Iryna Dobronravova

Leonid Finkel

Dynamic Chaos in Ukrainian Society as a Medium of Social Self-Organization.

A synergetic consideration of social self-organization has demonstrated its productivity in political theory, sociology and economy during the past years. [Bevzenko, 2002; Urry, 2003; Fuchs 2003, Kapiza, 1997] However, this consideration is not always provided with understanding that self-organization as a becoming of integrative complex structures takes place just in chaotic media. Furthermore, there are different kinds of self-organization within different kinds of chaos. [Peitgen, Richter, 1986]] Human strategies in attempts to provide the desirable alternatives of self-organization or to avoid the dangerous ones are special for different kinds of chaos and different phases of self-organization. Therefore, it is so important to be able to recognize if a social medium is chaotic and to distinguish what kind of chaos takes place: statistic or dynamic one.

Statistic (thermodynamic) chaos exists in linear media, for example as thermal chaotic motion of molecules in a gas or in a liquid. The suitable image within a social medium could be numerous people walking lazily. A scientific description of such media is connected to statistics: with the applicability of average means of parameters and with the fast destroying of fluctuations (small deviations from average means) by chaotic motion of elements of the medium (molecules or idlers).

If there is a source of non-linearity inside or outside of a medium, in other words, if there is a way to move a medium to a far-from-equilibrium state (to heat a liquid, to cry “Fire!” in a crowd), the medium becomes non-linear. Then a choice by chance among two (or few) possible variants of collective behavior of elements happens, i.e. self-organization takes place.

Dynamic chaos differs from statistic one by previous self-organization, it is one of the phases of self-organization. There are complex structures with scale invariance (fractals) in such a chaotic medium.

It will be shown in this article how to interpret in a synergetic way the self-organizing structures visualized with mathematical processing of data of election, and by which mathematical criteria it can be conceived that a social medium is in a state of dynamic chaos.

Now let us consider the visualization of performed choices in a space of choice during the Ukrainian parliament elections in 1998 and 2002. An analysis of the election results with methods of visualization of multidimensional measurements [Ajivazyan, 1989] have found out that a certain kind of distribution of points in space of choice is steadily replicated.  In other words, there is a specific geometry of such a distribution.


We mean that the candidates of choice (candidates, parties and blocs of parties) settle the space of choice and are regarded as coordinates of this space.  The coordinates are naturally ordered by quantity of the voters in favor of the candidates – p1, p2, p3… We consider as a point of choice the choice performed by one group of electors (this group is defined by the administrative principle, connected with an election campaign). This point of choice is defined by coordinates in the space of choice performed by this group. We regard the configuration of relative position of the collection of these points as geometry of their distribution. Besides, the scales are established in order to consider the results of the election. In our case they are:


·        election in 1998 by list of parties in main administrative territorial units of Ukraine (27 units at all – 24 regions, Autonomic Republic of Crimea and cities Kiev and Sevastopol)


·        election in 2002 by list of parties in main administrative territorial units of Ukraine


·        election in 2002 by list of candidates in electoral 150th district  (200 polling-districts of 150th district)


·        election in 2002 by list of candidates in electoral 151st district  (200 polling-districts of 151st  district)


·        election in 2002 by list of parties in electoral 150th district  (200 polling-districts of 150th district)


Every point of choice (representing a group of electors) is defined by the quantities of those who have voted in favor of one of the candidates from the series p1, p2, p3…

Thus, we regard the points of choice as points in the space of coordinates of choice: a group of electors (for example, a region or district) – is represented by one point. First of all, these points are considered (visualized) in a two-dimensional space of the two candidates with the highest rating (p1, p2), then we consequently increase the dimensions of space – (p1, p2, p3); (p1, p2, p3, p4)…

Surely, for dimensions greater than three (N>3), we cannot anymore observe distribution of points directly. To look at geometry in such spaces we attend to mathematical methods of visualization of multidimensional measurements. These methods are able to project the points on the plane of observation, saving geometry of relative disposition of points, (for such projecting close points stay close and far distant points stay distant in a sense of Euclidean metric). The coordinates introduced are computing as the weighted sums of initial values of coordinates and are interpreting as computed associations of coordinate of choice.


Distances between points are naturally interpreted as measure of similarity (difference) in the choice performed. The closer the disposition of points (less distance between points of choice in space of choice), the more similar the choices are. Farther disposition of points means less similarity or more difference in the choices performed.

The main consequence we have got as a result of this visualization of relative disposition of points is that the geometry of relative disposition of points steadily saves certain peculiarities: there is a core with two branches coming out of it, and they are co-oriented with the coordinates (see graphs at the end of the text). The branches are naturally interpreted as the main directions of choice (Graphs 1-6).

There is a theoretical representation of geometry of catastrophe of pleat kind on Graph 7 [Poston, Stewart, 1978]. Such geometry of relative positions of points of choices in the space of choices reproduces on all levels of consideration. Below we enumerate mathematical properties of the found phenomenon. All the statements about them have strict mathematical interpretation in terms of mathematical statistics. These properties altogether constitute the reason to understand the discovered phenomenon (geometry of distribution of points of choice) as appearance of the fact that the social medium in process of election is in the state of dynamic chaos. These properties are:


·        In a certain formal sense the geometry of distribution is not dependent on social and economic peculiarities of the territory for which the point of choice is defined (region, district, polling district). The geometry of distribution reproduces steadily on all observable levels, appearing features of fractal geometry. Though this geometry is not directly connected with social and economic features, this interpretation could be given for a few regions. These are the regions with traditionally bright-appeared political orientation (for example, west Lviv and east Donetsk regions). However, consequent disposition of regions along the branches does not correlate with the social and economic indexes. Moreover, similar geometry of distribution of polling districts inside of different districts could not be understood in terms of social and economical terms.

·        Stability in the appearance of geometry of performed choices for different schemes of elections (for the election by list of parties and for the election of a certain person as a candidate on a single-mandate district) 

·        The presence of one core and two branches and a normal logarithmic distribution along the direction of these branches, which is similar to the geometry of bifurcation.

·        Continuity in transition (ordering) of points belonging to each of branches. This continuity can be represent as a consequence of the


disposition of the points along the branches – from the core, where preferences are balanced to the ends of branches, where preferences themselves appear in most contrasting way.

·        Invariance of general features of the geometry during the sequential raising of the dimension of the space of choices (from the first two candidates by rating to the first N>2 candidates). Besides during the raise of dimensions the geometry of branching gets features, which are typical for the geometry of pleat.


In the above we have formulated the arguments in favor of our statement that a medium of realization of choice during election in Ukraine could be mathematically defined as a medium which is in a state of dynamic chaos.

Complexity of interpretation in this case and necessity to attend to synergetic methodology and its philosophical foundations are connected to the fact that the point of bifurcation and the following branching cannot be seen as the branching of chronologically unfolding variants of events. Poly-variance was discovered for the one-moment cut of realizing of choices by the electorate of Ukraine in the parliament elections of 1998 and 2002 [Dobronravova, Finkel 2003]. Then it was interpreted as an indication that Ukrainian society has become a whole, with understanding of integrity as unity of diversity. Below we will show on what methodological and philosophical foundations our interpretation was based.

Methodologically, synergetic models are needed on a stage of theoretical interpretation to process empirical data and to transform the protocol statements in scientific fact. Besides, scientific fact can be expressed as a statement about reality only in an adequate context. This is the context of foundations of science: scientific world picture, norms of scientific researches and the philosophical foundations of science.

The effectiveness of synergetic models in the area of life science and social science is connected with the synergetic capacity to grasp the definitive features of life and society: integrity and ability for self-organization. Understanding of integrity in philosophical foundation of science is most important because self-organization itself is a becoming of the new whole [Prigogine 1980]. In Synergetics the integrity is regarded as a process. It could be a process of a self-organization as the becoming of a new whole, making its parts out of elements of a non-linear medium (the becoming of the parameters of order) [Dobronravova, 1990, 1997, 2003]. It could be a dynamically stable periodical process of reproduction of a self-organized, self-sustained whole (the formation of the limit cycle in phase space, created by two periodically moving parameters of order). It could be a transition to chaotic behavior by a parameter of order.

Altogether, the phases of complex system evolution mentioned above create the unity through diversity by themselves, because of different


features of this process on its different phases. Let us remind the high sensitivity of a system in the bifurcation point to any small accidental influence, which can define its further destiny. Nonetheless, on the next phase a much more intense disturbance cannot destroy the dynamically stable existence of a self-organized system.

At the same time, though the choice in the bifurcation point is a choice by chance, it is a choice between two certain possibilities (a set of a few certain possibilities for critical point in general). These possibilities are defined by attractors of the non-linear medium. And, just the appearance of such a choice is by itself a sign of the integrity of the medium, on which the self-organization takes place. This choice emerges together with the emergence of a parameter of order and just for it.

As a rule they speak about the so-called “far order” (Prigogine, 1980), characterizing the long-scale fluctuations. These are not those small fluctuations as the derivation from average values for microscopic characteristics of medium elements, which are destroyed by collisions between near-by neighbors in their chaotic movement, that provide respective macroscopic parameters of previous state, described by thermodynamic curve till critical bifurcation point. There are no average values in the critical point at all, only two possible variants of coherent movement of the elements, corresponding to different values of a becoming parameter of order.

Thus, integrity is an immanent feature not only of a self-organized system, which comes into being as a result of historical choice of either variant of coherent movement of non-linear medium elements. The formation of inherent in the non-linear medium variants of choice, the emergence of a set of possibilities for the becoming parameter of order is a sign of integrity in synergetic way.

So, integrity and presence of alternates do not exclude, but pre-suggest each other. It is more evident for media, on which both variants of choice can coexist. For each element of a medium only one choice is carried out (one choice between vortexes for the molecule, one between political parties for the citizen), but all possible variants can be realized on the entire medium. It is important that not any possibility exists within a non-linear medium, but strictly certain ones, and this is what the emergence of integrity means.

By the way, we cannot help to associate such understanding of integrity with a similar one within quantum mechanical systems. In both cases, we can theoretically reconstruct the set of possibilities, characterizing the system as a whole. In both cases, the possibilities themselves appear in further measurements or in further realization of non-linear dynamics. In both cases, we have to deal with a choice by chance. However, in quantum mechanics the choices are within the limits of statistical laws, whereas


synergetic events of choice exist besides the regularities and define the choice between the regularities.

 The integrity, characterized by the presence of a parameter of order, saves its meaning within all phases of the self-organization processes. Even after the transition of a parameter of order to chaotic behavior, it has remained the parameter of order [H.Haken, 2000]. It continues to define the coherent movement of many elements of the medium, though on the surface of being we can watch the destroying of previous integrative structures. Moreover, complex systems (fractals) are forming only within dynamic chaos. So, the initial point of emergence of novelty is the emergence of the medium integrity. It appears as an emergence of a set of possibilities for further choice by long-scale fluctuations.

The human strategy for sustaining of preferable possibilities, of preferable variants of self-organization, has to be special on different stages of self-organization. The most preferable variants are those of relatively stable existence of structures. There are two main kinds of dynamic stable self-organized structures: open dissipative structures and fractals. As to the ecological and economic situation, it seems that it is possible to sustain the dissipative structures rather locally. Globally, mankind can hope on complex fractals as result of concurrence of different attractors. To consider the variants of dynamic stability in the political medium on the instance of elections in Ukraine we will make a few preliminary assumptions.

To discuss self-organization in non-linear medium adequately, it needs to define what medium is meant to take into account the specific level in the hierarchy of self-organization. Then we can specify the transition processes on one level, keeping homeostasis on the other one. Then we can consider control parameters and find the ways of influence on the process of self-organization on one level from other level.

In reality, it seems natural to regard separate human beings as elements of a social medium and to connect non-linearity of such a medium with human mental states, emotions, interests and so on. It is absolutely true if we are talking about the basic level of self-organization. However, even here we can not help to take into account historical and cultural definiteness of such mental states, emotions and interests. The subject of panic, tension or indignation in one crowd will not disturb the other crowd.

For higher kinds of social self-organization, the elements of a self-organizing medium are political parties, states and other forms of previous self-organization. So why do political processes in post-socialist countries not look like processes in western democracies? It is because these processes take place on different media. Self-organization of political parties creates elements of the medium in which the desirable self-organization is principally capable – in a hierarchy of media elements


of the meso-level based on previous self-organization on the micro-level and exist in conditions created by the mega-level. Taking this into account we can distinguish states of these media and consider self-organization on them in a proper way [Budanov, 2000].

The scale invariance of visualized structures discovered all over the country and also on discrete districts chosen by choice demonstrates that in 1998 and 2002 in Ukraine we have to deal with dynamic chaos at least within the medium of political self-organization.

Habitual evaluation of consequences about integrity and chaos are contradicted: it seems that integrity is good and chaos is bad. However, they are two sides of one coin. A really discrete set of possible choices is settled by previous political self-organization. The branching on the political medium constituted by political parties does witness that the country is in vicinity to a point of bifurcation. Chaos can not be canceled, so far as only chaos is the condition for order.

Thus, we have to deal with scientific facts, which are achieved as a result of synergetic interpretation of mathematically processed data about the elections. This mathematical processing operates with methods, which do not distort the data but discover the structures about which the data implicitly witness.

A further political interpretation of visualized structures of data has to take into account the disposition of regions along the branches. Some facts seem to become obvious in these graphics: the western regions are rather for “Our Ukraine”, whereas the eastern regions are rather for the Communists. However, the discovered scale invariance of the distribution shows that a primitive division by region is unacceptable. Ukraine became politically integrative in the sense of unity of diversity of the elections realized on relatively small districts in different regions. We would like to stress that integrity could not be reduced to unity as monotony; that was a thinking habit in the times of the totality regime. Integrity is unity through diversity; this is habitual for the image of a normal democracy.  

As to importance to differ kinds of chaos, we show how this distinction works for the evaluation of the role of the Communist Party, which saves its place in post-totalitarian political space of Ukraine. It is a pity, that Ukraine did not exclude Communist parties, at least temporally, from the political life, as other post-socialist European countries did. For a long time, the danger of returning to the past is conserved because the supporters of the past were politically present. Such a situation can be possibly interpreted as staying in the neighborhood of a first bifurcation where chaos does not condition the order yet and where the smallest disturbance can turn the country back.


Citizens of Ukraine and Russia very well remember how politic technologists used their misgivings to make them vote for presidents that they did not like anymore.

We think that by the use of mathematical methods, this analysis of the features of dynamic chaos in the political medium of Ukraine witnesses the few steps of political self-organization that are realized already. We think that we are now in close neighborhood of a bifurcation that it is comparable to those that emerge on certain phases of nonlinear chaotic dynamics. Unfortunately scenarios of self-organization are not obligatory favorable.

Dynamically stable structures emerge within chaotic media on the frontier of concurrency of different attractors. The “win” of one of them is a blow regime, which is not useful for their supporters. A balance of interests (but not their equilibrium) creates the complex structure of political life.  Long before Synergetics, the fathers of democracy and defenders of freedom understood it, when they fought for rights of minorities with which positions they did not agree. Devotion to this understanding creates a medium in which democratic structures originate.

The experience of democratic movement we have got in time of the velvet “orange” revolution in Ukraine. We felt us as participants of this social self-organization. Patience and tolerance are immanent to Ukrainian people and the process of the political self-organization was peaceful and fruitful. Now democratic Ukraine is on the way. We know that only democracy creates a political medium on which complex structures can emerge and be stable. The feeling of dignity made millions of men and women of different age and social status to attend political meetings and to claim the political strike for civic resistance. You could not notice any aggression on the democratic meetings, only inspiration and gladness for great human solidarity.

Specificity of social self-organization is that the thoughts and feelings of people, their cultural habits and moral references work as the control parameters of self-organizing processes. Exemplary behavior of people in Maydan was possible as a phenomenon of social self-organization, supported by moral law. This experience of tolerance to other opinion and understanding of its necessity is just an experience of democracy. Synergetic theoretic model of this non-linear situation became a rational foundation of pragmatic reasons to embody democratic values in life.

Let us come back to objective analysis of president election results in third round with method described above (first and second rounds were falsified and cannot be a subject of scientific consideration). Visualized structure of these results is represented on Graph 8 in the end of the article.

Relative position of Administrative Units of Ukraine in respect of election one or the other candidates in space, defined by these candidates, looks like preceding graphs. There is one difference, which regards the appearance of


two additional branches, “arising” from core, common for many administrative units of Ukraine.

We suggest to interpret this change as a step in becoming of complex fractal structure in area of concurrence of attractors. These attractors are connected with former existence of Ukraine in different empires: Austrian – Hungarian one and Russian one (last seven decades it was Soviet one). Their heirs, European countries on the west and Russia Federation on the East continue their cultural and social influence on Ukrainians now. Respective administrative units of Ukraine (Lviv and Donetsk with their most definitive choice of Jushchenko and Janukovich demonstrate this influence especially brightly. However, there was not unanimous choice in these units. What more various choices are in central and south units.

Further destiny of Ukraine as integrative in diversity entity needs to have a concurrence of above-mentioned or other attractors in free informational and democratic political space. Synergy of all various creative forces of Ukrainian people can prolong the self-organisation of complex fractal structure and its dynamically stable existence as a mature political nation. This synergetic model of events in context of democratic and national values is a ground of our hope for prosperity in democratic Ukraine.

There is an opportunity of such favourable scenario of development in Ukraine, however it is not only possible. Moreover, even on the way in right direction bifurcations are very probable. Really, there are “windows of transparency” in circumstances of dynamic chaos, when cascade of bifurcations can be observed again like as during penetration in dynamic chaos firstly.

The question arises: how can such cascade of bifurcations be found with method of visualization considered here? After all, we have the one moment cut of electoral will each time.  Nonetheless, unfolding of process in time you can watch if to compare respective graphs for election of different years. During this comparison of Graph 8 with Graphs visualised the results of parliament election in 1998 and 2002 years you can see the new bifurcation on both branches of the Graphs 1-6. There are two branches arising from the point of Micolaiv unit: one branch with Zaporozje, Odesa, Charkov and Dnepropetrovsk administrative units and other branch with Avtonomic republic Crimea, Lugansk and Donetsk units.  You can admit also the bifurcation of branch, leading to Lviv in point of Ivano-frankivsk to two branches to Lviv and Kyiv. We can watch further development of Ukrainian society visualising with the same method the results of parliament election in 2006.



Graph 1.

Relative position of Administrative Units of Ukraine in election to Supreme Council (Ukrainian parliament) in 2002 

Administrative Units are designated by numbers (see Table 1)

Space of selection is defined by two parties first by quantity of votes “for”

p3 – electoral “Victor Jushchenko’s bloc of parties “Our Ukraine”

p1 – Communist Party of Ukraine











Graph 2

Relative position of Administrative Units of Ukraine in election to Supreme Council (Ukrainian parliament) in 1998 

Administrative Units are designated by numbers (see Table 1)

Space of selection is defined by two parties first by quantity of votes “for”

s2 – “ People Movement of Ukraine” (“Ruh”)

s1 – Communist Party of Ukraine



















Graph 3

Relative position of Administrative Units of Ukraine in election to Supreme Council (Ukrainian parliament) in 2002 

Administrative Units are designated by numbers (see Table 1)

sp1, sp2, sp3 – associations of parties and blocks of parties – these are weighted sums of quantity voted “for” one or the other party or bloc

(evaluation of coefficients of weighting have been carried by regressive method of estimate of factor values)





















Graph 4

Structure of relative position of Administrative Units in election to Supreme Council (Ukrainian parliament) in 1998  

Administrative Units are designated by numbers (see Table 1)

sp1, sp2, sp3 – associations of parties and blocks of parties – these are weighted sums of quantity voted “for” one or the other party or bloc

(evaluation of coefficients of weighting have been carried by regressive method of estimate of factor values)





















Graphs 5,6

Relative positions of two hundreds of poll-districts in 151st district in space of choice of single mandate district and by list of parties in election to Supreme Council of Ukraine in2002. It is the same for both variants of election (let us remind that there was mixed system of election in Ukraine in 2002: by list of parties at all districts and choice of one of candidates at single-mandate district).

As1 As 2 – associations of two first by rating candidates of choice 

(evaluation of coefficients of weighting have been carried by regressive method of estimate of factor values)
























Graph 7      

Linear image of singularity (catastrophe) of pleat kind

























Structure of relative position of Administrative Units in 3rd tour president election results in 2004 in Ukraine


Ju3 – quantity voted for Jushchenko

Ja3 - quantity voted for Janukovich








Table 1

Numbers of Administrative Units of Ukraine used in article on Graphs 1,2,3,4

First column of table is number and code of Administrative Unit,

Second column is name of Administrative Unit


1 arkr


Autonomous Republic of Crimea

2 vinc


Vinnizka Oblast

3 voln


Volinska Oblast

4 dnpr


Dniproprtrovska Oblast

5 donc


Donetzka  Oblast

6 ztmr


Jitomirska Oblast

7 zkrp


Zakarpatska Oblast

8 zapr


Zaporizka Oblast

9 ivfr


Ivano-Frankivska Oblast

10 ky ob


Kyivska Oblast

11 kirv


Kirovogradska Oblast

12 lugn


Luganska Oblast 

13 lviv


Lvivska Oblast

14 mikl


Mykolaivska Oblast

15 odes


Odeska Oblast

16 pltv


Poltavska Oblast

17 rivn


Rivnenska Oblast 

18 sms


Sumska Oblast 

19 tern


Ternopilska Oblast

20 char


Ckharkivska Oblast

21 chrs


Hersonska Oblast

22 kmel


Hmelnitzska Oblast 

23 srks


Cherkaska Oblast

24 srnv


Chernivezka Oblast 

25 srng


Chernigivska Oblast

26 mkv


city Kyiv 

27 msv


town Sevastopol


Aivazyan S.A. and others (1989) Applied Statistics. Principles of Modeling and Processing of Data. Moscow: “Finances and Statistics”. (In Russian)

Bevzenko Lyubov (2002) Social Self-Organization. Kyiv: Institute of Sociology of NANU Press.

Budanov, Vladimir, Savicheva Natalia (2003) Principles of Synergetics. In: Causality, Emergence, Self-Organization. Moscow: Nia-Priroda, pp. 167-181

Dobronravova  Iryna (1990) Synergetics: Becoming of Non-linear Thinking. Kyiv: “Lybid”. 

See also on: (In Russian)



Dobronravova Iryna (1997) Dialectic as a Means of Understanding Nonlinear Science.  In: Dialectic, Cosmos and Society. 10 (1997), pp. 7-15. See also on:

Dobronravova Iryna (2003) Emergence of Cause or Cause of Emergence? In: Causality, Emergence, Self-Organization. Moscow: Nia-Priroda, pp. 19-22 See also on:

Dobronravova Iryna, Finkel Leonid (2003) Interpretative Capability of Synergetics. In: Actual Problems of Sociology, Psychology. Kyiv: Kyiv Shevchenko University Press. Issue 1, pp. 4-12.

Fuchs, Christian (2003) Globalization and Self-Organization in the Knowledge-Based Society. In: tripleC (, Vol. 1, No. 2. pp. 105-169.

Haken H. (2000) The Main Notions of Synergetic. In: Synergetic Paradigme, Moscow:

Progress  - Tradition, pp.28-56. (In Russian)

Kapiza Sergiy, Kurdyumov Sergiy, Malnezky Georgiy (1997) Synergetics: Prognozes of Future.

Moscow:”Nauka”. See on (In Russian)

Peitgen, Heinz-Otto, Richter, Peter (1986) The Beauty of Fractals. Berlin-Heidelberg: Springer-Verlag. 

Poston. Tim, Stewart Ian (1978) Catastrophe theory and its applications. NY: PITMAN

Prigogine, Ilya (1980) From Being to Becoming, San Francisco : W.H.Freeman and Company.

Urry John (2003). Global Complexity. Cambridge: “Polity Press”. 


Cultural Context of Social Self-Organization. Proceedings of scientific conference. - Kiev, 2006