Parts and Elements of the Wholes in Synergetics.

**Irina Dobronravova **

Kiev National Shevchenko University (Ukraine)

Twenty five years have elapsed since then German Haken used a term 'synergetics' to emphasize the role of coordinated motions of medium elements in such cooperative phenomena as non-equilibrium phase transitions, hydrodynamic vortices, autocatalystic reactions, dynamics of populations and other examples of self-organization as spontaneous becoming of space-time structures in open, far-from-equilibrium mediums.

There are some apparent analogies in non-linear mathematical models of the processes of becoming and dynamically sustainable existence of self-organized systems of different nature. The respective mathematical apparatus and methodological principles of its application, such as subordination principle, principle of coherence and so on, constitute 'the hard core' of synergetics as over disciplinary scientific research program of theoretical description of self-organizing complex systems.

Such systems are the typical subjects of the modern non-linear science, to which the notion of "a whole" seems to be naturally applicable. The most famous chemical clock based on Belousov-Zhabotinski reaction is of that kind. It is an example of a catalystic reaction, which presents bifurcations far from equilibrium. In the appropriate conditions we see the liquid becoming red, then blue, and red again within a period of about one minute. The reaction proceeds in a time-periodic fashion, involving billions and billions of molecules. In solution with space inhomogeneity (without mixing) we can watch the spreading of chemical wave in the two-dimensional system of Belousov-Zhabotinski reaction. (Fig.1)

Nobel prize-winner Ilya Prigogine used the notion of 'a whole' to describe such behavior of a far-from-equilibrium medium: "near the critical point the chemical correlations became the large-scale ones. The system behaves as a whole in spite of the short-range character of chemical interactions."

Such a combination of notions of 'system' and 'a whole' contains the following methodological problem: if we regard a system as a whole, what relations exist between the elements of a system and the parts of the self-organized whole? Can we consider the elements of a system as the parts of a whole?

In fact, the correlated notions of 'system and its elements' are usually associated with the reduction principle, at least in physics: to be explained, a system has to be reduced to its elements and their interactions. But in synergetics the contrary methodological principle works: that of subordination. According to it, a self-organizing system could be understood as a whole that subordinates the behavior of non-linear medium elements. Mathematically, in non-linear dynamics it is expressed by existence of attractors on the nonlinear medium in condition of critical values of the control parameter. For dynamically stable dissipative structures, periodical trajectories (the so called limit cycles) in phase space are attractors. On this figure (Fig.2) you can see the arising of the limit-cycle in the brusselator (the simplified model of the chemical clock) in the phase space of the two independent variables (X and Y present the density of the intermediate components in the 'chemical clock.') Whatever the initial conditions, the system involves towards a 'limit cycle.' It expresses the dynamically stable existence of the structure as a whole.

I will use as an easy-to-grasp example the so called Benard structures. You can watch their self-organization if you heat for a while a frying-pan with a layer of liquid (oil or water). Then the surface of this liquid will remind a honeycomb with hexagonal cells. (On Fig.3 sight from above.) This space structure is formed by convective flows of liquid created by coordinated motion of molecules.(Fig.4) The motion of particles in every other vortex has the opposite direction of rotation (clockwise or anti-clockwise). Every vortex contains many billions of molecules, which move in the correlated, coherent way. This dynamic order exists due to the dissipation of energy of heating. (That is why such structures are named the dissipative structures.) The size of each Benard cell is about one millimeter, though intermolecular force's space scale is million times less. Reduction to molecular level can not serve as the principle of scientific explanation of dissipative structure. To know the elements of medium is only the first step, because molecules are the same in chaotic motion before critical value of temperature and after its threshold and becoming of self-organized structure. To understand how the molecules act together under certain conditions is to define self-sustainement of dynamic stability of the dissipative structure as a whole. The use of notion of 'a whole' here is in accordance with dialectic tradition to regard the stability of 'the organic whole' as steadiness of inner changes (Schelling). Here I can refer also to Hegel: "not a result is a real whole, but the result with its becoming."

Thus, we can consider the dissipative structures like Benard ones as the wholes, but can we regard the elements of medium as the parts of the wholes?

To clarify this question, let us attend to the experience of biology, which already applied the system approach to the live organism traditionally considered as a whole. This approach represents the organic whole as a functional system and associates the parts of a whole with certain subsystems, which realize certain functions in the whole. Thus in biology a difference between parts and elements is assumed. In this respect not such elements of living matter, as cells, play the role of parts of the organism as a whole, but rather organs or their systems.

Physics and chemistry usually considered as an element the components to which a system can be decomposed through the methods of physical or chemical analysis. Classical science does not deal with system as a whole and regards only the conditions of a system's stability (the energy of internal interactions exeeds the energy of outside disturbance). Irreducible integrity of systems appears within the limits of quantum mechanics applicability. However, the notion of a part here is not used or, if it is, it is identified with the notion of an element.

Hans Reichenbach considered physical and biological systems from the united position, when he wrote about the difference of substantial and functional genetic identity in his book "The Direction of Time." Organism interchanges its material substance and can be regarded as a functional but not substantial identity. But substantial identity is idealization in general. It becomes evident from the quantum mechanics point of view (corresponding to principle of elementary particle's identity) and any physical system saves only functional identity.

Dissipative structures are the open systems which change energy and matter with medium. It is naturally to consider a dissipative structure as a functional genetic identity (let me remind that the live organism is also an open system and can be considered as a hierarchy of dissipative structures). But the functional genetic identity of a system presupposes the identity of the elements with which it interchanges with medium, meaning their interchangeability.

However, the philosophical tradition defines the whole as the unity of variety (different variety, as Hegel admitted, i.e. parts). Thus if we want to think about dynamically stable self-organizing systems as the wholes (and in this case only we express the reductionistic explanation known to be insufficient), we have to discuss the problem of distinguishing the parts of dissipative structures not identifying them with elements of nonlinear medium, where self-organization takes place.

In a whole live organism as in hierarchy of dissipative structures certain dissipative structures realize certain functions and play the role of part of the organism.. There are many models of self-organization processes in living systems. As example I can mean the self-organization of ion transport through the membrane of a live cell in organism, model of palpitation of the heart as auto-oscillation, spread of autowaves in cortex and so on. But the relationship of dissipative structures as themselves to their elements and parts is the same problem for biological mediums as for any others.

When self-organization forms space or space-time structures like the Benard ones or autowaves, the visual image can suggest us variants of partition. In these cases we can say about the whole, which forms its parts from the medium elements in the process of self-organization and sustains its existence as periodic cooperative motion of elements in frame of self-organized whole. Such description obviously corresponds to Benard cells as dissipative structures which form the convective flows (their walls) as their parts from molecules as medium elements.

However, the association of phase portrait of self-organized system with macroscopic space-time structures is not always so simple. There is not suitable image even if limit cycles have time but not space-time projections (autooscillations as time structures). For example, what parts can we intuitively distinguish in oscillation of the size of populations in the system 'the predator and its prey'? Moreover, even if the visual image can work, conjecture about distinguishing of parts is tentative and has to be made valid in a more general way.

For this it is necessary to overcome the intuitive use of notions 'a part' and 'a whole' to using these notions in terminological correct way in frame of synergetic approach, so as the notions 'element' and 'system' are defined in system approach. Precondition for such a transition is the definition of the observer position. Let me remind that just the analysis of the observer position by Einstein and Bohr helped them to create and to interpret the theory of relativity and quantum mechanics during the previous scientific revolution.

First of all, to see the self-organized structure as a whole, one need to choose the right space-time scale. Thus, observer can see the 'honeycomb' Benard structures with coherency of the vortices only in macroscopic scale. Prigogine writes in his recent book "Time, Chaos and the Quantum": 'a demon, who would be able to observe an instantaneous state of the Beanard system would be unable to distinguish it from an equilibrium state. In both cases he would 'see' the same buzzing confusion of molecules hurrying in any direction. The coherence of the Benard structure creates a specific scale for space and time.'

In his earlier book "Exploring Complexity", Progogine also wrote about an imaginary miniature observer, who can observe the coherent moving of molecules and is able to discover the breaking of space symmetry, travelling from the vortex with clockwise rotation of molecules to another one with anti-clockwise direction of rotation. Evidently, he meant here the corresponding interval of time, long enough to observe the coherence of molecular motion and the integrity of each vortex. Nevertheless, the coherence of all the vortices in the entire volume and the changing of the sizes of vortices with changing of the size of this volume (i.e., behavior the system as the whole) can be seen only by the macroscopic observer in the right interval of time.

We do not always occupy this comfortable position of quasi classical observer as in case of observing the Benard cells on the frying pan. Often, we find ourselves in the position of that miniature observer watching the self-organizing processes in ecological or social systems as if from inside. At best, we can notice then a tendency for the coherent motion of elements. It would be integrity, rather than the entire whole and its parts.

Here we could use the analogy with the right distance we need to keep from a picture to start perceiving the portrait rather than the strokes of the painter's brush. However, this analogy works only for certain values of control parameters (temperature, or concentration, or coefficient of increment of population). There is no such stability in many natural nonlinear processes. Control parameter changes with time, and observer who watches the processes long enough can see how the equilibrium state is changed by relatively stable periodical structures. In their turn, dynamic stability of the wholes will be changed by the state of dynamic chaos. Thus all this rather looks like a moving pictures, movies, cinema.

Let me give as an example the Ferhulst's nonlinear dynamic of population (taking into account the limits of ecological environment). For small coefficient of increment of population its size increases till optimal value; for coefficient more, than 200 per cent, stable oscillation appears between two sizes, more and less, than optimal one; for coefficient more, than 245% oscillations take place between 4, then 8, then 16 different sizes of population; for the coefficient more, than 257% the dynamic chaos became. (Axes abscissa: parameter of increment, axes ordinate: size of population.)

And the most important point: to understand a certain phenomenon as a self-organizing process, we need to make the theoretical reconstruction of the former. For instance, not always the flame of a candle could be understood as a solitary wave. To see the Benard cells on a frying pan does not mean to understand them as the self-organized dissipative structures as the wholes. By the way, Benard discovered his effect in 1900, but it was theoretically interpretated as an example of the self-organization only in seventyths.

Sometimes only in the theoretical reconstruction we can see the feature of integrity of structure. As an example I will give the Lottka - Volterra model for the predator - prey system. Here you can see the periodical solutions for different initial conditions. (X - quantity of preys; Y - quantity of predators.) This is a visualized representation by computer simulation of the complex behavior of this system for different values of parameters. Different colours demonstrate the fields of attraction by different attractors (yellow - for periodical trajectories, red - for invariant circle, other trajectories tend to infinity.)

To describe such structures with their features of integrity one can use not only the notions of 'a whole and parts' and 'a system and elements,' but also the notion of 'global and local' (as Prof. Peruzzi suggests), or 'central and peripheral' and so on. However, we have to apply all these notions only in theoretical way. Thus, the position of quasi classical observation as if from outside is possible, when we are able to reconstruct by the means of nonlinear dynamics the choice of evolution of a system in bifurcation points of nonlinear equations' solutions. The look as if from inside can mean the attempt to 'unpack' the bifurcation point and to find the right scenario of the going of a system out of chaos.

Just this theoretical position ensures the right application of such notion as 'a whole' in respect with dialectic tradition that understands 'a whole' as a concrete unity of variety in its becoming and transiency and dynamically stable existence. Numerical solutions of nonlinear equations with computers or computerized realization of great number of iterations give us not the general analytical description of essence and application of it to appearances, but theoretical reconstruction of concrete phenomena, that can exist in certain systems with certain values of parameters. Some of these phenomena can be understood as the existence of the wholes. I think, there are different ways of adequate notion analysis of those wholes. I am afraid, there is not the only possible decision of the problem I meant from the very beginning. Moreover, the increasing of variety of notions, capable to help us to understand the existence of the wholes of different kinds in the world of non linearity, leads to the increasing of quantity if such problems. But these are the problems of creation of adequate philosophical foundations of nonlinear science. I hope, now and here we are just taking part in this process.