Conceptions, role and chaos opportunities in dynamical system theory
Table of contents: The KazakhAmerican Free University Academic Journal №6  2014
Author: Akhmetova Dana, East Kazakhstan State University in honor of S. Amanzholov, Kazakhstan
The huge force of
science is put into its abilities to establish connection between cause and
effect. For example, laws of gravitation allow predicting eclipses on thousands
years forward. Other natural phenomena don't give in to so exact prediction.
Currents in the atmosphere as strictly submit to physical laws, as well as
movements of planets, nevertheless weather forecasts still have probabilistic
character. Both weather, and a current of the mountain river, and movement of
the thrown playing bone have unpredictable aspects in the behavior. As in these
phenomena it isn't visible to accurate communication between cause and effect,
say that at them there is an element of chance. However until recently there
were few bases to doubt that in principle it is possible to reach exact
predictability. Was considered that for this purpose it is necessary to collect
and process information enough only.
Such point of view was
changed coolly by amazing opening: the simple determined systems with small number
a component can generate casual behavior, and this accident has basic character
 it is impossible to get rid of it, collecting more information. Accident
generated in such a way began to call chaos.
The seeming paradox
consists that the chaos is determined  is generated by certain rules which in
itself don't include any elements of chance. In principle the future is
completely defined past, however in practice small uncertainty grow and
therefore the behavior allowing the shortterm forecast, for long term is
unpredictable. Thus, in chaos there is an order: at the heart of chaotic
behavior graceful geometrical structures which create accident in the same way
as creates its handing over cards lie, shuffling a pack, or a mixer, stirring
dough for a cake.
Discovery of chaos
generated a new model of scientific modeling. On the one hand, it entered new
basic restrictions on possibility of predictions. On the other hand, the
determinism put in chaos showed that many casual phenomena are more
predictable, than was considered earlier. Information collected in the past
seeming casual and sent to shelf as too difficult, now received an explanation
by means of simple laws. The chaos allows to find an order in so various
systems, as the atmosphere, a leaking water tap or heart. This revolutionary
opening mentioned many areas of science.
What sources of casual
behavior? Brownian motion is a classical example. The mote considered in a
microscope makes the unceasing and chaotic dance under the influence of thermal
movement of molecules of water surrounding it. As molecules of water are
invisible, and their number is huge, exact movement of a mote is absolutely
unpredictable. Thus, the web of causal influences of one parts of system on
others can become so confused that the final picture of behavior will be
absolutely casual.
The chaos of which it
will be a question in article, isn't connected neither with a large number a
component, nor with their invisible influences. Existence of casual behavior in
very simple systems forces to look in a new way and at such big systems, as the
atmosphere.
Why to expect currents
in the atmosphere it is much more difficult, than movements in Solar system?
Both that, and another are made of many parts, and both submit to the second
law of Newton of F = ma which can be considered as the simple instruction for a
future prediction. If m of force of F operating on mass are known, it is known
and acceleration a. So it is happened that as soon as situation and speed of
any object are measured at some set moment, they are unambiguously defined
forever. The idea was so strong that the French mathematician of the XVIII
century Pierre Simon Laplace declared once that if for each particle in the Universe
situation and speed was set, it could predict the future on all rest of the
time. And though on a way to achievement of this purpose set by Laplace there
are the obvious practical difficulties, more than hundred years as though there
were no reasons to doubt that at least in principle Laplace is right. Literal
distribution of this statement of Laplace on the social phenomena led to a
philosophical conclusion about full predefinitions of people’s 't exist.
Views of two great
scientists of accident and probability are absolutely opposite. The French
mathematician Pierre Simon Laplace believed that laws of the nature mean a
strict determinism and full predictability though imperfection of supervision
and demands probability theory introduction. Poincare's statement anticipates a
modern view according to which as much as small uncertainty in a condition of
system can amplify over time and predictions of the longterm future can become
impossible.
The science of the XX
century finished with a Laplace determinism. The first blow to it was struck by
quantum mechanics. One of the main provisions of this theory  the principle of
uncertainty opened by Heisenberg who claims that at the same time situation and
speed of a particle can't be precisely measured. The principle of uncertainty well
explains, why some casual phenomena, such as radioactive decay, don't submit to
a Laplace determinism. The kernel so isn't enough that the principle of
uncertainty comes into force, and precisely to know processes occurring in a
kernel it is essentially impossible, and therefore however many it was
collected about it information, it is impossible to predict precisely when it
will break up.
However the
unpredictability source for largescale systems should be looked for in other.
One largescale phenomena are predictable, others  No, and quantum mechanics
here at anything. For example, the trajectory of a baseball ball in principle
is predictable, and each player intuitively uses it every time when catches a
ball. On the contrary, the balloon trajectory when from it air escapes, is
unpredictable: it heels and randomly spins at any moments and in any places
which can't be expected. But after all this balloon submits to the same laws of
Newton, as a baseball ball; why to predict his behavior it is more difficult?
The classical example
of similar double behavior gives a liquid current. Under one circumstances it
is laminar  equal, steady, regular  and is easily predicted by means of the
equations. Under other circumstances the current becomes turbulent  rough,
unstable, irregular  and difficult predictable. Transition from laminar
behavior to the turbulent is familiar to everyone who at least once flied in
the plane to quiet weather and then suddenly got to a thunderstorm. Than to explain
an essential difference between laminar and turbulent flow?
Better to understand in
what here a riddle, let us assume, that we decided to sit at a mountain stream.
Water is turned in whirlpools and laps so as if at own will that rushes there,
here. But after all stones in line with a stream strongly lie on a place, and
inflow of water is almost identical. Than casual nature of its movement is
caused?
The Soviet physicist L.
D. Landau offered in due time an explanation of casual movement of liquid which
many years dominated. It consisted that in turbulent flow there are many
various independent fluctuations (whirlwinds). At increase in speed the current
becomes even more turbulent and gradually, on one, new frequencies will
increase. Though each separate fluctuation can be simple, their complex
combination leads to movement which can't be predicted.
However concerning
Landau's theory there were doubts. The casual behavior is shown even by the
systems not differing neither special complexity, nor uncertainty. At a turn of
a century it was realized by the French mathematician Henri Poincare, having
noted that the unpredictable phenomena arising "happygolucky" are
inherent in rather such systems where little changes in the present lead to
noticeable changes in the future. We will imagine a stone at hill top. Having
slightly pushed him in this or that party, we will force it to slide down on
absolutely different ways. But, if the stone is sensitive to small influences
only when it is at hill top, chaotic systems are sensitive to them in each
point of the movement.
To show, some physical
systems how sensitively react to external influences, we will give a simple
example. We will imagine a little idealized billiards in which spheres slide on
a table and face among themselves so, that losses of energy can be neglected.
The player does one blow, and a long series of collisions begins; naturally, he
wants to know that will follow its blow. For what term the player, in
perfection controlling blow can, to predict a trajectory of a sphere which it
pushed with the cue? If he neglects even so small influence as a gravitational
attraction of an electron on the edge of a galaxy, the forecast will be
incorrect in one minute!
To show, some how
sensitively react Rapid growth of uncertainty is explained by that spheres
aren't ideal, and small deviations from an ideal trajectory in a blow point
with each new collision increase. Growth happens exponential just as bacteria
in the conditions of unlimited space and a food stock breed. With each new collision
of a mistake collect, and any even the smallest influence quickly reaches the
macroscopic sizes. This one of the main properties of chaos.
The exponential
accumulation of mistakes peculiar to chaotic dynamics, became the second
stumbling block for a Laplace determinism. The quantum mechanics established
that initial measurements always aren't certain, and the chaos guarantees that
these uncertainty will quickly exceed predictability limits. There is no chaos,
Laplace could play with hope that mistakes remain limited or at least will
grow rather slowly, allowing to do the longterm forecast. In the presence of
chaos reliability of forecasts quickly falls.
The concept of chaos
belongs to the socalled theory of dynamic systems. The dynamic system consists
of two parts: concepts of a state (essential information on system) and
loudspeakers (the rule describing evolution of system in time). Evolution can
be observed in space of states, or phase space  abstract space in which as
coordinates state components serve. Thus coordinates get out depending on a context.
In case of mechanical system it there can be a situation and speed, in case of
ecological model  population of various species.
Good example of dynamic
system  a simple pendulum. Its movement is set by only two variables:
situation and speed. Thus, its state  is a point on the plane, which
coordinates  the provision of a pendulum and its speed. Evolution of a state
is described by the rule which is output from laws of Newton and expressed
mathematically in the form of the differential equation. When the pendulum
shakes backwardsforward, its state  a point on the plane  moves on some
trajectory ("orbit"). Ideally a pendulum without friction the orbit
represents a loop; in the presence of friction the orbit twists on a spiral to
some point corresponding to a stop of a pendulum.
Fig. 1
The phase space gives convenient means for
evident representation of behavior of dynamic system this
abstract space coordinates in which are degrees of freedom of system. For
example, movement of a pendulum is (above) completely defined by its initial
speed and situation. Thus, to its state are answered by a point the planes
which coordinates are situation and pendulum speed (below). When the pendulum
shakes, this point describes some trajectory, or "orbit", in phase
space. For an ideal pendulum without friction the orbit represents the closed
curve (below at the left), otherwise the orbit meets on a spiral to a point
(below on the right).
The dynamic system can develop or in
continuous time, or in discrete time. The first is called as the stream, the
second  display (sometimes the cascade). The pendulum continuously moves from
one situation to another and, therefore, is described by dynamic system with
continuous time, i.e. a stream. Between drops from a leaking water tap it is
more natural to describe number of the insects born every year in a certain
area, or a period system with discrete time, i.e. display.
To learn how the system from the set
initial state develops, it is necessary to make infinitesimal advance on an
orbit, and for this purpose it is possible to use dynamics (the movement
equations). At such method the volume of computing work is proportional to time
during which we want to move on an orbit. For simple systems like a pendulum
without friction it can appear that the equations of movement allow the
decision in the closed form, i.e. there is the formula expressing any future
state through an initial state. Such decision gives "a way straight",
i.e. simpler algorithm in which for a prediction of the future only the initial
state both final time is used and which doesn't demand pass through all
intermediate states. In that case the volume of the work spent for tracing of
movement of system, doesn't depend almost on final value of time. So, if the
equations of movement of planets and the Moon, and also situation and Earth and
Moon speed are set, it is possible, for example, for many years forward to
predict eclipses.
Thanks to successful finding of decisions
in the closed form for many various simple systems at early stages of development
of physics there was a hope that for any mechanical system there is such
decision. Now it is known that it, generally speaking, not so. The
unpredictable behaviour of chaotic dynamic systems can't be described the
decision in the closed form. Means, at establishment of their behaviour we have
any "no way straight".
And nevertheless the phase space gives a
powerful tool for studying of chaotic systems as it allows to present their
behaviour in a geometrical form. So, in our example of a pendulum with friction
which eventually stops, its trajectory in phase space comes to some point. It
is a motionless point; as it attracts nearby orbits, it call an attracting
motionless point, or an attractor. If to report to a pendulum a small push, its
orbit will return to a motionless point. To any system which comes eventually
to a condition of rest, the motionless point in phase space answers. This
phenomenon has very the general character: energy losses because of friction
or, for example, viscosity lead to that orbits are attracted to the small set
of phase space having smaller dimension. Any such set is called as an
attractor. Roughly speaking, the attractor answers the installed behaviour of
system  to what it aspires.
Fig. 2
Attractors  it is the geometrical
structures characterizing behavior in phase space after a long time. Roughly
speaking, an attractor  it to what the system seeks to come to what it is
attracted. Here attractors are shown by blue color, and initial states  red.
Trajectories, having left initial states, eventually come nearer to attractors.
The simplest type of an attractor  a motionless point (at the upper left).
Such attractor corresponds to behavior of a pendulum in the presence of
friction; the pendulum always comes to the same position of rest irrespective
of the fact how it started fluctuating (see the right half of fig. 2). The following,
more difficult attractor  a limit cycle (above in the center) which has a form
of the closed loop in phase space. The limit cycle describes steady fluctuations,
such, as pendulum movement in hours or heart beat. To difficult fluctuation, or
quasiperiodic movement, there corresponds an attractor in shape a Torah (at the
upper right). All three attractors are predictable: their behavior can be predicted
with any accuracy. Chaotic attractors correspond to unpredictable movement and
have more difficult geometrical form. Three examples of chaotic attractors are
represented in the bottom row; they are received (from left to right) by E.
Lorentz, O. Ryesler and one of authors (Shaw) respectively a solution of simple
systems of the differential equations with threedimensional phase space.
Fig.
3
The chaotic attractor has much more complex
structure, than predictable attractors  a point, a limit cycle or Torus. In a
vast scale the chaotic attractor is a rough surface with folds. Stages of
formation of a chaotic attractor on the example of Ryesler's attractor (on the
right) are shown. At first close trajectories on object disperse exponential
(at the upper left); the distance between the next trajectories increases
approximately twice. To remain in final area, the object develops (below at the
left): the surface is bent and its edges connect. Ryesler's attractor was
observed in many systems, from liquid streams before chemical reactions; this
fact illustrates Einstein's maxim that the nature prefers simple structures.
Some systems don't stop after a long time,
and cyclically pass some sequence of states. Example  a pendulum clock which
are got by means of a spring or weights. The pendulum again and again repeats
the way. In phase space to its movement there corresponds a periodic
trajectory, or a cycle. No matter, as the pendulum is started  eventually it
will start moving besides to a cycle. Such attractors are called as limit
cycles. Other system familiar to all with a limit cycle is heart. The same
system can have some attractors. If this is so, different entry conditions can
lead to different attractors. The set of the points leading to some attractor,
is called as its area of an attraction. The system with a pendulum has two such
areas: at small shift of a pendulum from a rest point it comes back to this
point, however at a big deviation hours start ticking, and the pendulum makes
stable fluctuations.
More difficult attractor the Torus (a
reminding surface of a bagel) has a form. Such form answers the movement made
of two independent fluctuations  to socalled quasi periodic movement.
(Physical examples can be constructed by means of electric ostsillyator). The
trajectory is cast on Torus in the phase space, one frequency is determined by
time of a turnover of a small circle a Torah, another  by a big circle. For a
combination more than two rotations by attractors there can be multidimensional
Torus.
Important distinctive property of quasi periodic
movement consists that, despite difficult character, it is predictable. Though
the trajectory can precisely repeat never (if frequencies are incommensurable),
movement remains regular. The trajectories beginning nearby one from another on
a Torah, and remain nearby one from another, and the longterm forecast is
guaranteed.
Existence of chaos mentions a scientific
method. The classical way of verification of the theory consists in making a
prediction and to verify it with experimental data. But for the chaotic
phenomena the longterm forecast in principle is impossible, and it should be
taken into account at an assessment of advantages of the theory. Thus,
verification of the theory becomes much thinner procedure relying more on
statistical and geometrical properties, than on a detailed prediction.
The chaos throws down a new challenge to
supporters of a reductionism who consider that for studying of system it is
necessary to segment and study it each part. This point of view kept in science
thanks to that there are many systems for which the behavior as a whole really
consists of behavior of parts. However the chaos shows us that the system can
have difficult behavior owing to simple nonlinear interaction only several
component.
This problem becomes sharp in the wide
range of scientific disciplines, from the description of the microscopic
physical phenomena and before modeling of macroscopic behavior of biological
organisms. The huge step forward in ability in detail to understand is in
recent years taken, what structure of this or that system, however ability to
unite collected data in an integral picture reached a deadlock due to the lack
of the suitable general concept within which it would be possible to describe
behavior qualitatively. For example, having even the full scheme of nervous
system of any simple organism like the nematode studied by S. Brenner from Cambridge university, it is impossible to bring behavior of this organism out of it. The
point of view is in the same way unreasonable that the physics is settled by
clarification of the nature of fundamental physical forces and elementary components.
Interaction a component in one scale can cause difficult global behavior in
more vast scale which generally can't be brought out of knowledge of behavior
separate a component.
Chaos often consider in the light of
restrictions imposed by its existence, such, as lack of predictability. However
the nature can use chaos structurally. Through strengthening of small
fluctuations it, probably, opens to nature systems access to novelty. Perhaps,
the victim which has escaped a predator not to be grabbed, used chaotic
adjustment of flight as a surprise element. Biological evolution demands
genetic variability, and the chaos generates casual changes of structure,
opening thereby opportunity to put variability under evolution control.
Even process of intellectual progress
depends on emergence of new ideas and finding of new ways to coordinate old
ideas. Congenital creative ability, perhaps, hides for itself chaotic process
which selection strengthens small fluctuations and turns them into macroscopic
connected conditions of mind which we feel as thought. It can sometimes be any
decisions or that is realized as will manifestation. From this point of view
the chaos provides us the mechanism for manifestation of free will in the world
which copes the determined laws.
BIBLIOGRAPHY
1. Packard N.H., Crutchfield J.P., Farmer J.D. and Shaw R.S. Geometry from a time series. In: Physical
Review Letters, 1980, v. 45, N 9, pp. 712–716.
2. Abraham R. and Shaw
C. Dynamics: the geometry of behavior. Aerial Press, P.O. Box 1360, Santa Cruz, Calif. 95061; 1982–85.
3. Schuster H.G.
Deterministic chaos: an introduction. VCH Publishers, Inc., 1984.
4. Bass T. A. The eudemonic
pie: or why would anyone play roulette without a computer in his shoe?
HoughtonMifflin, 1985.
5. Dimensions and
entropies in chaotic systems. (Edited by G. MayerKress). SpringerVerlag,
1986.
6. Sinaj Ja. G. Sluchajnost' nesluchajnogo
// Priroda, 1981, № 3, s. 72–80.
Table of contents: The KazakhAmerican Free University Academic Journal №6  2014

