On Dynamical Systems for Transport Logistic and Communications

In this paper a discrete dynamical system is considered . There is a dial with N positions (vertices) and M particles. Particles are located in vertices. Each particle moves, at every time unit, in accordance with its plan. The plan is logistics, given through a real number which belongs to the segment [0, 1]. The number is represented in positional numeral system with base N equal to the number of vertices. A competition takes place if particles must move in opposite directions simultaneously. A rule of competition resolution is given. Systems characteristics are investigated for sets of rational and irrational plans. Some algebraic constructions are introduced for this purpose. Probabilistic analogues (random walks) are also considered.


Introduction
A dynamical system, named bipendulum, was introduced and investigated in (Kozlov, Buslaev, & Tatashev, 2015c).This system is the simplest version of the transport-logistic problem.The geometric interpretation of this system is the following.Each particle is located in one of two vertices.A channel connects the vertices.Particles must move or not move at each discrete time instant in accordance with plans.Delays are due to the condition that the particles cannot move towards each other.A competition takes place if two particles attempt to move in opposite directions.A rule of competitions resolution is given.The implementation of plans of losing competitions particles is delayed.
Plans of particles are given by binary representation of numbers which belong to the segment [0, 1].A rational number determines a plan, if the representation of this number is a periodic positional fraction or a common fraction.In the first of these cases, we assume that the particle plan is written on the tape, and the particle reads digits of the plan successively.The transition to implementation of the next digit of the plan is equivalent to the following.Digits of plans are shifted to the left onto a position, and the digit to the left of the point is excluded.Therefore this transition is equivalent to multiplication of a number by 2 and exclusion of the integral part.This operation is called the Bernoulli shift (Schuster, 1984).Let us consider a set of plans defined by proper common fractions with the same denominator N.An algebra W N of elements of this set with respect to the operation of Bernoulli shift is introduced in (Kozlov, Buslaev, & Tatashev, 2015c), (Buslaev, & Tatashev, 2016).The algebra W N , depending on N, is represented uniquely as a sum of connected components which are subalgebras W N i , where N i are divisors of N. In this case, dynamics of particles can be interpreted as motion of particles on the subalgebra graph, where weights of arcs generate logistics.The concept of particle tape velocity was introduced.Thus this algebra is an important component of the dynamical system called bipendulum.
A system with any number of vertices and particles is considered in this paper.The system is called real-valued pendulum.The plan of each particle is given by a number which belongs to the segment [0,1].This number is represented in the numeral positional system with the base equals to the number of vertices.If particles try to move towards each other, then delays take place as in the case of bipendulum.Transition to the next digit of the plan is equivalent to applying of the generalized Bernoulli shift operation to the number which determines the plan.The generalized Bernoulli shift is defined as follows.The number is multiplied by N and the integral part is excluded.The base of positional system is equal to the number of vertices.Thus algebra of proper fractions generate real-valued N−pendulum as Bernoulli algebra is the abstract basis for bipendulum.Some discrete dynamical systems with symmetric periodic structures were introduced and investigated in (Kozlov, Buslaev, Tatashev & Yashina ,2014) (Kozlov, Buslaev & Tatashev, 2015b), (Kozlov, Buslaev & Tatashev, 2015a).
Suppose there are N vertices V 0 , . . ., V N−1 and M particles P 1 , . . ., P M , Figure 1.Each particle is in one of N vertices at every moment of time.There are logistic plans of particles dislocations.Each plan is a real number a ( j) belonging to the segment [0, 1].The plan of a particle determines a sequence of indexes of vertices in which the particle must be located at successive moments of time.This sequence is given by N-ary representation of the number a (i) .The vertices are connected by channels.If two particles try to move towards each other, then a competition takes place.In this case only particles, trying to move in one of two directions, realize the attempt.This direction is chosen in accordance with a given rule.Implementation of plans of losing competition particles is delayed.The main essence of this system is total planning of particles dislocation.
The system state can be described for one of special cases with a binary matrix.The rows of the matrix correspond to particles.Each row contains a single "one".The index of the column, containing this "one", is equal to the index of the vertex, containing the particle at present time.This index is equal to one of numbers 0, 1, 2, . . ., N − 1, Figure 1.

Plan Logistics
A real number a (i) is given.This number is called the plan of the particle P i , i = 1, 2, . . ., M. This number is represented in N-ary system where each digit value is equal to one of the numbers 0, 1, . . ., N − 1.We assume that the number a (i) is recorded on the tape of the particle P i , i = 1, . . ., M. Each particle reads a digit recorded on its tape at every discrete time moment T = 1, 2, . . . .This digit determines the index of the vertex such that the particle tries to occupy this vertex, Figure 2.
Figure 2. Turing tapes and plan logistics

Random Walk Logistics
Random walk is considered as an alternative for the logistics plan.In the case of random walk, the next planned dislocation of particle P i (T + 1), i = 1, . . ., M, will be determined, when P i (T ) has been realized.Each digit of the number a (i) is played before the particle reads this digit, and the value of the digit is equal to j with probability p i, j , j = 0, . . ., N − 1; the numbers p i, j are given, 0 < p i, j < 1, p i,0 The finally configuration is formed if possible competitions have been taken into account.

Rules of Activity and Competition for Plan Logistics
The particle P i must be located in the vertex V a jT in accordance with the plan of this particle, at the moment T, i = 1, . . ., M, T = 1, 2, . . .If the particle P i is in the vertex V a (i)   T at the moment T, then at the moment T + 1, T = 0, 1, 2, . . ., this particle will be in the vertex V a (i)   T +1 except the case of competition.A competition takes place if, at present moment of time, there are particles which try to come from the vertex V k to the vertex V l , and particles which try to come from the vertex V l to the vertex V k , k l, 0 ≤ k, l ≤ N − 1.If s k,l particles try to move from the vertex V k to the vertex V l and s l,k particles try to move from the vertex V l to the vertex V k , then all particles, trying to move from the vertex V k to the vertex V l , win the competition with probability, Figure 3, If a competition takes place at the moment of time T , then all competing particles, winning the competition, will be, at the moment of time T + 1, in the vertex, determined by the plan, and losing particles do not move.The tape of winning particle will read next digit at the moment of time T + 1.The losing particle will read at the moment of time T + 1 the digit corresponding the moment of time T. Initial conditions are given.These conditions determine the location of particles at the moment of time T = 0. Denote by D i (T ) the number of transitions of the particle P i tape during the time interval [0; T ], i = 1, . . ., M; T = 1, 2, . . .; H(t) is the number of competitions during the time interval (0; T ]; H i (t) is the number of lost competitions of the particle P i during time interval (0; T ], T > 0, The limit is called the velocity of the particle P i tape, i = 1, . . ., M, if this limit exists.
The limit h = lim is called the intensity of competitions if this limit exists.
The limit is called the intensity of lost particle P j competitions, if this limit exists.
The limits (3)-( 5) depend on the process realization.These limits can exist or not exist depending on the realization.
The system is in the state of synergy after a moment of time T syn if, after the moment of time T syn , no competitions take place.

Rational Logistic Plans
In Section 3, we introduce concepts related to real-valued pendulum, and prove theorems about particles tapes velocities.
If the number a (i) is rational, then this number can be represented as a periodic fraction where l j is the length of the aperiodic part of the number a (i) representation, and m i is the length of the repeating part of the representation, i = 1, . . ., M.
The system is a Markov chain (Feller, 1970), (Kemeny, & Snell, 1976), (Borovkov, 1986) with states (i 1 , i 2 ), 0 ≤ i 1 , i 2 ≤ 2, where i j is a digit such that the particle P j , j = 1, 2, reads this digit.There are 9 states States S 2 , S 3 , and S 7 are inessential states of the chain.The other 6 states form a set of essential states.This set is divided into two subsets: The chain goes from any state of the set G 1 to a state of the set G 2 , and from any state of the set G 2 to a state of the set G 1 .If the system is in a state of the set G 1 , then there is no competition.If the system is in a state of the set G 2 , then there is a competition.Thus, h = 1 2 , w 1 = w 2 = 3 4 .Theorem 1. Suppose M = 2. Then the greatest possible value of the competitions intensity is equal to 1 2 , and the least possible value of the tape velocity is equal to 3 4 .Proof.It follows from Lemma 2 that, if M = 2, then plans of particles can be given such that h = 1 2 , v = 3 4 .If M = 2, then plans cannot be given such that the competitions intensity is more than 1 2 .Indeed, if there is a competition at moment of time T, then there will be no competition at moment of time T + 1. Theorem 1 has been proved.

Phase Pendulums
Suppose N = M = 2, and we define the plan a (2) , shifting plan a (1) for a fixed number of positions.
and we get the plan a (2) , shifting plan a (1) onto c positions where addition is meant modulo m.Then the system comes to the state of synergy after a time interval with finite expectation.
Proof.The system is a Markov chain (Feller, 1970), (Kemeny, & Snell, 1976), (Borovkov, 1986) with m 2 states.The system comes to the state of synergy with a positive probability, and the system cannot leave the state of synergy.Therefore a state is essential if and only if the system is in the state of synergy.The Markov chain comes from any inessential state to the set of essential states after a time interval with finite expectation.Lemma 3 has been proved.
Theorem 2. Suppose a , and we get the repeating part of a (2) , shifting the repeating part of a (1) onto a fixed number of positions.Then the system comes to the state of synergy after a time interval with finite expectation.
Proof.Both particles will read digits of repeating parts of their plans after a time interval with finite expectation.Theorem 2 follows from Lemma 3.

Functional Interpretation of Bipendulum
In Section 4, we consider functional representation of plans.We define variation of proper fraction .
Suppose the binary representation of the number j/K is Suppose The variation of the representation of the number j K is defined as where the addition, in indexes, is meant modulo m.The limit is called the variation of the binary representation of a real number a ∈ [0, 1), a = 0, a 1 a 2 . . .a k . . .if this limit exists.It is evident that V(a) = Var(a) for rational number a. Suppose ) .
Proof.If θ(i) = 0 and a particle reads digit a i at time T, then the particle will be read digit a i+1 , i = 1, 2, . . ., at time T + 1.If θ(i) = 1, the particle reads digit a i at time T, and no competition takes place, then the particle will read digit a i+1 at time T + 1, i = 1, 2, . . . .Suppose θ(i) = 1, the particle reads digit a i at time T, and there is a competition at this time.Then, at time T + 1, with probability 1 2 , the particle reads digit a i , and with probability 1 2 , the particle reads digit a i+1 .Therefore, the average time of particle going from digit a i to digit a i+1 is not greater than Corollary 2. Suppose N = M = 2. Then the following inequalities are true

The Concept of Generalized Bernoulli Algebra
Algebras of proper common fractions with respect to the operation, called Bernoulli shift, (Schuster, 1984), were considered in (Buslaev, Tatashev, 2016).If this operation is applied to a fraction, the fraction is multiplied by 2, and the integral part of the product is excluded.We introduce algebras with operation of Bernoulli shift.If this operation is applied, then the fraction is multiplied by the natural number N, and the integral part is excluded.Consider the set of proper fractions where [N x] is the integral part of N x.
Suppose W N K is the algebra with the operation FB N on the set G K .The number i/K (0 ≤ i ≤ K − 1) generates an algebra.Denote by W N K (i) the subalgebra generated by the element i/K, i = 0, 1, . . ., K − 1. Elements i/K and j/K are called communicating with each other if j/K ∈ W N K (i) and i/K ∈ W N K ( j).The element i/K is called inessential if there exists an element j/K such that j/K ∈ W N K (i) and i/K W N K ( j).The other elements are essential.
The set of essential elements can be divided into disjoint sets such that any two elements of the same set communicate with each other.These sets are called classes of communicating essential elements or orbits.If an element i/N satisfies equation then this element is called absorbing.
If i/K and j/K communicate with each other, then , we divide the algebra W N K into disjoint subalgebras.Each of these subalgebras contains a class of communicating essential elements and a set of inessential elements.This set of inessential elements can be empty.These subalgebras are called connected components of the algebra.
This theorem gives an approach which allows to find the N-ary representation of proper fraction b 0 = j K .The value of the first digit a 1 of the representation equals the integral part of Nb 0 .The fractional part of the product is the value of b 1 .The value of the digit a j equals the integral part of the product b j−1 and N. The fractional part of this product equals the value of b j , j = 1, 2, . . .Suppose numbers l (l ≥ 1) and m (m ≥ 1) are such that b l = b l+m , and there are no numbers i and j such that i, j < l + m and b i = b j .Then we have a i = a i+m for any i ≥ l.Hence the N-ary representation of the proper fraction s/K contains an aperiodic part with length l − 1 and a repeating part with length m.We write s K = 0.a 1 . . .a l−1 (a l . . .a l+m−1 ).
We have formulated an approach to find the binary representation of a proper fraction.This approach is similar to the approach of converting the decimal representation of a proper fraction to the N-ary representation, (Broido & Ilyina, 2006).It is easy to reconstruct the trajectory of the element s/K of the algebra W N K if we know the N-ary representation of the number s/K.Some problems of number theory, related to N-ary representation of numbers, are considered in (Uteshev, Cherkasov, Shaposhnikov, 2001).

Absorbing Elements of Algebra W N K
We shall prove theorems about absorbing elements of algebras W N K .Lemma 4. If N = 2, then the element 0 K is a unique absorbing element of the algebra W 2 K .Proof.It is obvious that 0 K is an absorbing element of the algebra W N K for any N and K. Since, then there exist no other absorbing elements of the algebra W 2 K .We shall give some examples.These examples show following.If N ≥ 3, there exist algebras which do not contain elements not equal to 0 N , and algebras which contain absorbing elements not equal to 0/N.Example 1. Algebra W 3 5 contains an absorbing element 0 N and an orbit with period 4. Example 3. Algebra W 3 6 contains two absorbing elements 0 6 and 3 6 .The other elements of the algebra are inessential.Lemma 5. Suppose N = K + 1.Then all elements of the subalgebra W N K are absorbing.Proof.If N = K + 1, then we have

Structure of the Algebra W N
K , Where K is Coprime to N In Subsection 5.3, we prove theorems about generalized Bernoulli algebras in the case of coprime K and N.

Preliminary Definitions and Results
Suppose the canonical representation of the number K, K ≥ 2, has the form where 2 ≤ p 1 < • • • < p l are prime numbers, s 1 , . . ., s l are natural numbers.We write K = p s , where p = (p 1 , . . ., p l ), s = (s 1 , . . ., s l ).Denote by E(K) the Euler's function (Buchstab, 1972), Euler's theorem on numbers, (Buchstab, 1972).The value of E(K) is equal to the number of positive integers that are coprime to N and less than K This function can be also represented as ) .
Euler's theorem on divisibility, (Buchstab, 1972).Suppose N and K are coprime natural numbers.Then K is a divisor of N E(K) − 1.
Fermat's little theorem is a special case of Euler's theorem on divisibility.
Fermat's little theorem, (Buchstab, 1972).If p is prime, and p is not a divisor of N, then p is a divisor of N p−1 − 1.
Denote by LCM(a 1 , . . ., a L ) the least common multiple of numbers a 1 , . . ., a l .Suppose K has the form (6).
It is obvious that any divisor of e(K) is a divisor of E(K).
The generalized Euler's divisibility theorem, (Buchstab, 1972).Let N and K be coprime positive integers.Then K is a divisor of N E(K) − 1.
According to this theorem, if N and K are coprime, then there exists a positive number l such that K is a divisor of N l − 1.
The smallest positive integer l such that K is a divisor of N l − 1 is called the order of K modulo N, (Buchstab, 1972).Denote by m(K, N) the of K modulo N.
The generalized Euler's order theorem, (Buchstab, 1972).Let N and K be coprime positive integers.Then m(K, N) is a divisor of e(K).
The elements of the algebra W N K can be divided into two classes.They are the class of noncancelable fractions and the class of cancelable fractions Consider the set of all elements j/K of the algebra W N K such that j/K is a noncancelable fraction.The number of these elements is equal to the number of positive integers less than K and coprime to N, i.e., this number is equal to Euler's function.Suppose In accordance with section 5.1, an orbit of period d is a set of elements of the algebra Denote by O(K, d) the number of orbits with period d (d ≥ 1).

Simple Case
Suppose K = p s , where p ≥ 2 is a prime number, the number p is not a divisor of K, s is a natural number, Theorem 4. The following formula is true The proof is based on Lemma 6.
Lemma 6. Suppose r ≤ s, j/p r is an element of the algebra W N p r such that it is a noncancelable fraction.Then this element belongs to an orbit of period m(p r ).The set of cancelable fractions of the algebra W N p r is a subalgebra isomorphic to the algebra W N p r−1 .Proof.Suppose the number a satisfies the equation The number a is natural if and only if the number K is a divisor of N m − 1.We take into account that j and K are coprime, and j < K. Applying to a noncancelable element the generalized Bernoulli shift operation m(K, N) times, we obtain the same element.If the generalized Bernoulli shift is applied to this element less than m(K, N) times, then the same element cannot be obtained.It is obviously that the subalgebra of cancelable fractions of the algebra W N p r and the algebra W N p r−1 are isomorphic to each other.Lemma 6 has been proved.

General Case
Suppose K is coprime to N.
Lemma 7. Any element j/K of the algebra W N K belongs to an orbit, and, if j/K ∈ W N,copr K , i.e., the element j/K is a noncancelable fraction, then the period of this orbit equals m(K, N).
Proof.Apply the Bernoulli shift operation to the element j/K i times.We obtain the same element if and only if the equation contains natural number a.If i = e(K) in ( 8), then the number a is natural.We take into account that K is coprime to N.
Hence there exists an orbit with period not greater than e(K) such that this orbit contains the element j/K.If the fraction j/K is a noncancelable fraction, then i = m(K, N) is the smallest value of i such that the number a is a natural number.From this Lemma 7 follows.
Lemma 8. Let K be a natural number, and K is coprime to N. Then two elements a ∈ W N,copr K cannot belong to the same orbit.Proof.If an element is cancelable fraction, and we apply the generalized Bernoulli shift operation to this element any times, then we can obtain no noncancelable fraction.From this, Lemma 8 follows.
Denote by δ d,K the number of orbits with period d in the subalgebra W N,copr K .In accordance with Lemmas 7 and 8, .
If d m(p r , N), then δ d,p r = 0.
Theorem 5. Suppose O (K, d) is the number of orbits with period d where δ(d, p r ) is calculated in accordance with (7).
Therefore no orbit can belong to both W N,copr K 1 and W N,copr K 2 . From this fact and Lemmas 7 and 8, Theorem 5 follows.

Algebra W N K in the Case of K
Where K 1 and N are Coprime In the subsection 5.4, we assume that K equals K 1 multiplied by N h , where K 1 and N are coprime.We shall prove a theorem about the form of generalized of Bernoulli algebras.

Theorem 6. Let h be a natural number, K
where K 1 = p s 1 1 . . .p s l l , p 1 < • • • < p L are prime numbers, and none of these numbers is not a divisor of N. Then the following is true.

1) Algebras W N
K and W N K/N h contain the same number of orbits with each period.2) The algebra W N K contains K 1 essential elements.The other K − K 1 elements are nonessential elements.
3) The graph of algebra W N K contains, as its subgraphs, K 1 trees, and the depth of each of these tree equals h.Each of K 1 essential elements of the algebra W N K is the root of precisely one tree, and the degree of the root equals 1.The degree of other vertices, for any tree, is equal to N except the vertices of the level h.Any tree contains N h elements.One of these elements is the root of the tree, and there are N i−1 elements that are vertices of the level i for any i = 1, . . ., h.
The proof of Theorem 6 is similar to proofs of Theorems 5 -7 in (Buslaev, & Tatashev, 2016), where the case of N = 2 was considered.Corollary 3. Let d be a divisor of the number K. Then the number of orbits with period m, the number of connected components and the number of elements of the algebra W N K are not less than in the algebra W N d .

Rational Pendulums and Generalized Bernoulli Algebras
In section 6 we consider connection between generalized Bernoulli algebras and real valued pendulums.We show that the algebraic construction of rational logistic plans has an effect on properties of real valued pendulums.

Generalized Bernoulli Algebra and Real Valued Pendulum
Consider a real valued pendulum with rational particles plans given by the rational numbers a Suppose k is the least common multiple of numbers k 1 , k 2 , and The generalized Bernoulli algebra W N k contains all subalgebras (orbits), satisfying initial conditions (10).

Subalgebras, Orbits, and Velocity of Turing Tapes
To each initial condition a (i) , i = 1, . . ., M, assign vector of elements b (i) , i = 1, . . ., M, where a (i) = b (i) if a (i) is an element of the orbit of a subalgebra, or the first element of the orbit of the subalgebra on the path if a (i) is an inessential element.
Lemma 9. Suppose M ≥ 2, and a (i) , i = 1, . . ., M, are plans.Then, with probability 1, numerical characteristics of the real valued pendulum are the same as in the case of plans b (i) , i = 1, . . ., M, and the elements of the pendulum, belonging to the same orbit, can be not taken into account.
The proof is evident.
Theorem 8. Suppose M ≥ 2, and plans of particles a (i) , i = 1, . . ., M, belong to the same subalgebra of the algebra W N k .Then the system comes to the state of synergy after a time interval with finite expectation.
Proof.Suppose M = 2.We can get the periodic part of plan a (2) , shifting the periodic part of the plan a (1) onto a fixed number of positions.There will be a finite number on competitions before the the system comes to the state of synergy, or the periodic parts of the plan a (1) and a (2) coincide after a time interval with finite expectation, and the system also come to the state of synergy.From this the theorem follows.
In the general case, each particle will read digits of periodic parts of plans after an instant with finite expectation.Suppose that, from time T 1 , the set of competing particles, containing the particle with the least index, wins each competition.The particle P 1 does not lose competitions.The particle P 2 can lose competitions with the particle P 1 .However there can be no more than d −1 competitions of these particles.The particle P 2 cannot compete with the particle P 1 after a finite instant T 2 .Similarly, there are T 3 , . . ., T M (T 1 ≤ • • • ≤ T M ) such that, after time T i none of particles P 1 , . . ., P i can compete with any other particle of this set.There will be no competitions after time T M .Thus, with positive probability, the system comes to the state of synergy after time bounded above.From this, the theorem follows.

The Velocity of Turing Tapes with Plans from Different Subalgebras
Suppose there 2 particles and plans of particles belong to different subalgebras.
Particles will read repeating parts of their plans after a time interval with finite expectation.Therefore it is sufficient to know to which subalgebras plans belong.Let (i 1 , i 2 ) be the system state such that the particle P j reads the i j -th digit of the period, j = 1, 2.
We introduce Markov chain X(t ′ ) obtained from the original chain as follows.The process is considered only at instants t ′ such that competitions take place at these instants.New time scale is obtained from the original chain, excluding all instants such that there are not any competitions at these instants.
We can find transition probabilities between all states.There are no more than two positive elements at each row of the transition probability matrix.Break of the chain is the state of synergy.The time of going to the next state is calculated.This time depends on what state will be the next state.The problem of calculation of transition probabilities is connected with analysis of linear Diophantine equations with two variables.When we have found to what class of communicating essential states the initial state belongs, we find the system of equations for steady state probabilities and calculate the steady probabilities.When we have found steady state probabilities, we calculate the average duration between competitions, and calculate the velocity of particles tapes.
Following theorems give conditions of synergy in the case such that the plans of particles belong to different algebras.
We introduce the operation +, * of bitwise addition and multiplication of binary representations of rational plans a (i) .Then the equality a (1) * a (2) = a (1) is true if the set of positions with 'ones' of number a (1) belongs to the set of positions with 'ones' of the number a (2) .Denote this relation by a (1) ≤ a (2) .From a (1) + a (2) = 1 it follows that a (1) * a (2) = 0.
Then the system comes to the state of synergy after a time interval with finite expectation.
Proof.Suppose a , where b (1) i = a (1) i+1 , i = 1, . . ., d (addition is meant modulo d), i.e., we get the plan b (1) , shifting the plan a (1) onto a position to left.Consider bipendulum with plans b (1) and a (2) .If a (2)  i = a (2) i+1 (addition is meant modulo d), then the first competition cannot take place at time T = i, i = 1, . . ., d.If a (2) i a (2)  i+1 , then b (1) i = a (1) i+1 = a (2) i , and, therefore, the first competition also does not take place at time T = i.Thus the bipendulum is in the state of synergy from the initial instant.From this and Lemma 3, it follows that the original bipendulum comes to the state of synergy after a time interval with finite expectation.Theorem 9 has been proved.Theorem 10.Suppose M = 2, N = 2 a (1) ≤ a (2) .
Then the system comes to the state of synergy after a time interval with finite expectation.
The proof is evident.
Remark 1.The following example shows that the synergy can take place for plans given by fractions with coprime denominators. Suppose The number a (1) can be represented in the form a 1 = 0.(001001).Therefore no competition can take place at the first 6 steps.Hence no competition can take place in the future.
Remark 2. The following example shows the following.It is possible that plans of all particles are given by fractions with the same denominator such that this denominator is a prime number, and the system cannot come to the state of synergy.
The system cannot come to the state of synergy.
Indeed, if we shift the repeating part of a plan onto any number of positions, then a competition takes place after a finite time.

Interpretation of Real Valued Pendulum in Terms of Bernoulli Algebra
We can assume, in the general case, that there are N vertices and M particles, then the plans of the particles P j , are given by fractions such that these fractions generate orbits.This means that the process of cyclic shift of the periodic sequence a ( j) = s j k j , 0 ≤ s j < k j , j = 1, . . ., M, takes place.We introduce the system such that this system is equivalent to real valued pendulum, Figures 4, 5.There are M particles P 1 , . . ., P M and k vertices, where k is the least common multiple of numbers k 1 , . . ., k M .Each vertex corresponds to an element of the algebra W N k .The initial state is given such that the particle P j is in the vertex, corresponding to an element of the subalgebra generated by the element a ( j) = s j k j , 0 ≤ s j < k j , j = 1, . . ., M. The vertex such that this vertex corresponds to the element with representation α = 0.(a 1 , . . ., a m ) is labeled with the value of digit a m .At each discrete instant, the particle located in the vertex, corresponding to the element x, goes to the vertex corresponding to the element y = FB N (x), if no competition takes place.A competition takes place if, for some i, j (0 ≤ i, j ≤ N − 1), there are particles, trying to come from a vertex with the label i to a vertex with the label j, and particles, trying to come from a vertex with the label j to a vertex with the label i.If a competition takes place, then Let us get the plan a 2 , shifting plan a 1 onto a fixed number of positions.If plans are rational numbers, then the system comes to the state of synergy after a time interval with a finite expectation.If plans are irrational numbers, then the system comes to the state of synergy, with probability 1, after a time interval interval T syn with infinite expectation.The behavior of the bipendulum has been investigated with plans a 1 = √ 2 and a 2 such that we get a 2 , shifting a 1 onto a fixed number of positions.The dependence of the average velocities on the time interval (0, T ) is shown in Figure 7.We suppose that the system comes to the state of synergy with probability 1 for a time interval.However the duration of this interval can be large.9. Comments, and Further Research (9.1)General transport-logistical problem is described in article (Buslaev, & Tatashev, 2015).
(9.3) Algebras with rational logistical plans are described in (Buslaev, & Tatashev, 2016).The Bernoulli algebra divides the triangle of rational numbers into subalgebras.If the plans belong to the same algebra, then the tape velocity of the bipendulum equals 1 (the synergy).
(9.4)There is remarkable distinction between behavior of rational pendulums and behavior of irrational pendulums.In particular, the estimation of irrational bipendulum velocity over half a million characters is equal to the random walk bipendulum velocity.
(9.5) Irrational phase bipendulum converges also to synergy but with remarkably lower velocity.
(9.6)It is interesting to find exact lower bound of a real valued pendulum numeric characteristic when quantities of vertices and particles are equal to each other.In our paper, this problem has been solved in the case of 2 particles and any number of vertices.
(9.7)It is interesting to find the intensity competitions in the case of random pendulum with any number of particles.In our paper, this problem has been solved in the case of 2 particles.
(9.8)It is proved from random walk theory that on two dimension lattice any point is reached with 100% probability during infinite expectation time.In case of a random walk on a lattice with dimension more than two, the particle returns to given point with probability less than one.We may assume that, with positive probability, phase irrational pendulum with more than two particles is not synchronized for finite time.
Figure 3. Competition resolution K and W N d , Where d is a Divisor of N Consider algebras W N K and W N d , where d is a divisor of the number N. In Subsection 5.5, we prove theorems that alow to compare the algebras W N K and W N d .Theorem 7. Let d be a divisor of the number K. If the element i/d of the algebra W N d belongs to the orbit with period m, then the element K d i K of the algebra W N K belongs to an orbit with period m.If the elements i/d and j/d of the algebra W N d belong to the same orbit, then elements algebra W N K also belong to the same orbit.If the elements i/d and jd of the algebra W N d belong to different orbits, then the elements to the different orbits.If the element i/d of the algebra W N d is inessential, then the element algebra W N K belong to the same connected component if and only if the elements i/d and j/d of the algebra W N d also belong to the same connected component.Proof.The value of the element K d i K of the algebra W N K and the value of element i/d of the algebra W N d are the same for any i = 1, 2, . . ., d − 1.From this, Theorem 7 follows.