables which could play an important role in the Belousov-Zhabotinski reaction as well as in other chemical systems are the tempera- ture and the electrical field. The latter could be especially crucial in the generation of electrical waves capable of interfering with or even triggering the chemical waves. Other possibilities of wave forms having a symmetry less than the symmetry of the spatial domain include secondary bifurcations or multiple stable solutions as described at the end of subsection 3(i). At present it appears to be very difficult to investigate the stability of these solutions although one can construct them explicitly using bifurcation theory. 5. TRAVELLING WAVES The requirement of temporal periodicity imposed on the solutions calculated in the previous section turned out to be incompatible with the existence of a well-defined speed of wave propagation although as we saw expressions such as eqn (4.1) could approximate propagating fronts as closely as may be desired.In this section we review briefly the problem of existence of travelling waves by requiring solutions of eqn (2.1) and (2.2) of the formf(v-uvt). For details we refer the reader to a recent paper by Auchmuty and Nico1is.l’ Work in this direction in the case of unbounded media has been reported recently by Kopell and Howard l8 and by Ortoleva and Ross. SPATIO-TEMPORAL DlSSIPATIVE STRUCTURES 5(i). WAVES IN RINGS Let 2n be the length of the ring. The Laplace operator reduces to a single derivative along the ring.We want to find solutions of the form &(r t) =L(r-of)=A(<> (5.la) (5.1 b) The reaction-diffusion equations take the form . n). (5-2) -l;f,/(<) = Difl(t)+vi((fj>) (i = 1 We wish to determine those values of u for which there are non-trivial periodic solutions of (5.2) Ni = N;+ui exp(im<) 112 = 0 1,. . . (5.3) Note that both (5.16) and (5.2) are compatible with the existence of a constant speed of propagation. A linear analysis of eqn (5.2) confirms this point and provides expressions for the critical values of the bifurcation parameter as well as for the speed of propagation. The solutions can then be constructed using bifurcation the0ry.l’ As in section 4 for systems with two variables the first bifurcation occurs at rn = 0.Thereafter the system undergoes a uniform limit cycle type oscillation. For three or more intermediates the first bifurcation can lead to a non-trivial spatial dependence in the form of a propagating pulse. 5(ii). MORE GENERAL ONE-DIMENSIONAL SYSTEMS We now consider a system with open ends subject to the boundary conditions Ni(O t) = Ni(L t) = NP (5.44 or dNi dlvi -dr (0,t) = -(L 1) = 0 (5.46) a). where NP are the values of the (uniform) chemical concentrations on the thermo- dynamic branch. It can easily be seen l7 that such boundary conditions rule out automatically the existence of solutions of the form (5.14 with v constant. We relax these conditions and seek solutions of the form A detailed analysis shows l7 that a solution of the form (5.5) can emerge beyond asecondary bifurcation of a steady-state dissipative structure.The solvability condition of the bifurcation equations will provide us with a set of partial differential equations which determine the propagation velocity v. The ability of these waves to arise in a non-uniform background provides a mecha- nism of localization of these waves as well as a possible formation of leading centres. 5(iii). TRAVELLING WAVES IN SPACES OF HIGHER DIMENSION By an argument similar to that developed in the previous subsection one can show that plane waucs of the form f(k Y-cot) are ruled out in two or higher spatial G. NICOLIS AND I. PRIGOGINE dimensions.The possibility remains however of having superpositions of such waves resulting in a wave packet of the form Ni(r t) = S dt d~ dCf(S V C) exPti(k. r-u(k)t)l (5.6) where k = (c q 5). The group velocity drldk of this wave packet will play the role of the propagation velocity. Again when the geometry of the problem is specified one can look for specific wave forms like cylindrically or spherically symmetric waves spirals and so on. 6. CONCLUDING REMARKS From this analysis of nonlinear chemical systems we have seen that a wealth of structures presenting a wide variety of sizes and shapes becomes possible beyond an instability of the thermodynamic branch. It is remarkable that in spite of the extreme diversity of these far-from-equilibrium phenomena it has been possible to reach general conclusions on two fundamental aspects of dissipative structures.On the one side nonlinear thermodynamics and bifurcation theory have enabled us to provide a rough classijication as well as to construct explicit expressions for the various possible types of organized states. On the other side nonlinear thermodynamics supplemented with fluctuation theory has enabled us to get a better understanding of the mechanisms prevailing in the transitions between states. Finally it is important to stress the analogies between these and similar phenomena encountered in various branches of natural sciences such as fluid dynamics shock waves and detonations ecology and even social behaviour. One can hope that the progress accomplished during the last few years in the field of nonlinear chemical kinetics will provide the tools necessary to tackle complex problems in these fields which until now have eluded quantitative analysis.It is a pleasure to acknowledge that a substantial part of the mathematical results of this paper have been obtained in collaboration with Prof. J. F. G. Auchmuty. We are also indebted to Dr. Herschkowitz-Kaufman Dr. Lefever Mr. Malek-Mansour and Mr. Erneux for their important contributions in the research reported in this paper and for many stimulating discussions. The financial support of the “ Ministhe de I’Education Nationale et de la Culture fraqaise et le Ministke de I’Education Nationale et de la Culture NCerlandaise ” from Belgium and of the Welch Foundation Houston Texas is gratefully acknowledged.’I. Prigogine Bull. had. Roy. Belg. 1945 31 600. P. Glansdorff and I. Prigogine Thermodynamicsof Structure Stability arid Fluctuations (Wiley-Interscience New York 1971). For a historical survey of the concept of dissipative structure and of its precursors we refer to a recent review paper by R. Glansdorff and I. Prigogine Bull. Acad. Roy. Belg. 1973,59 672. G. Nicolis and J. Portnow Chem. Rev. 1973,73 365. G. Nicolis M. Malek-Mansour K. Kitahara and A. Van Nypelseer Phys. Letters 1974 48A 217. ti G. Nicolis J. Stat. Phys. 1972 6 195 ;G. Nicolis and I. Prigogine Proc. Natl. Acad. Sci. 1971 68 2102 ;A. Nitzan P. Ortoleva and J. Ross this Symposium. ’J. Portnow and Svobodova Phys. Letters 1975 51A 370.* M. Marek Biophys. Chem. 1975 3 263. G. Nicolis and J. F. G. Auchmuty Proc. Natl. Acad. Sci.,1974,71 2748. lo D. Sattinger Lecture Notes iit Mathematics vol. 309 (Springer Verlag Berlin 1973). A. Babloyantz and J. Hiernaux Bull. Math. Biol. 1975. SPATIO-TEMPORAL DISSIPATIVE STRUCTURES l2 A. Goldbeter Proc. Nutl. Acud. Sci. 1973 70 3255 M. P. Hanson J. Chem Phys. 1974 60,3210. l3 J. F. G. Auchmuty and G. Nicolis Bull. Math. Biol. 1975 37,323. l4 M. Herschkowitz-Kaufman,Bull. Math. Biol. 1975. J. Boa Ph. D. Dissertation (Calif. Inst. of Technology 1974). l6 A. M. Zhabotinski and A. N. Zaikin J. Theor.Biol. 1973 40 45. J. F. G. Auchmuty and G. Nicolis Bull. Math. Bid 1975. l8 N. Kopell and L. Howard Studies in Appl. Math. 1973 52 291. l9 P. Ortoleva and J. Ross,J. Chem. Phys. 1974,60 5090.