Conceptions, role and chaos opportunities in dynamical system theory

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 short-term 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 long-term 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 large-scale systems should be looked for in other. One large-scale 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 thunder-storm. 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 "happy-go-lucky" 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 long-term forecast. In the presence of chaos reliability of forecasts quickly falls.

The concept of chaos belongs to the so-called 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 backwards-forward, 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 three-dimensional 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 so-called 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 long-term 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 long-term 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.

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