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International Journal of Engineering and Applied Sciences (IJEAS)
ISSN: 2394-3661, Volume-4, Issue-6, June 2017

e3

In the work, as a generalized criterion proposed spectrum
of dimensions Poincare return time, describing how
geometric, information, and dynamics characteristics of the
transient process. That is generalized criterion is formed on
the basis of synergetic principles. Spectrum of dimensions for
~ is a functional dependence
the Poincare return time α

- transition system to a non-equilibrium state.
Graph of the system shown in figure 1.

α~q dimension type characteristics from the scale q

Fig 1.

expressed in terms of the Hausdorf dimension dim H [9]:
(10)
~q  dim H x 1  q / h  ,
where q - scale, x - invariant set,
entropy.
Shown [10] that the transport exponent

( )

lim h(μ) = h d f

relationship

and

In the first mode, the system is time
1  min e1 ,e3 ,



h - topological

)

[10]:

P1  t  1  P e1  t P e3  t  1  1  e1 t  1  e3 t  ,

lim h(μ) = h d f + 1

respectively.
Noted

that

 opt  d f  2 

μ opt

log 2
,
2log 2  3  d f 

defined

as

(14)

that is, there is a mode e1 until it will go off
mode.
Random variable θ randomly allocated:

μ defined by the

(

















or



e3

(15)

where P e  t   e t  - arbitrary distribution
1
1

(11)

function



ηTE

(testing equilibrium);



P e3  t  e3 t  -

where d f - fractal dimension.
However, in practical, is used
μ opt ≈ d f + 2 .

e2

function

(12)

ηTE

(testing equilibrium).

Then expected value for

In addition, knowing that h ≤ λ [11], where λi ∑i



λi >0



arbitrary distribution

θ

1







(16)

M1   tdP1  t   1   e1 t  1   e3 t  dt .

Lyapunov exponents, spectrum dimensions of the Poincare
return time will look:

0

Let

(13)
~q~  d f  X 1  q / h   2Q 0 h .
Thus, there is functional dependence of the memory of the
transport exponent mixing heterogeneous chaotic maps, as
well as the Lyapunov exponent.

θ2

0

e2 .

- the time of the test works,

The distribution function of time

P 2  t  Ge1 t 

θ3

(17)

- the time in the system e3 .

The distribution function for

P3  t  Grel t  .

III. FORMULATION OF THE PROBLEM
Definition. 6 If the system is not only all parameters are
constant over time, but there is no steady flow through the
action of any external source, such a state is called the
equilibrium.
Period of time during which the system returns to the
equilibrium state, called the relaxation time.
Definition 7. A system in a non-equilibrium state means
that unbalanced potential exist within the system.
Let the functioning of fractional order chaotic system is
carried out with the efficiency control system. That is, we can
assume that in addition to a complex system of tasks by the
main industrial activity goes into the efficiency control regime
(a transition from a non-equilibrium state to an equilibrium,
the equilibrium state of test mode, etc).
Thus, it is possible to present the interaction of a complex
system with a control system as the efficiency of functioning
of the structure with a variable operating mode. Known
[12,13] that after nating operation successfully describes
semi-Markov processes (SMP).

θ3

is:
(18)

Hence the expectation of relaxation time


M3   tdGrel t 

(19)

0

Thus, the transition of the states at the system:
and

e3

e2

in e1

to e1 completely recovery work.

The task is to determine the mathematical expectation of
the time in a state

e1 , e1

state probabilities and the

mathematical expectation of the number of states e3 .
In accordance with work V.S.Korolyuk semi-Markov
process (SMP) with a finite number of states E  e , e , e



is completely determined by residence times

1

2

3



 i  1,,3
i

with distribution functions

P{θi < t }

of the transition

ei

is in the state

e j , provided that

probabilities of the states
the SMP was in

ei

state for a time

t

[14].

Let τ1 the time before reaching into e3 , beginning e1 ;

3.1. Model I.
Let the system works in there successive modes:
e1 - normal operation;

τ2

- the time before reaching into

e3 ,

beginning with

e2 .

e2 - transition system in the equilibrium state of test mode;

87

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