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In the paper is presented the model of insulated deterministic dynamic structure with the finite number of conditions. It is shown that structures of such type have attractors in the form of stationary points or cycles. Maximum attractors of such structures are describing by mathematically models, equivalent to the theory of finite groups. Different forms of presentations of under study structures dynamics are built..

We shall begin the mathematical modeling of self-organizing structures and systems from the most simplest case, which will easy be realized on the computer. Suppose that we are to create a simple mathematical model of insulated structure, which can potentially exist for infinitely long time [1].

Let collection of generalized coordinates of this structure forms certain given list of parameters, describing certain condition. For the propose that this model could be realized on the computer, number of conditions of structure, corresponding to different lists, must be equal to finite value.

We enter notion of autonomous dynamic structure (or dynamic process ) with finite number of conditions. For this purpose we shall suppose that dynamics of investigating structure is described as follows: Alongside with the finite number of lists (conditions) there is unequivocal not changing in the dynamically process operator, displaying one condition on the another.

If we introduce in consideration closed chain of such consequent displays, than the process becomes counting-endless. The autonomous structure, describштп by such model process, potentially can exist an unlimited time.

If chain of displays is organized so, that the process continues infinitely, then finally it comes on one of the lists, which was already passing once upon a time. From this step of iterations (moment of the time) process, describing dynamics of structure, begins to repeat - appears a cycle. If by certain iteration number a list is displaying onto itself, the cycle degenerates in certain stationary condition, which autonomous structure already never can leave, if anyone will not to act on it from outside.

We shall consider originally mutually unequivocal images. In this case each unequivocal function , describing image of ensemble of lists on itself (operator), has an unequivocal inverse image. If we will give to all available lists (elements of finite ensemble of lists) numbers, then the collection of transitions from one list to other, defined by display , will be equivalent to certain transposition of numbers. Hereunder, we may identify each of incorporated by us operators with the element of group of substitutions from lists (conditions). This group is in turn equivalent to finite group of order [2].

Dynamics of autonomous structure, having finite number of possible conditions and defined by the mutually unequivocal operator , is describing completely by the mathematical theory of finite groups. As is well known, each such group element, corresponding to unequivocal display , forms closed cyclic orbits. For given all lists (elements of ensemble of lists) are dividing on classes, into which elements can cyclic pass friend in the friend and are not to move over to the other class of elements. All getting thereby cycles turn out to be deterministic and strictly separate all ensemble of lists (conditions). Each cycle turns out to be autonomous and it is possible to consider that for it corresponds the dynamics of certain autonomous structure.

Reversible algorithm of transition, uniquely fixing condition of transition from one element to the another, acting on the end ensemble of lists and realizing potentially endless process, can generate under its realization only dynamic structure models, which any condition infinite number returns on its place.

Let function , being unequivocal, is not mutually unequivocal function. Then, using it on the first iterations (first step on a time) to each element of ensemble of lists, as a result of this iteration we shall receive , that number of lists, which will participate in the second step of dynamic process, will decrease. After the finite number of iterations will occur so, that -th iteration already does not reduce number of lists, and all further iterations will deal with that elements, for which function will turn out to be mutually unequivocal. Thereby, from all possible conditions of structure (lists of data) as a result of finite number of iterations will be chosen (will survive) certain collection (subset) - maximum attractor of conditions, which dynamics is equivalent to the dynamics of collection of lists, defined by the mutually unequivocal function (finite inertial set) [3]. Being narrowed on the set-attractor, function becomes mutually unequivocal.

This maximal attractor can consist of one list, and then formed by it structure can be considered as the stationary point.

It can consist also from the single cycle, including all elements of attractor, then it can be called the single attractor of the limiting cycle type. The whole collection of elements (lists), assign originally, in the result of certain finite number of iterations reconverts to the limiting cycle and can be called the basin of its attraction.

However, limiting collection of lists can form, in turn, several stationary points and cycles, each of which defines not only corresponding to it class of lists, in it participating, as well as its basin of attraction, the collection of lists (conditions), which as a result of certain sequence of iterations reconvert exactly to the given cycle (or stationary condition).

For each number of lists (conditions), equal , can be count total number of possible functions and be determined not only each of them, as well as that dynamic system, which it describes.

And consequently, theoretically or by means of computer modeling can be realized full categorization of all possible functions on the finite ensemble and all autonomous dynamic systems with the finite number of elements.

While to each mutually unequivocal function corresponds one transposition, the number of such functions is defined by the number of possible transpositions from elements and is !

Collection of kit of lists (kit of conditions) and unequivocal deterministic algorithm of transition from one condition to other forms a phase space of the model, describing dynamics of autonomous structure. Phase trajectory, characterizing the path of dynamic system in the space of conditions, depends on that condition, from which begins a motion (first iteration).

In the case of mutually unequivocal function initial condition or is a stationary point, or belongs to the cycle.

Each element of group of transpositions of finite number of lists is attractor of whole class of algorithms of compressible not mutually unequivocal displays for elements, where .

Suppose now, that dynamics of autonomous structure with not mutually unambiguous display function, is realized in the manner of certain program on the computer. For each list enter indication of corresponding to it condition -. Start our program times, assigning every time different values of . On the first step of iterations a number of a get lists will decrease. This process will last until on anyone step a number of lists, participating in iteration process, does not become equal , where upon most further reduction of number of lists stops.

Hereunder we shall receive new mutually unequivocal function from elements, rested in the dynamic process. All another lists will disappear. Staying ensemble of lists and will be a maximum attractor.

Phase space of deterministic autonomous structure with the finite number of conditions can be described also by the directed graph, which knots present conditions (lists or kits of data), but lines define a dynamics of displays. Herewith all paths of this direct graph (phase trajectories of dynamically system ) are terminated or by stationary points, or by cycles.

Each cycle has its basin of attraction. In general case all phase space is dividing into unconnected between itself directed graphs, each of which finishes by the limiting cycle (or stationary point). Collection of unambiguous chains of displays separates all ensemble of lists on limiting cycles and basins of their attraction, each of which can be considered as a model of dynamically separated autonomous structure.

Inwardly each such model, containing elements we can select paths, beginning from each element of ensemble of lists. All these paths possess that characteristic, that, beginning from certain member, self for each path, trajectory falls into attractor - a cycle, one for all paths.

Beginning of motion is defined from outside - by the external system, which defines an initial list (condition). It is possible to consider that initial list element is defined by the certain external controller [4].

"Infinity" of process is defined by the external field, in which will be assign infinite natural series of numbers, defining infinite sequence of iterations.

Thereby, even on this - deterministic consideration level with the finite number of conditions dynamic system restrain three elements, which in more complex systems are reveal more vividly.

It is possible another way of description of such type systems. Collection of possible system conditions can be presented in the manner of the condition vector, which components take values a zero and unit. To the each system condition corresponds, by such description, a single vector, which component, corresponding to present state, is an unit, but rest components are a zero. Algorithm of transition from one condition in other in this case is described by the operator in the form of square matrix of order , consisting from zeroes and units. If structure is described by reversible display function, the matrix looks thereby, that in each its line and in each column turns out to be on one unit. Consequent multiplying of such matrixes describes a cyclic process of such structure dynamics.

Dynamics of any deterministic autonomous structure, taking finite number of conditions, can be asymptotically described by certain periodic function from natural series of numbers. By the shift of zero on the number of iterations aside great values of numbers an area of its task can be extended on all whole numbers, as positive, so and negative.

For the external description of such structure is reasonable to carry in consideration external field with continual presentation of time. In this case discrete series of consequent occasions of turning a structure from one condition to other is possible to present as an step-like function from an external continual time, which possible is to consider as an attribute of field. From continual axis of time is possible to choose certain series and to each member of this series to put in the correspondence certain value , equal to the number of iterations in considered by us dynamically model.

Gaps are to be as noneven, so and even. Last means that dynamics of structure is synchronized with the dynamics of external field. If - is a constant value, number of iterations is connected with corresponding to it gap of time with the help of the single-line correlation . Last formula characterizes a synchronizing of dynamics of structure and external field. Entering of external time allows to describe simultaneously different dynamic processes in one external field.

It is possible to carry in consideration one more value, characterizing сyclic structure, defining accurate to the scale a number of cycles, which are passed by the structure from a moment of counting out beginning, which we shall call action of structure in the external field . Suppose, that for the passing of one cycle an action of structure changes on the certain constant value . Changing of an action of structure for cycles . Time of passing by the structure of one cycle defines in the case of constant value on the formula , where - is a number of steps, corresponding to one cycle. General time of passing of cycles

Whence

. (1)

If we shall suppose that at the time between steps is a smooth function from , then

. (2)

Here - is an energy of cycle.

The parameter, characterizing unceasing changing of condition of autonomous structure (wave function) may be presented in the complex form.

. (3)

Enter indication - a circular cycle frequency. Then we shall receive relationship between energy and cycle frequency

. (4)

Any deterministic structure with the finite number of conditions is cyclic and can be described by the function.

, (5)

where

- quantum of action, corresponding to passing by the structure of one cycle;

- energy of structure, defining velocity of passing of cycle;

- number of elements in the cycle;

- elementary time cell, corresponding to one step of the process.

Within the framework of our description a multiplying of and on is possible present as action of some operators

. (6)

Using to function differentiation operator on variable , we shall receive

(7)

This correlation allows, in analogy with quantum mechanics, identify an operator with the operator. In this case action of operator is possible to increase on any function from and get commutative correlation.

or

(8)

Structures trajectory can be presented also by separate points on screw spiral of single radius.

On axis of spiral line will be remitted points, which characterize a time, but length spirals characterizes in non which scale an action of structure. Attitude characterizes a value of energy of structure.

Thereby, dynamics of deterministic processes with the finite number of possible conditions can be described by the way of entering a chain of displaying a one on the another discrete values of certain complex function with the module equal 1.

However, at we desire to analyze conditions in which can turn out to be a structure more detailed, such consideration turns out to be insufficient.

We suppose that amongst conditions, determining dynamics of structure, there is a certain class of conditions, transition between which is defined by some identical image (for instance, transition from one natural number to nearest). In this case for the given class can be installed universal algorithm of transition, which can be described by the united image, i.e. this algorithm does not depend on the element, to which it is using. Hereunder, all ensemble of lists is dividing on classes inwardly of each of which acts its universal algorithm. For some elements of class an using of this algorithm brings about that, that result is beyond the scope of class. Similarly occurs and with entering in the class.

Consider originally mutually unequivocal function. In this case inwardly given class exists two types of paths:

1. Pass paths, which enter and come out of the given class.

2. Closed inwardly class paths - cycles.

Class elements (points), in this case are to be divide into two types:

а) internal points.

б) border points.

Border points are to be divided into

а) entering points,

б) leaving points.

In the partiqular case, if enter spot is simultaneously and leaving point, corresponding to it pass path consists of one point.

If consider a dynamics of autonomous structure, described by not mutually unequivocal function, the number of types of paths turns out to be more.

In this case not all enter paths come out of the given class. Part from them, entering into the class, close on certain cycle, part - has as entering, so and leaving points. Besides, inwardly of the class are to inhere initial points of paths, which or close on the cycle, or come out of the class.

Dynamics of such structures is modeling on any standard computer in the manner of programs with single input, containing cycles and marks. Changing kits of data and operators , we are to prototype a broad class of deterministic infinite mutually unequivocal processes.

Operator of turning a system from one deterministic condition in other can be multifunction, consisting from several consecutively executed operators. In this case appear intermediate conditions, transition which one in the another depends not only from the source condition, as well as from that, what element of general algorithm we realize. Herewith on intermediate stages returning a structure to one and same condition can become irregular. However in the case limbs of steps, to which is split main algorithm and limb of number of possible conditions , periodicity of process , understood now in the broader sense, is saving.

In computer interpretation this statement sounds thereby.

1. Basin M. A. Computers. Vortexes. Resonance's. Wave theory of structures and systems interaction. Part 2. SPb.: "Norma". 2002. 144с.

2. Chebotarjov N. Bases of Galua theory . Part first. "ML ONTY GTTI" 1934. 222p.

3. Malinetskii G.G., Potapov A.B. Modern problems of nonlinear dynamics. М.: Editorial URSS. 2000. 336 p.

4. Basin M.A Waves. Quanta. Occasions. Wave theory of structures and systems interaction. Part 1. SPb.: "Norma". 2002. 144с.

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