Dynamics of Synergetic Systems

According to its definition Synergetics is concerned with the cooperation of the individual parts of a system that produces macroscopic spatial, temporal or functional structures. Quite in the spirit of our previous meetings the symposium at Bielefeld brought together scientists from various disciplines. The topics treated at this symposium and being dealt with by the articles of the present volume mirror quite well the present status of synergetics, in particular with respect to theoretical developments. A survey is given in fig.1.

The opening talk of an interdisciplinary meeting should ideally start at “square one”. In the present case this means I should assume no previous background in the field of equilibrium phase transitions. Although everyone in the audience has some background in this field, the background of no two people is identical. Hence I begin with a brief introduction to phase transitions. Accordingly, I shall organize this talk around three simple questions:

The basic phenomenon of condensation appears in a great many different but related forms throughout the sciences. After we briefly mention some of the physical properties of the special type of phase transition known as Bose-Einstein condensation (BEC) in direct connection with some of the main themes of this symposium in synergetics, we shall proceed to describe the particular collective phenomena relating to the relativistic generalization [1,2] of this problem in d spatial dimensions.

The creation of ordered structures in nonlinear many body systems far from thermal equilibrium is a subject of increasing interest. In particular, the vast field of nonlinear optics has provided many examples of systems showing the effects of cooperative behaviour. The best known is the laser where the analogy with phase transitions at thermal equilibrium was first pointed out by Graham and Haken [1] and Degiorgio and Scully [2]. A more recently discovered phenomena is optical bistability which occurs when light interacts with a nonlinear absorptive or dispersive medium inside an optical cavity [3]. In this case the systems involved exhibit a behaviour analogous to first or second order phase transitions depending on the nature of the relaxation mechanism [4].

In the last ten years the properties of highly excited semiconductors have been studied thoroughly [1,2,3,4]. If the frequency of the exciting laser light is sufficiently high, electrons from the valence band are lifted into the conduction band. Thus, in the process of optical excitation one. generates electron-hole pairs (e-h pairs) i.e. electrons in the conduction band and holes or missing electrons in the valence band. In the low density regime electrons and holes form bound pair states, called excitons, due to their attractive Coulomb interaction. In an intermediate density regime one observes besides free carriers and excitons also higher bound states such as trions (e-e-h or e-h-h), biexcitons and even multiexcitons. In the high density regime bound states are no longer stable, because the attractive interaction between an electron and a hole is screened by the other electronic excitations. In this limit a plasma is formed.

It is a very general observation that the environment of nonequilibrium systems, be they natural systems or laboratory systems, is at best only constant on the average. The environment is very complex and thus unavoidably its temporal behaviour appears to be random. In other words, the system perceives the environment as a noise source. A good illustration of this is the population of, say, some animal species. Its growth is determined by factors such as the birth rate, mortality, abundance of predators etc. which are strongly influenced by climatic conditions and therefore show a more or less pronounced variability. More generally, the fluctuations of the environment, which are called external noise, can be understood as the expression of a turbulent or chaotic state of the surroundings.

The Zhabotinskii system [1], [2], is an excellent example of chemical synergetics [3]. When four or five chemical compounds are mixed in the appropriate concentration ranges and at the appropriate temperature, the Zhabotinskii system spontaneously organizes itself into temporal or spatio-temporal dissipative structures of macroscopic dimensions [4]. In this chemical reaction, at least twenty intermediates are formed. The chemical mechanism involved is so complex that almost all theoretical work is performed on models rather than on the best rate equations available today [5]. A first type of model involves “macrokinetic” steps rather than elementary ones. This type includes the model of ZHABOTINSKII and his collaborators [1] which attempts to reproduce both waveforms and periods of oscillations. It includes also the many versions of the Oregonator [6] which are designed to reproduce the waveforms of a few intermediates. Other models are of the heuristic-topological type, according to the PACAULT [7] classification. Two of them are the well-known PRIGOGINE-LEFEVER model [8] (or Brusselator) and the analytic BAUTIN system [9], [10] (or DREITLEIN-SMOES model). It is unnecessary to emphasize here the role played by the PRIGOGINE-LEFEVER model as a research tool in the theory of dissipative structures. The BAUTIN system is less well known in spite of several attractive features: this system is solvable in closed form; it exhibits a limit cycle and bistability; there is a saddle-node transition between steady-state and finite amplitude oscillations [2].

The discovery of propagating waves of various types in chemical reagents has provoked a great deal of research, during the last ten years, into the phenomenology and the underlying mechanisms for such wavelike activity. The research has been performed by natural scientists and mathematicians alike. Most of it has been experimental, but much computer simulation and mathematical analysis has also been done. Chemical wave activity is believed to be prevalent in biological organisms, but the most readily accessible reagent for laboratory study is that discovered by Belousov and Žabotinskiĭ. This mixture has oscillatory or excitable kinetics, depending on the concentrations of the various chemicals in the solution. Both of these regimes have at least two natural time scales: During one period of an oscillation or during one excited “excursion”, most of the variation in the concentration of the reactants occurs within a brief interval of time. The time scale associated with this brief spurt of activity is much shorter than that associated with the slow variation which occurs before and after. This is well known from experiment and computation, and is evident from scaling analyses of model kinetic equations performed in [1] and elsewhere. Spatial structures are also prevalent in unstirred layers of this reagent ([23]; [2], [22], [24], and references therein). Target patterns (expanding concentric circular waves) are among the most prevalent of these structures. Here again, disparate space and time scales are evident from computer simulation of propagating waves [3].

There are two main principles according to which chemical reactions in a spatial domain are modeled:

The term “synergetics” which is the title of the present meeting has been invented by Haken to describe the field of non-equilibrium cooperative phenomena; the idea being that such phenomena in largely different systems should be amenable to analysis within a common mathematical formalism. With this in mind, considerable work has been done within the past decade [1] in an effort to reduce the nonlinear equations of motion (EOM’s) describing the dynamics of systems of interest, to as few as possible common forms. This of course cannot be done under the most general circumstances. However, for systems near the critical point of their non-equilibrium transitions (e.g. the system described by Fig.1), multiple time and length scales procedures were used to reduce the EOM’s to the corresponding time dependent Ginzburg Landau (TDGL) equations for the appropriate order parameter characterizing the transition. Once such a TDGL equation is derived, the powerful machinery developed for equilibrium critical phenomena can be used to make definite predictions about the critical behavior of the given system. In particular the idea of universality implies that non-equilibrium critical phenomena will fall into a few universality classes with systems belonging to the same class showing the same type of critical behavior (e.g. the same critical exponents).

It is almost a common belief today that there exist such macroscopic phenomena that are certainly governed by deterministic chaos. Many of the chaotic phenomena appear as spatio-temporal chaos, and their adequate mathematical modeling often calls for a set of partial differential equations. In the past, the studies of spatio-temporal chaos have largely been concerned with a variety of turbulent phenomena in fluid systems [1,2] obeying the Navier-Stokes equation. Recent studies made it clear that a much simpler class of partial differential equations called reaction-diffusion equations is also capable of showing complicated space-time behavior. This was demonstrated for the Brussels model [3–5], the Rashevsky-Turing model [6], and also for a still wider class of systems consisting of diffusion-coupled oscillators [3–10]. In view of the surprising richness of patterns which reaction-diffusion equations can exhibit, it is quite natural to expect that chemical turbulence of diffusion-induced type could hardly be confined to the kinds discovered so far. The purpose of the present paper is to explore the possibility of a novel class of chemical turbulence, in particular, the one associated with the wavefront behavior of some chemical waves in media with the spatial dimension two.

A special type of turbulence, called boiling type turbulence, can be obtained by coupling a set of identical single-threshold relaxation oscillators in such a manner that the slow variables are locally cross-inhibitory. If the cross-inhibition is overcritical, leading to ‘morphogenesis’ between the slow variables, a sudden slackening occurs wherever the the slowly moving-up ‘skyline’ hits the threshold (with the consequence of a rearrangement of the skyline); and so forth. A simple equation is considered on a ring in a cellular approximation. One cell is not chaotic; two are capable of chaos; three apparently produce higher chaos of the first order; and so forth. In the 3-cellular case, geometrical arguments backed by simulation suggest the presence of a cross-section which is folded over (between one passage and the next) in several independent directions. By implication, the formation of singular sets of Alexandrov type can be expected.

Autocatalysis is a rather exceptional phenomenon in chemical kinetics and, when it appears in reaction networks, various characteristic phenomena like oscillations, spatial pattern formation or eventually chaotic behaviour are commonly observed. Biochemistry and biology, in contrary, have to deal with a class of molecules for which selfreplication became obligatory. These molecules, autocatalysts in a sense, are the polynucleotides, the nucleic acids or, later on in evolution, the genes. Self-organization, the most interesting attribute of the biosphere, appears to be essentially based on the capability of self-replication.

The emergence of life can be studied under two aspects, as a historical investigation or as an engineering problem.

Since the pioneering work of TURING [1] there has been great interest in the study of selforganizing systems. These systems present coherent behaviour that will not be possible when considering only the properties of the subunits of the ensemble. The relevance of these problems to the study of biological systems has been viedly recognized. These questions have been reviewed in several monographs [2,3].

The immune system regulates in a remarkably concerted way the production and properties of an extraordinary diverse collection of cells and molecules. The purpose of this activity is to protect the organism against various forms of aggression. This requires the circulation of cells and molecules between different organs, the discrimination between self and non self antigens, the capability to deliver highly specific responses, the capability of learning and of memorisation (for review articles, see [1[).

The concept of “synchronization” is employed as an index of good organization in usual (“preallocation”) communication networks; it should be relaxed though and enlarged in cognitive networks (which are hierarchical structures) in order to incorporate dynamical deliberations taking place in “dynamic allocation” or packet — switching networks, finite state automata as well as biological, ecological economic and social systems.

These processes may lead via destabilization of the old control strategy to the abrupt emergence of a metalanguage or a new decision making algorithm among the members — “users” of the system at the expense perhaps of temporal or spatial coherence among the interacting partners. The result of such a “catastrophe” is a richer perceptive and behavioral repertoire of the system and an amelioration of his adaptability to “noisy” environments. In that sense the system “Soviet Union” is more synchronized but the system “United States” is more organized. In short the way we envisage the role of organization in a “cognitive network” of interacting variables (organisms) — performing under conditions of uncertainty and conflict has not so much to do with the congruence between the sequences of behavioral mode turnover of the individual subsystems constituting the networks: it has rather to do with the solution of a dual objective optimization problem compromizing defacto conflicting factors such as homeostasis (e.g. persistence at a given state) and adequate crosscorrelations with the partner — subsystem.

The whole idea is fully developed in the present paper in a simple network consisting of two interacting hierarchical subsystems.

Each subsystem — organism posessessin turn two irreducible hierarchical levels playing the role of “store — forward” and “traffic” processes or “behavior” and “experience” respectively.

The evolution of the system is pursued in a unifying contentless formalism of inductively played nonnegotiable games between the hierarchical levels of each partner — subsystem.

A detailed isomorphic model concerning nonresolvable ambiguity of communication dynamics (envisaged as bidirectional information transfer) between two hierarchical structures, is worked out.

The scope is to design feed forward control algorithms leading to limited resolution of the intrasystemic conflict via inter systemic communication.

Before developing a general frame for the quantitative description of dynamical processes in society, let us make some remarks about the structure of science into which our theory has to be embedded.

The predictions of many scientific models are highly sensitive to small perturbations of the equations. Topological study of these can clarify and improve robustness of the models, sometimes revealing new phenomena implicitly associated with them.

It is a well known fact that exact solutions of the Master Equation which describes the time evolution of the probability function P({X_i},t) in terms of the various gain and loss processes, are few and far between. Already in the case of one variable, the absence of detailed balance precludes any straightforward general solution, even for the stationary solution. (1,2)

I shall present here a review and updating of the work on Poisson Representations and, in particular, give the theoretical justification for the use of “negative noise”, which we originally introduced without rigorous justification a few years ago.