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Analysis Simulation of Interaction Information in Chaotic Systems of Fractional Order
In [14] shows that the expectation of

τ1 = Mτ1 satieties

M 1  M1 

 P  M

(20)

ij

j E 

j

M1 Э 

(21)

- means equality on distribution;



σ12

formula

of

total

-



0



0

M 2  M1  M 1  



T1 s  



M 1 

e1

e3

e1

0

e1

0

e3

(26)

0

2

 e1 x

xe

dx  1  e1 te

 e1 t

e

  e1 t

.

(27)

P1 s  

 e3 t

Ge1 t   1  e1 te

e

  e1 t

d e3 t   2e3 te

 e3 t

e

  e3 t

.

  e3 t

.

1  e1 s Ge1 s   e3 s 

(28)
.

dt .

P2 s  

(29)

(41)

s 1  e1 s Ge1 s 

Similarly

e3 t   1  e3 te

1   e1 s Ge1 s 

1  Ti s 
(40)
.
s
0
Transformation of L-S probability of normal works,
starting with e1 it is expessed as [15]:

In the case of the gamma-distribution with parameter

1

Ge1 s  e3 s 

(39)

m = 2 we have:
2e

 e3 s 

1   e1 s Ge1 s 

Pi s    e  st Pi t dt 

0

t

the formula [18]:

0

1    e1 t d e3 t 

 e1 t   

(38)

With the transformation of L-S probability of normal works
of the system, starting with the state i Pi ( s ) expressed by

 1   t 1   t dt   1  G t dt  t d t  .

  e3 s 

 e1 s Ge1 s   1
T2 s  

Here

(37)

Here



  1  Ge1 t  dt   e1 t d e3 t   M 1   e1 t d e3 t ,
0
0
 0

 
 
M 1  1    e1 t d e3 t    1   e1 t  1   e3 t  dt   1  Ge1 t dt   e1 t d e3 t .
0
0
 0
 0

(36)

T1 s   e1 s Ge1 s   1   e3 s 

(25)

j3

 T1 s    e1 s Ge1 s T1 s    e3 s 

0

{e1 , e2 }
(34)

or
T1 s    e1 s T2 s    e3 s 

Ge1 s T1 s   T2 s   0
From whence
T2 s   Ge1 s T1 s 

   e1 t d e3 t    1   e1 t  1   e3 t  dt 
0

e3

and
 T1 s   P12 s T2 s    P13 s 

1221 s T1 s   T2 s   0

(24)



M 1   1   e1 t  1   e3 t  dt   1  Ge1 t dt  M 1  
0
0

j1,2

Where the average time of the system in the state e1 has



Ge1 s    e st dGe1 t   P21 s .

Then on the basis of [17] we have the system of equation:
(35)
 Pij s   ij Tj s    Pij s , i  1,2

(23)

the form [14]:

(33)

e3

0

M 1   1   e1 t  1   e3 t  dt    e1 t d e3 t M 2

0
0

M  1   t  dt  M .
e1
 2 
0



(32)

i -th state has the form:

1 e1  e1
.
e1 e1  e1

Ti s    e  st dP~1  t.

Then the system of algebraic equations for determining the
mathematical expectation of the output time e3 , beginning
with

(31)

beginning with e1 :

1

0

L-S transform distribution time to SMP output

(22)

1  Ge t dt   1 .

e3

  1   e1 t  1   e3 t  dt    e1 t d e3 t M 2 ,
0

Grel s    e st dGrel t   P31 s ;

  1   e1 t  1   e3 t  dt   P e1  t d e3 t M 2 
0

 e3 s    e st d e3 t   P13 s ;

e3

0

3

 e1 s    e st d e1 t   P12 s ;

probability

  1   e1 t  1   e3 t  dt  P e1  e3 M 2 

1

M 1  M1  M 12 M 2 

e e  e

transition indicator.
According to the

2e1

3

=

Laplace-Stieltjes transform (L-S) for distribution values
[16]:

 2  2   ,
where



In the case of the exponential distribution equation for
Mτ1 has the form [15]:

Then the system of stochastic equations for time reaching
in state e3 will have the form [14]:

 1  1   12 2

4e1 e3  1.38e1 e3  2.62e1  2e1 3e3  e1 .
2e1 2e3 3e1  e3

M1  

the system of algebraic equations of the form:

1  e1 s Ge1 s   Ge1 s e3 s 

s 1  e1 s Ge1 s 

(42)

Let

(30)

 e3 s    e  st

Then the equation for Mτ1 will be form [15]:

0

88

me

3

t m 1e

m 

e3 t

dt 

me
.
s   m
3

(43)

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