Creative accent of a term “chaos” and its management
Table of contents: The Kazakh-American Free University Academic Journal №5 - 2013
Murzagalieva Ajauzhan, East Kazakhstan State University in honor of S. Amanzholov, Kazakhstan
Akhmetova Dana, East Kazakhstan State University in honor of S. Amanzholov, Kazakhstan
Bektasova Gulsum, East Kazakhstan State University in honor of S. Amanzholov, Kazakhstan
Physics came into a need first to study chaos when Brownian motion
(1827) - random motion of tiny particles suspended in a liquid or gas molecules
under the influence of environmental impacts- was discovered. It was chaos equilibrium
that classical physics studied, based on deterministic laws. Physics gave the
description of the molecular (thermodynamic) chaos without breaching the idea
of deterministic one-to-one characteristics of laws of nature. In this physics
got help from probability theory is the theory of random, non-predictable
events. It was found that the average values of the interdependence of system
performance with the chaotic state of equilibrium, still obey deterministic
laws. The theory of equilibrium does not predict (ranked and ordered) any evolutionary
transformation and it is confirmed by numerous experiments.
In natural and social sciences since the XIX century, there were
attempts to study the processes occurring under the conditions of
non-equilibrium, open systems, under conditions of intensive interaction with
their environment, in which there is a constant exchange of energy, matter,
information between the system and the environment. Experimentally, it was
found that most of these processes can occur in two qualitatively different
modes: regular (right) and random (catastrophic).For example, in hydrodynamics
it is a laminar and turbulent flow of liquid and gas; in biology - ecological
balance and environmental crisis, the stable coexistence of species and flash
excessive reproduction (locusts, rodents, etc.) epidemic; in nuclear physics -
controlled nuclear reaction and nuclear explosion, in the economy - a stable,
functioning economy and the economic crisis; in space - a stable movement of
the planets and the collapse of the planetary system, etc. .
In leap from orderly to chaotic motion of the system, researchers
saw some resemblance of physical phenomena of phase transitions, such as the
transformation of a crystalline solid with a strict order of atom arrangement
to the melt, in which there is no strict configuration of atom arrangement.
This analogy did not bring a great success, but it helped to make
two conclusions: first, that one can try to study the transitions
"order-disorder" by the theory of phase transitions. As a result
conversion modes at the bifurcation points of evolutionary diagrams of
Synergetics were called no equilibrium phase transitions [2, p. 146].Second,
existence of critical values of the external and internal parameters (one or
more), at which the transition occurs, was thought through. Determination of
quantitative parameters and their number systems of different nature - is in
every case the research task. Parameters defining the transitions are called
controls [3; 2].
The transition from laminar to turbulent flow was most persistently
investigated [4, 5, 6]. Most flows in liquids and gases are turbulent as in
nature (air movement in the Earth's atmosphere, water in rivers and seas, the
gas in the solar atmosphere, etc.), and in technical devices (pipes, channels,
streams, etc.). Therefore, the central problem in hydro- and aerodynamics,
meteorology, chemical technology is the problem of turbulence. The term
"turbulence" is perceived almost synonymous with the term
"chaotic." It is possible to confidently say that the turbulent
motion is difficult to describe deterministically; its hydrodynamic fields are
treated as random functions of coordinates and time. Naturally, in the
framework of statistical theories attempts were made to describe the transition
from laminar to turbulent flow.
Conclusion of the theory of turbulence based on the establishment of
statistical regularities of the need of a large number of degrees of freedom to
describe the chaos is consistent with the conclusion that was made earlier in
molecular physics: that the chaotic behavior of particles (molecules of an
ideal gas), in principle, can be described by deterministic laws, rather than
probabilistic, if they get a computer with enormous computing power to account
for each particle change its motion parameters with time.
From the point of classical physics view, the above examples of
chaos in natural and technological systems provide a basis to conclude the
destructive role of chaos in the process of creation. Chaos destroys the
ordering of any nature, it carries a negative impact on the structures,
including functional ones. Chaos serves as a negative factor in the evolution
of the world. This view of the chaos in Western culture has not changed
fundamentally since the first paradigms. Only Synergetics forced researchers to
take a fresh look at the chaos.
In recent decades, the synergy is actively studied a new kind of
chaos that is different from the molecular - the so-called deterministic chaos.
The name is composed of unusual combinations: determinism associated with
certainty, order, stability, and chaos - it was an accident, confusion,
instability. Nevertheless, the existence of such chaos found not only in the
theoretical models, but also in the behavior of experimental and natural
objects. Chaos was found where it seemed fundamentally impossible to be found -
in deterministic systems with a small number of degrees of freedom.
It is possible to give brief acquainted with the concept of
deterministic chaos on biological example of population growth dynamics . A
mathematical model of population growth can be represented as follows. If - the initial
population size, and - population
in n number of years and growth rate of the number of individuals for 1 year R
a constant r, then the law of the growth dynamics expressed by the equation = f ()= (1 + r) . In this version, the law of growth is linear, while
the population size grows exponentially. At long observation times the number
of individuals will increase to infinity. But for population growth there are
always limitations. Population size, filling a certain ecological niche, can
not be more than a certain maximum value of x. Therefore, P.Ph. Pherhulst in
1845 formulated the law which contains a restriction on a population growth. He
suggested that growth rate, which is dependent on the size of population, is
proportional to the (1- , ie R= r(1-), and then the
law that governs the dynamics of the Pherhulst process will now look like this:
хn+1 = f
(хn) = (1 + r) хn - r, (1)
where r - growth parameter .
In this case, the process becomes nonlinear (increase in the number
of individuals +1 at time n
+1 depends nonlinearly on the number of individuals at previous
point of time n) and radically changed its dynamic behavior.Growth parameter
rink this example of one-dimensional nonlinear dynamic process plays a role of
a control parameter. Its change in particular will determine the evolution of
the system. In a detailed mathematical analysis of the dynamics of the
nonlinear iterative process the nature of its stability regimes is investigated
. Figure 1 shows the evolutionary changes of the system dynamics
(population) for certain values of r. The vertical axis represents the value of
x (t) - population size at certain times.
а) r = 1,8 б) r = 2,3 в) r = 2,5 г) r = 3
1 - The effect of the control parameter r of formula(1) on the nature of
population x (t)change over time [7, p.40]
With the r=1.8 population size though increasing at first, reaches a
maximum value of X, which conventionally is taken as one. As can be seen from
the graph (Fig. 1a) , the value of X is stable - the population retains its
strength, but with increasing r, caused by the influence of the environment
(change in the quantity and quality of feed, the number of enemies, appearance
or disappearance of epidemics , etc.), the state with a constant value x (t)
becomes unstable and at r = 2,3 the system enters anew state: there are periodic
oscillations in it between the two values of x (t) ( Fig. 1b). At the value of
r = 2.5 process comes to a new stable oscillations of the values of x (t) with
a period of 4, which means that period has doubled in the system –with which
came another mode of behavior of the system (Fig. 1c). Further increase in r
leads to an unexpected result: the r 2,57 process
ceases to be periodic - system becomes chaotic(Fig. 1d) . Relatively simple
deterministic model spawned behavior that does not have any obvious regularity
- spawned chaos, which is called deterministic.
From this example follows not less methodologically significant
conclusion of a scenario in which the order turns into chaos. In the literature
it is called period-doubling scenario.
In a more complex dynamic models, representing a system of 2-3
nonlinear differential equations,above mentioned scenario is also observed.
Currently several scenarios of transition "order-disorder" in
mathematical and experimental models are described [2, 3].
In addition to the aforementioned features of deterministic systems
(the appearance of chaos in which the transition from order to disorder
complies a particular scenario) in some of them chaos found to have a peculiar
order. For these systems, chaos order can be visually identified by the
abstract phase space system. Search for matching coordinates of the phase space
is often a difficult mathematical problem. Its coordinates may be either a
single characteristic of the system, or several, with a certain mathematical connection.
Theorist-meteorologist Lorenz first discovered the dynamic mode with the
orderly chaos in 1961, when he tried to describe the hydrodynamic flows in the
Earth's atmosphere using a non-linear system of three differential equations,
which in the literature on the synergy became known as a Lorenz model. The
system had the peculiarity that in the phase space it had an unusual type of
attractor. Attractor - a geometric object in the phase space, which attracts
the phase trajectories. It can be a point of a closed curve, torus for different
states of the system. But in the Lorenz system it was located in a limited
volume of the phase space, in which all the phase trajectories were
concentrated. This attractor is called strange attractor. To date several types
of strange attractors were discovered (Lorenz attractor, O. Ressler, M. Henon,
R. Show and others).
In order for the strange attractor to exist, the system must possess
the phase space with not less than three dimensions. Presented in Figure 2 are
the attractors of chaotic systems in three-dimensional phase space.
Chaos, which is described a strange attractor, is not the result of
accidental exposure to unpredictable environmental factors or for quantum
mechanical systems - the result of the Heisenberg uncertainty relations. Its
origin is related to an intrinsic property of a dynamical system. In nonlinear
systems (they in particular exhibit strange attractors) two closely spaced in
phase space figurative points (two almost similar system state) in a short
period of time move almost parallel, but over time their paths diverge
exponentially. As strange attractors occupy a limited region of phase space,
the evolutionarily divergent trajectories cannot last indefinitely, moving away
from each other. Attractor should form folds inside, while trajectories will
fold according to certain rules. As a result compact coil of randomly mixed
trajectories is formed. Details of this process are described analytically and
geometrically . Geometry of strange attractors has a fractal (self-similar)
The chaotic character of a strange attractor is observed with the
pictorial point (state system) randomly jumping from one region of phase space
to another. Within the scope of the attractor pictorial points motion is
chaotic. Orderliness of chaos manifested itself in the fact that the phase
trajectories are not distributed over the entire phase space, but are grouped
in the borderline of its domain.
Fourier analysis of strange attractor chaos allows for each
attractor to define its fairly wide frequency band width, not infinite, as with
white noise - a truly random process without any ordering. Using Fourier
spectra it is possible to distinguish the chaos of the strange attractor from a
white noise, as well as to determine in a complex stochastic process stochastic
process corresponding to a strange attractor.
Processes corresponding to the strange attractors are found in the
convective fluid flows, in fluctuations in the concentration of substances in
chemical reactions, in the reduction of heart cells of a chicken, in
oscillatory processes in electrical, radio, mechanical installations [3, 8]. In
dynamic systems of a strange attractor, there exists a "strange
relationship" between chaos and order: they are closely intertwined with
each other, they cannot be separated from one another, cannot oppose each
other. They are inherently simultaneously present in one and the same object,
in the same mode. Chaos mode with a strange attractor is not similar to the
thermodynamic (molecular) chaos. Here he has a new image, new features.
Parallel branch in the description of chaos theory is the theory of
disordered spatial distributions fields (electron density distribution in
crystals, the density of matter in galaxies, etc.) . This theory uses the
same basic concepts as in the theory of dynamical chaos , such as a dynamic
system , the phase space , the phase trajectories and others with relevant content,
which should help in describing the spatial arrangement of the elements of irregular
or complex object structures . In this theory, a new concept of a finite spatial
disorder appears, which is defined by the authors of the work  qualitatively
and quantitatively. From the perspective of this concept molecular chaos seems
to be an infinite spatial disorder.
Those can be called a disorder in the arrangement of some of the
structural elements of a rigid body in equilibrium: the distribution of defects
in the crystal structures, magnetic domains in ferromagnets, liquid crystals -
the orientation of the molecular axes. To the description of spatial disorder
authors came from nonlinear dynamics point of view, not from the standpoint of
statistical concepts, as it always has been. In this theory model nonlinear
differential equations (Swift-Hohenberg , Ginzburg -Landau theory) are used
with a small number of degrees of freedom with which in particular cases it is
possible not only set the spatial distribution of elements in a given time, but
also to trace the evolution of the distribution in the unstable states system.
These examples emphasize, first, the universality of chaos, its
inevitable presence in all material systems, and secondly, it reveals a
hierarchical structure, tracing how it manifests itself at different levels -
from the micro to the macro level, and thirdly makes note of the fact that
chaos may be a manifestation of temporal and spatial.
As a result, a close analogy between the temporal (dynamic) and
spatial disorder can speak of a finite disorder (provisional) in dynamical
systems and synergy.
"Why is the system, developing on certain laws, behaves
chaotically? Influence of extraneous noise sources, as well as quantum
probability in this case is not relevant. Chaos is generated by its own dynamics
of a nonlinear system: its property to exponentially develop arbitrarily close
trajectories. The result strongly depends on the shape of the trajectories on
the initial conditions "[10, p. 46] - the emphasis here is immanent chaos
for nonlinear systems.
"The strange attractor is a very stable structure, - writes the
same author - despite the extreme sensitivity of the individual trajectories.
Dynamic chaos can be regarded as a model of disorder, but at the same time and
as of a stability, orderliness on different levels, in different scales"
[10, p. 49-50] - just like stability and instability, chaos and order flexibly
combined in one and the same process. As a generalization and demonstration of
the presence of chaos in the form of material and ideal systems can serve
proposed a scheme that reflects the universality of chaos, its multi-level
character (reflecting the hierarchy of levels) and the nature of chaos at
different levels (time and space , of course - and infinite). This diagram
shows the new natural- scientific ideas about the forms of presence of chaos in
natural systems, which are partly mentioned in research works [12-11].
Since synergy brought in new scientific ideas about chaos, a new
status of a chaos in natural systems is expected. In the fundamental work of
the founders of Synergetics I. Prigojine and Haken, attention was drawn to a
natural phenomenon, in which from chaos ordered structure is born[13, 14, 15].
In other words chaos acts as a necessary condition for the birth of an order,
as up building order of the beginning. They and their colleagues proposed a
series of mathematical models to describe the ordering processes. Possibility
of generation of order out of chaos can be seen from an example of Benar cells,
known since 1901. They can be observed experimentally under simple conditions
in relatively thin layers (from a few millimeters to a few centimeters) for
liquids with sufficiently high viscosity (in whale fat, in oils - engine,
automotive, silicone, vegetable, etc.).
Scientific concepts used in the construction of theories, models,
methods, many highly specialized disciplines (natural, human, social).
Currently chaos as a scientific concept took place scientific concept. The
proof is its extensive use in modern fundamental scientific constructs
(physics, mathematics, biology, etc.) and applications (in medicine, electrical
and radio engineering, physiology, etc.) scientific concepts can acquire the
status of a philosophical category, when it will be used in the theories of the
highest level of generalization, ie in worldview.
Currently, the concept of "chaos" is developing towards
the acquisition of the status of philosophical categories. Comprehension goes
properties matching concept main features of philosophical categories and their
relationship with the properties of the elements of the conceptual apparatus of
1. Kurdiumov SP, GG Malinetskii Synergetics -
self-organization theory / / Computers, models, computer experiment. - Moscow, 1988. Pp. 79-136.
2. Akhromeeva TS, SP Kurdyumov, Malinetskii GG,
Samara AA Unsteady structure and diffusion chaos. - Moscow: Nauka, 1992. - 544.
3. P. Berger, Pomo I., Vidal K. Order in chaos:
About deterministic approach to turbulence. - New York, 1991. - 368.
4. Turbulence: Principles and Applications. / Under.
Ed. W. Frost, T. Mould . - New York: Wiley, 1980. - 535.
5. LD Landau Collected Works .V.1. - Moscow: Nauka, 1969. - 511.
6. Physics. Encyclopedic Dictionary. -M.: Great
Russian Encyclopedia, 1999.
7. Paytgen HO, Richter PH The Beauty of Fractals:
Images of complex dynamical systems. - New York, 1993. - 176.
8. Crutchfield JP, Farmer, JD, NH Packard, R. Shaw
Chaos / / In the world of science , 1987. Number 2. Pp.
9. MI RabinovichFabrikant AL, LS Tsimring
Finite- dimensional disorder. / / UFN, 1992 .T.162. Number
8. - 176.
10 .Dmitriev chaos, fractals, and information
technology / / Science and Life. 2001. Number 5. Pp. 44-52.
11. Bektasova GS Methodological aspects of the
synergetic approach to the role of chaos in the evolution of natural systems. /
/ Materials of the international scientific-practical conference "Amanzholovskie
Readings - 2007" - Ust-Kamenogorsk, 2007. Pp. 148-150.
12. Bektasova GS About a modern interpretation of
chaos in the evolution of the relationship of natural systems. "Science
and Education of Kazakhstan on the threshold of the third millennium" / /
Abstracts of the conference: - Ust-Kamenogorsk, 2000. Pp. 11-12.
13. Klimontovich NY Without formulas of synergy. -
Mn., 1986. - 223 p.
14. G. Nicolis and I.
Prigogine, Self - organization in no equilibrium systems. - New York: Wiley,
1979. - 512.
15. Prigogine I. From Being to Becoming: Time and
complexity in the physical sciences. - Moscow, 1985. - 327.
Table of contents: The Kazakh-American Free University Academic Journal №5 - 2013