New Developments in Pure and Applied Mathematics
Generalized Real Numbers Pendulums and Transport Logistic
Applications
A. P. Buslaev and A. G. Tatashev
Abstract— A discrete dynamical system, called a real numbers
bipendulum, is considered in the paper. This system is the model
of relocation of particles on an abstract graph. The behavior
of the system is formalized. Competitions resolution rules and
movement schedules, i.e. logistic, are given. The movement
schedules are given with aid of real numbers, represented
in system of base, equal to the graph cardinality. The main
problem is investigation of the pendulum behavior depending
on rationality or irrationality of logistics. A chaotic pendulum is
considered parallel to the logistic pendulum.
In the case of the chaotic pendulums, the plan of tomorrow
behavior of a particle is played today. A special case of the
egoistic pendulum is considered. In this case plans of the
particles are time shifts of N −ary representation of a number
called a phase pendulum.
Keywords: Discrete dynamical systems; Markov processes; Number theory; Ergodic theory; Classical Russian
literature.
1. F ORMULATION OF PROBLEM
1.1. ”You are sitting wrong”
Suppose there are N vertices V0 , . . . , VN −1 and M particles P0 , . . . , PN −1 , Fig1. Each particle is in one of N
vertices at every time instant. At the next time instant, after
the supreme verdict ”you are sitting wrong”, particles are
trying to change seats, Fig1. If no conflicts take place,
then the seats are changed in accordance with a given rule.
Here there are logistic plans of particles. This is the so
called democratic jumping. Otherwise conflicts are solved in
accordance with given rules. We have a dynamical system.
Basic essence of this system is an endless search of the
correct dislocation. However one of the Russian literature
classics [1] dared to say that any music orchestra cannot be
obtained in this manner. Formally speaking, the system state
can be described 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 particles at present
time. This index is equal to one of numbers 0, 1, 2, . . . , N −1.
1.2. Plan logistics
A real number aj is given. This number is called the
Pj particles plan, j = 0, 1, 2, . . . , M − 1. This number is
This work was supported by Ministry of Education and Science of the
Russian Federation, project No. 14.740.11.0397
A. P. Buslaev is with the Department of Mathematics, MADI, Russian
Federation [email protected]
A. G. Tatashev is with the Department of Math. Cybernetics, Moscow
Tech. Univ. of Communications and Informatics, and the Department
of Mathematics, Moscow Automobile and Road State Tech. Univ.
[email protected]
ISBN: 978-1-61804-287-3
388
V11
P4
V10
P0
V9
V0
P1
V1
P2
V2
V8 P3
P7
V7
P5
V3
P6
V4
V6
V5
Fig. 1.
Round table and dislocation of particles
represented in N -ary system
(j) (j)
(j)
aj = 0.a1 a2 . . . ak . . . ,
where each digit value is equal to one of the number
0, 1, . . . , N − 1. We assume that the number aj is recorded
on the tape, correspondingly to the particle Pj , j =
0, 1, . . . , M − 1. Each particle reads a digit recorded on its
tape at every discrete time instant T = 1, 2, . . . This digit
determines the index of the vertex such that the particle tries
to pass to this vertex, Fig.2.
1.3 Democratic chaos
Random walks are considered instead of the destroyed
system of logistics. In the case of these random walks the
next dislocation of particles is determined flip the rest of
their coins.Each digit of the number aj is played before
the particle reads this digit, and the value of the digit is
equal to i with probability Pi,j , j = 0, 1, . . . , M − 1; the
numbers Pi,j are given, 0 < Pi,j < 1, 0 ≤ j ≤ M − 1,
P0,j + P1,j + ... + PN,j = 1. The main problem of our paper
is to investigate the behavior of the dynamical system in the
cases of given strategies and rational or irrational plans.
1.4 Simulation of activity
At the initial time instant T = 1, the particle Pj reads the
first digit of its tape at the initial time instant T = 1. The
New Developments in Pure and Applied Mathematics
today
1.5 Quantitative and qualitative characteristics
time table
(0)
1
(0)
2
(0)
3
(0)
4
(0)
5
(1)
1
(1)
2
(1)
3
(1)
4
(1)
5
(2)
1
(2)
2
(2)
3
(2)
4
(2)
5
(3)
1
(3)
2
(3)
3
(3)
4
(3)
5
a a a a a
a a a a a
a a a a a
a a a a a
Fig. 2.
Turing tapes and plan logistics
(j)
particle is in the vertex with index a1 , j = 0, 1, . . . , M − 1.
If no conflict competition takes place at time T = 1, then
each particle will be, at time T = 2, in the vertex such that
the index of this vertex is equal to the second digit of the
plan. If a conflict takes place at time T = 1, then one of
the competing particles, winning the competition, will be, at
time T = 2, in the vertex, determined by the plan, and the
losing particle, does not move. The tape of winning particle
will read the third digit at time T = 2. The behavior of the
system at time T ≥ 2 is similar. A competition takes place if,
at present time, there are particles which try to come from
the vertex Vi to the vertex Vj , and particles which try to
come from the vertex Vj to the vertex Vi , 0 ≤ i, j ≤ N − 1.
If si,j particles try to move from the vertex Vi to the vertex
Vj and sji particles try to move from the vertex Vj to the
vertex Vi , then the particle, trying to move from the vertex
Vi to the vertex Vj , win the competition with probability
si,j
.
si,j + sj,i
Denote by Di (T ) the number of transitions of the particle
Pi tape on the time interval [0; T ], i = 0, 1, . . . , M − 1; T =
1, 2, . . . ; H(t) is the number of conflicts, on time interval
(0; T ]; Hi (t) is the number of conflicts, losing by the particle
Pi in time interval (0; T ], T > 0.
The limit
Di (T )
wi = lim
, i = 0, 1, . . . , M − 1,
T →∞
T
is called the velocity of the particle Pi tape, i =
0, 1, . . . , M − 1 if this limit exists.
The limit
H(T )
, i = 0, 1, . . . , M − 1,
h = lim
T →∞
T
is called the intensity of conflicts if this limit exists.
The limit
Hi (T )
hi = lim
, i = 0, 1, . . . , M − 1,
T →∞
T
is called the intensity of conflicts losing by the particle Pj
if this limit exists.
The limits (1) − (3) depend on the process realization.
These limits can exist or not exist depending on the realization. The system is in the state of system after a time instant
Tsyn if, after the instant Tsyn , no conflicts take place, and
each particle comes to the next state at every instant.
2 R ATIONAL OR IRRATIONAL LOGISTIC PLANS
If the number aj is rational, then this number can be
represented as a periodic fraction
(j) (j)
(j)
(j)
(j)
(j)
a(j) = 0.a1 a2 . . . akj (akj +1 akj +2 . . . akj +lj ),
where kj is the length of the aperiodic part of the number
aj representation, and lj is the length of the repeating part
of the representation, j = 0, 1, . . . , M − 1.
2.1. An example of a rational bipendulum
Consider an example. Suppose N = M = 2; a0 and a1
are the numbers 1/3 and 1/5, which are represented in the
binary system as periodic fractions
a0 =
1
= 0.(01), a1 = 0.(0011).
3
Proposition 1. Suppose
a0 = 0.(01), a1 = 0.(0011).
2
p= 3
P0
Limits (1), (2) and (3) exist with probability 1, and
h=
P2
P1
Proof. Consider a Markov chain are vectors (i0 , i1 ), where
ij is the number of the period digit, which the particle Pj
reads, j = 1, 2, 1 ≤ i0 ≤ 2, 1 ≤ i1 ≤ 4. There are 8 states
of the chain
p= 13
Fig. 3.
2
1
4
, h1 = h2 = , w1 = w2 = .
5
5
5
Conflict resolution
E1 = (1, 1), E2 = (1, 2), E3 = (1, 3), E4 = (1, 4),
ISBN: 978-1-61804-287-3
389
New Developments in Pure and Applied Mathematics
E5 = (2, 1), E6 = (2, 2), E7 = (2, 3), E8 = (2, 4).
2.3 Generalize pendulums fluctuations with a phase shift
Denote by pij the probability of transition from the state Ei
to the state Ej , 1 ≤ i, j ≤ 8. The transitions from the state
Ei to the state Ej , 1 ≤ i, j ≤ 8. The transition probabilities
matrix has the form


0 0 0 0 0 1 0 0
 0 0 0 0 0 0 1 0 


 0 0 0 0 0 0 0 1 
 1


0 0 0 0 0 0 12 
2

P = (pij ) = 
 0 1 0 0 0 0 0 0 .


 0 1 0 0 0 0 1 0 
2
2


 0 0 0 1 0 0 0 0 
1 0 0 0 0 0 0 0
Common pendulum is considered, provided that logistical
plans of particles are time shifts generated by the same
number (father, mother, source).
(1) If father is rational with probability is equal to one and
finite expectation time pendulum is going to synergy state.
(2) Synergy expectation time of a rational bipendulum can be
lower estimated by border reaching problem during random
walk of an integer valued cell on a right line.
(3) Irrational bipendulum of wellfamous irrational numbers,
as can be seen below, are well defined by democratic chaos
structure.
The state E3 and E5 are inessential. The other states form
single communication class. Hence steady probabilities of
these states exist, [3,4]. Denote by pi the steady probability
of the state Ei , i = 1, 2, 4, 6, 7, 8. The steady probabilities
satisfy the system of equations
p4
p6
p1 =
+ p8 , p2 = ,
2
2
p6
p4 = p7 , p6 = p1 , p7 = p2 + ,
2
p8 = p4 /2, p1 + p2 + p4 + p6 + p7 + p8 = 1.
3 A PPROXIMATION OF IRRATIONAL BIPENDULUMS WITH
RANDOM WALKS BIPENDILUMS
Suppose N = M = 2,
p00 = p01 = q, p10 = p11 = p.
1
1
, p2 = p8 =
.
5
10
Suppose the chain comes to the state Ei in the time
interval (0, T ]. The value θi (T )/T tends to steady probability
of the state with probability 1 [4]
θi (T )
lim
= pi .
T →∞
T
Conflicts take place only in the states E4 and E6 . The
number of conflicts in the time interval (0, T ] equals to
Therefore any digit of each tape is equal to 1 with probability
p, and the digit is equal to 1 with probability q, p + q = 1.
Consider the stochastic process X(T ), T = 2, 3, . . . , which,
at each time instant, is in one of 5 states Gi , i = 1, 2, . . . , 5.
The process X(T ) is in the state G1 if both the particles are
in the vertex P0 , and no conflict took place at time T − 1.
The process X(T ) is in the state G2 if both the particles are
in the vertex P0 , and a conflict took place at time T − 1.
The process X(T ) is in the state G3 if both the particles are
in different vertices. The process X(T ) is in the state G4 if
both the particles are in the vertex P1 , and no conflict took
place at time T − 1. The process X(T ) is in the state G5 if
both the particles are in the vertex P1 , and a conflict took
place at time T − 1.
Suppose pi (T ) is the probability of the stochastic process
X(T ) is in the state Ei , and
H(T ) = θ4 (T ) + θ6 (T ).
pi = lim pi (T ), i = 1, 2, . . . , 5.
The solution of this system is
p1 = p4 = p6 = p7 =
Therefore, with probability 1,
H(T )
2
h = lim
= p4 + p6 = .
T →∞
T
5
Since the particle Pi loses each conflict with probability 1,
then we have
Hi (T )
4
lim
= , i = 1, 2.
T →∞ H(T )
5
Proposition 1 has been proved.
2.2 Properties of the real value pendulum
(1) For some states of a real value bipendulum generated by
irrational numbers (irrational bipendulum) limits (1) - (3) do
not exist.
(2) Numeric characteristics of a rational real valued pendulum (1) - (3) are defined correctly. The proof is immediate
from the definition so that these characteristics on the interval
[0.5, 1]. Better estimations is a problem to be solved.
(3) For irrational pendulums typical behavior system is
described by democratic chaos.
ISBN: 978-1-61804-287-3
390
The following theorems have been proved.
Theorem 1. The stochastic process X(T ) is a Markov
chain. There exist steady state probabilities p1 , p2 , . . . , p5 .
These probabilities satisfy the system of equations
p1 = q 2 p1 + q 2 p3 + q 2 p4 ,
p2 =
pq
· p3 ,
2
(4)
(5)
p3 = 2pqp1 + qp2 + pqp3 + 2pqp4 + pp5 ,
(6)
p4 = p2 p1 + pp2 + p2 p3 + 2pqp4 + pp5 ,
(7)
p5 =
pq
· p3 ,
2
p1 + p2 + · · · + p5 = 1.
(8)
(9)
The average number of conflicts per a time unit is equal to
pqp3 . The average number of conflicts, where the particle
Pj loses, per a time unit, j = 0, 1, is equal to pqp3 /2.
Tapes velocity, cumulative moving average
New Developments in Pure and Applied Mathematics
Theorem 2. Suppose p = q = 1/2. Then the average
number of conflicts per a time unit equals 1/20, j=0,1.
The proof of Theorem 1 is based on representation of
the dynamical system in the form of the Markov chain.
Steady state probabilities of this chain have been found.
Having found these state probabilities, we can find the tape
velocities. Theorem 2 follows from Theorem 1.
4. I RRATIONAL PENDULUMS COMPUTER SIMULATION
1
shift = 20
shift = 100
0.99
0.98
0.97
0.96
0.95
0.94
0
4.1. Irrational plans
If plan of particles are irrational numbers, then the system
cannot be described with finite Markov chain.
Simulation experiments have been
√ implemented.
√ The plans
were irrational
numbers
such
as
2(mod
1),
3(mod 1),
√
π − 3, 5(mod 1).
0.952
Values
Average
0.9515
Tapes velocity
0.951
0.9505
0.95
0.9495
0.949
0.9485
0.948
0
Fig. 4.
20
40
60
80
100
120
Positions shifted
Phases logistic pendulum
√
140
2(mod1) –
√
160
180
3(mod1)
4.2. Chaotic behavior of the system in the case of irrational
plans
Let us describe results of experiments in the case N =
M = 2 (bipendulum). The results of experiments show that
the velocity of particles is equal to 19
20 as in the case of the
chaotic pendulum. The simulation experiments are stable in
the case of rational plans. The simulation experiments can
be unstable in the case of irrational plans.
4.3. Phase pendulums
Let us get the plan a1 , shifting plan a0 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 behave such as it comes to the state of synergy,
with probability 1, after a time interval interval Tsyn , but
the expectation of Tsyn is infinite.
The behavior
√ of the bipendulum has been investigated with
plans a0 = 2 and a1 such that we get a1 , shifting a0 onto
c positions. The dependence of the average velocities on the
time interval (0, T ) is shown in Fig. 5. We suppose that the
system comes to the state of synergy with probability 1 for
a time interval. However the duration of this interval is large
if c is large.
ISBN: 978-1-61804-287-3
50000 100000 150000 200000 250000 300000 350000 400000 450000 500000
Time
Fig. 5.
5. C OMMENTS ,
Phases shift
√
2 mod 1
DRAWBACKS AND FURTHER RESEARCH
(1) General transport-logistical problem is described in
article [5]. There were considered flows on linear networks
(neckless type) and plane flows (chainmail).
(2) Rational pendulum as N = M = 2 (bipendulum) is
introduced and considered in [8]. Some aspects are described.
(3) In [9] are described algebras connected with dynamic
system analysis problem (real valued pendulum) in case of
rational logistical plans. The Bernoulli algebra divide the
triangle of rational numbers to subalgebras such that the tape
velocity of the bipendulum equals 1 (the synergy). If the
plans belong to different subalgebras, then the tape velocity
200can be less than 1. This problem is studied.
(4) From computational experiments with irrational bipendulum can be seen remarkable behavior distinctions comparing
with rational pendulum. In particular, for classical irrational
examples velocity bipendulum estimation with half a million
characters is equal to correspond characteristic in case of
a democratic chaos. Behavior of a system is described
according to the next rules each character equiprobably is 0
or 1 independent from other characters.
(5) Phase pendulum fluctuations give foreseen results, so that
synergic state is always to happen, however in irrational case,
with remarkable lower velocity.
(6) It is interesting to find exact low border of a real valued
pendulum main numeric characteristic. For instance in the
case when quantity of vertex and number are equal. This
problem is solved only for rational pendulum.
(7) It is proved from random walk theory that on two dimension cell any point is reached with 100% probability during
infinite expectation time. In case of a random walk on a cell
with dimension more than two, the particle returns to given
point with probability less than one. We may consider that
phase pendulums with more than two particles and generative
irrational logistics are not going to be synchronized for finite
time.
(8) Since the plans are given with real numbers, we use
constructive approaches to determine irrational numbers.
These approaches to determine irrational numbers. These
methods were not used in analysis of models, investigated
in [5 - 7]. The theory of Markov chains and the theory of
391
New Developments in Pure and Applied Mathematics
Markov chains and the theory of random walks are used as
in [5 -7].
I. ACKNOWLEDGMENTS
This work was supported by the Ministry of Education and
Science of Russian Federation under Grant No. 2.723.2014/K
R EFERENCES
[1] Krylov I. A. Fables. Moscow, Detskaya literatura, 1989.
[2] Feller W. An introduction to probability theory and its applications.
Vol. 1. John Willey. New York, 1970.
[3] Kemeny J. G., Snell J.L. Finite Markov chains. Springer Verlag. New
York, Heidelberg, Tokio, 1976.
[4] Borovkov A. A. Probability theory. Moscow, Nauka, 1986.
[5] Kozlov V. V., Buslaev A. P., Tatashev A. G., Yashina M.V. Monotonic
walks of particles on a chainmail and coloured matrices. Proceedings
of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMSSE 2014, Cadiz Spain,
June 3 – 7 2014, vol. 3, pp. 801 – 805.
[6] Kozlov V. V., Buslaev A. P., Tatashev A. G. Monotonic walks on
a necklace and coloured dynamic vector. International Journal of
Computer Mathematics (2014). DOI: 1080 00207150.2014/915964
[7] Kozlov V. V., Buslaev A. P., Tatashev A. G. A dynamical communication system on a network. Journal of Computational and Applied
Mathematics, vol. 275 (2015), pp. 247 – 261.
[8] Kozlov V. V., Buslaev A. P., Tatashev A. G. On real-valued oscillations
of bipendulum. Applied mathematical Letters (2015) pp. 1 – 6. DOI
10.10.1016/j.aml. 2015.02.003
[9] Kozlov V. V., Buslaev A. P., Tatashev A. G. Bernoulli algebra on
common fractions. International Journal of Computer Mathematics, in
print, 2015, pp. 1 – 6.
ISBN: 978-1-61804-287-3
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