Key words: economic system, mathematical model, parametrical regulation, task of variational
calculation, extremal, functional, bifurcation.
Abdykappar ASHIMOV, Kenzhegaly SAGADIYEV, Yuriy BOROVSKIY, Nurlan ISKAKOV,
Askar ASHIMOV
ON THE MARKET ECONOMY DEVELOPMENT PARAMETRICAL REGULATION
THEORY
The paper offers the theory of a parametrical regulation of market economy development.
This theory consists of the following sections: forming of a library of economic systems
mathematical models; researches on rigidness (structural stability) of mathematical models;
development of parametrical regulation laws; finding of bifurcation points of the extremals of
one class of variation calculations tasks, etc. The report delivers some of the results obtained
in the course of developing the theory under consideration. The work contains also example
of a finding of bifurcation points for one model of economic system and the results of the
considered model’s rigidness study with and without parametrical regulation.
1. INTRODUCTION
Many dynamic systems (Gukenheimer, 2002) including economic systems of nations (Petrov,
1996), after certain transformations can be presented by the systems of non-linear ordinary
differential equations of the following type:
dx
 f ( x, u,  ), x(t 0 )  x 0 ,
dt
(1)
Here x  ( x1 , x 2 ,..., x n )  X  R n is a vector of a state of a system; u  (u1 , u 2 ,..., u l )  W  R l - a
vector of (regulating) parametrical influences; W, Х are compact sets with non-empty interiors Int (W ) and Int ( X ) respectively;   (1 , 2 ,, m )    R m ,  is an opened connected set;
f f f
maps f ( x, u,  ) : X  W    R n and
,
,
are continuous in X W   , [t0 , t0  T ] - a
x u 
fixed interval (time).
As is known (Pontryagin, 1970), solution (evolution) of the considered system of the ordinary
differential equations depends both on the initial values’ vector x0  Int ( X ) , and on the values of
vectors of controlled (u) and uncontrolled ( ) parameters. This is why the result of evolution
1
(development) of a nonlinear dynamic system under the given vector of initial values x 0 is defined
by the values of both controlled and uncontrolled parameters.
In the recent years the dynamic trends of such parameters are being actively researched as:
various tax rates, state expenditures, accounting rate, reservation norm, currency rate and others.
The influence of these parameters upon the evolution of economic processes is also in the focus of
researches. Thus, in (Chernik, 2000) econometric methods are used for modeling the dynamic lines
and statistical forecasting of tax revenues. In (Belenkaya, 2001) econometric methods are used for
the analysis of dependences between the parameters of cash and loan policy (refinance rate,
reservation norm) and economic development indicators (indexes of investment activity in the real
sector etc.). Paper (Petrov, 1996) considers the impact of state expenditures’ share in the gross
domestic product and of the state loans’ interest upon the average real income of the workers, upon
average state expenses in stable prices and on the average gross domestic product. This is based on
the mathematical model suggested by the authors (Petrov, 1996) and conducted after solving the
task of parametrical identification.
Currently, due to the development of the theory of dynamic systems (Andriyevsky, 2004), the
parametrical influence has become applied to the regulation of economic systems. The dynamics of
these economic systems, according to the opinion of a number of experts, is described with the help
of nonlinear models (Lorenz, 1997), whose behaviour can be chaotic. Thus (Yanovsky, 2002), the
parametrical influence that is defined by the Otto-Gregory-Yorke method (Otto, 1990), was used in
order to stabilize the unstable solutions in the models of a neoclassical theory of optimal growth.
In a number of papers (Popkov, 2005; Kulekeev, 2004; Ashimov, 2005a,b,c) the parametrical
influences are being offered in order to effectively regulate the market economy development in the
given range of the changes of main endogenic indicators of an economic system and to hinder its
expansion outside the given range. The suggested parametrical influences are the extremals of the
corresponding variation calculations tasks on the choice of optimal laws of parametrical regulations
within the given finite set of algorithms. The functionals in the given tasks of variation calculations
express certain (global, intermediate or tactical) goals of economic development. Phase restrictions
and permitted restrictions are represented by mathematical models of economic systems from
(Petrov, 1996). Mathematical models from (Petrov, 1996) contain a range of indexes, whose change
within a certain interval leads to the deformation (perturbation) of the considered variational
calculus tasks.
Nowadays, the parametrical deformation of the variational calculus tasks is being widely
researched. Thus, the parametric perturbation in (Ioffe, 1974) is used in order to obtain the
sufficient extremum conditions through construction of the corresponding S-functions and usage of
the principle of restrictions removal. In (Ulam, 1964) the problem of the terms of stability of
variations calculus tasks solutions is set (Ulam’s problem). The study of this problem is brought to
finding the terms of regularity under which the functional of the perturbation task has the point of a
minimum that is close to the point of a minimum of a non-perturbation task’s functional. In
(Bobylev, 1998) the theorem about the terms of bifurcation point existence for one variations
calculus task is proved. The functional of this task is considered in the Sobolev’s space
0
W pm () (2  p  ) and depends on the scalar parameter   [0,1] . Thus, it could be noted that
among the existing sources there has been no evidence as to the terms of existence of the solutions
to variational calculus tasks on the choice of optimal laws of parametrical regulation in an
environment of the given finite set of algorithms. There are no researches on the influence of
parametrical perturbations upon the solution of the given tasks.
Existence of the solutions to the variational calculus tasks on the choice of optimal laws of
parametrical regulation in an environment of the given finite set of algorithms is under study in
2
(Ashimov, 2006a,b,c). These research works also focus on the influence of parametrical
perturbation (changes of uncontrolled parameters) upon the results of solving the tasks under
consideration. In other words, the researches are conducted, in particular, as to the bifurcation of
extremals of these tasks under parametrical perturbations.
The approaches suggested in (Popkov, 2005; Kulekeev, 2004; Ashimov, 2005a,b,c) and the
results obtained in (Ashimov, 2006a,b,c) could be treated as components of the parametrical
regulation theory developed by the authors. Основными результатами доклада
2. COMPONENTS OF THE MARKET ECONOMY DEVELOPMENT PARAMETRICAL
REGULATION THEORY
On the whole, the following components could characterize the initial version of the market
economy development parametrical regulation theory:
1. Methods of formation of a macroeconomic mathematical models library. These methods
are oriented towards a description of a variety of certain socio-economic situations with the account
of environmental safety conditions.
2. Rigidness (structural stability) conditions assessment methods of mathematical models of
an economic system out of the library without the parametrical regulation. At this, it is studied if the
mathematical models under consideration belong either to the class of Morse-Smale system or to
 -rigid systems, or to the uniformly rigid systems, or to the class of У-systems and others.
3. Methods of control or suppression of non-rigidness (structural instability) of mathematical
models of an economic system. The choice (synthesis) of algorithms of control or structural
instability suppression of a relevant mathematical model of a country’s economic system (Popkov,
2005).
4. Methods of choice and synthesis of the laws of parametrical regulation of market economy
mechanisms based on the mathematical models of a country’s economic system (Kulekeev, 2004;
Ashimov, 2005a,b,c).
5. Rigidness (structural stability) conditions assessment methods of mathematical models of
an economic system out of the set/store with the parametrical regulation. At this, it is studied if the
mathematical models under consideration belong either to the class of Morse-Smale system or to
 -rigid systems, or to the uniformly rigid systems, or to the class of У-systems.
6. Methods of clarification of the limits on the parametrical regulation of market economy
mechanisms in case of a structural instability of mathematical models of a country’s economic
system, with parametric regulation. Clarification of the limits on the parametrical regulation of
market economy mechanisms.
7. Methods of study and research on bifurcation of the extremals of variational calculus tasks
on the choice of optimal laws of parametrical regulation (Ashimov, 2006a,b,c).
8. Econometrical analysis, political and economic interpretation and coordination of the
analytical researches findings and calculation experiments with the preferences of decision-makers.
9. Development of an informational system for the research and imitational modeling of
market economy mechanisms with parametrical regulation.
10. Development of recommendations on elaboration and implementation of an effective state
economic policy based on the theory of parametrical regulation of market economy mechanisms
with the account of certain socio-economic situations.
Putting this newly-developed theory of parametrical regulation of market economy
mechanisms into practice - elaboration and implementation of an effective state economic policy could be as follows:
3
1. Choice of a state economic development direction (strategy), on the basis of a relevant
assessment of a country’s economic status in the framework of economic cycle stages.
2. Selection of one or several mathematical models of economic systems that correspond to
the tasks of economic development out of the library.
3. Assessment of suitability of the mathematical models to the given tasks: calibration the
mathematical models (parametrical identification and retrospective prognosis according to the
current evolution indicators of the economic system) and an additional verification of the selected
mathematical models with the help of econometric analysis and political and economical
interpretation of the sensibility matrixes.
4. Assessment of structural stability (rigidness) of mathematical models without the
parametrical regulation according to the above given methods of assessment of rigidness conditions
(see item 2). Structural stability of a mathematical model reflects the stability of the economic
system itself. In this case a mathematical model can be used after the econometrical analysis,
political and economic interpretation of the rigidness research findings, in order to solve the task of
choice of optimal laws of regulating the economic parameters and forecasting the macroeconomic
indicators.
5. If a mathematical model is structurally unstable, it is necessary to choose the algorithms
and methods of stabilization of the economic system in accordance with the methods pointed out in
section 3 of the theory under development. The result obtained in the course of the corresponding
econometrical analysis, and political and economic interpretation could be approved and ready for
practical implementation.
6. Choice of optimal laws of regulating the economic parameters.
7. Assessment of structural stability (rigidness) of mathematical models with the chosen
parametrical regulation laws according to the above given methods of assessment of rigidness
conditions (it.2). If the mathematical model is structurally stable under the selected parametrical
regulation laws, then the results obtained in the course of the corresponding econometrical analysis,
and political and economic interpretation, and after coordination with the preferences of decisionmakers could be adopted for their practical implementation.
If the mathematical model under the selected parametrical regulation laws is structurally
unstable, then the decision on the choice of parametrical regulation laws is clarified. The clarified
decisions on the choice of parametrical regulation laws are also subject to be considered according
to the above given scheme.
8. The study of dependence of the chosen optimal laws of parametrical regulation on the
changes of uncontrolled parameters of an economic system. Here it is possible to replace some
optimal laws by the others.
The given enlarged scheme on making decisions in the sphere of elaboration and
implementation of an effective state policy through the choice of optimal values of economic
parameters should be supported with the modern informational technologies of research and
imitational modeling.
At the present time, the above given sections 1, 2, 3, 5, 6 of the theory of parametrical
regulation are developed in the framework of the modern approaches of the theory of identification
(Krasovsky, 1987; Samarsky,2002) and the theory of dynamic systems (Katok, 1999; Magnitsky,
2004).
This report presents some findings on the development and usage of the components of the
offered theory of parametrical regulation of market economy development as applied to an optimal
growth model.
4
3. RESULTS OF THE PARAMETRICAL REGULATION THEORY COMPONENTS
DEVELOPMENT
3.1 On mapping of the conjunction of some areas of a phased space for non-perturbed and
perturbed dynamic systems
The following theorem has been formulated and proved in the framework of sections 2 and 5
of the parametrical regulation theory.
Theorem 1. Let x0  X , u W and    be fixed, and the inequality f i ( x, u,  )  C  0 is
x (t ) be a solution to
fulfilled in area X of a phased space of system (1) for some i (1  i  n) . Let ~
Cauchy’s task (1), determined for t  [0, T ] and located in Int(X). Let f1 ( x) be a perturbed vector
field under X that meets the requirements laid out in the Introduction; f ( x, u,  )  f1 ( x)   for
x (t ) be the solution to the corresponding Cauchy’s task for the field
x  X and some   0 . Let ~
1
f1 ( x) .
~
~
~
x (t )  Int ( X ) and
Then (closed) areas X  X and X 1  X will be found, where ~
~
~
~
~
x1 (t )  Int ( X 1 ), 0  t  T , for which there exists a homeomorphism H : X  X 1 , that at the same
~
~
time realizes one-to-one correspondence between the phase curves in areas X and X 1 keeping
their direction (for the fields f and f1 respectively).
3.2. Conditions of existence of a solution to the one variational calculus task
The variational calculus task on the choice of an optimal set of parametrical regulation laws at
the multitude of combinations from p parameters on r in an environment of the given finite set of
algorithms and the assertion on existence of a solution to the variational calculus task in an
environment of the given finite set of algorithms looks like this:
Let x (t ) be a solution of the given task (1) in the interval [t 0 , t 0  T ] under the constant
values u W and    . Let x (t )  Int ( X ) . Let us mark the solution (1) for the selected
u*  (u*1 , u*2 ,..., u*l )  W through x* (t ) . Further u* is fixed.
Let us mark through  the closed set in the space of continuous vector-functions
C n l [t 0 , t 0  T ] , which consists of all the continuous vector-functions x(t ), u(t ) that satisfy the
following restrictions.
x  X , u W , x j (t )  x*j (t )  x*j (t ) , t  [t0 , t0  T ] , j  1  n,   0 .
(2)
Let {F i ( x) : i  1  r} and G ( x)  0 be the finite set of continuous for x  X real value
F i
are also continuous in X . An opportunity of choice of an
x j
optimum set of parametrical regulation laws at the set of combinations from the p parameters on r
and in the time interval [t0 , t0  T ] is explored in the following algorithms (laws of control):
functions. All the functions
U
ij

 kij F i ( x)  u*j , i  1  r, j  1  l .
(3)
5
Here, k ij  0 are adjusted factors.
Handling of the set of r ( 1  r  l ) laws U ij from (3) under the fixed k ij in the system (1)


means the substitution set of functions u js  U is js in the right parts of the equations of the system
j
for r different indexes js (1  s  r , 1  j s  l , 1  is  p ). Herewith, the rest values of u , where j
does not enter into the specified set of indexes j s , are considered as constant and equals to the
j
values u * .
For the solutions of system (1) under the usage of the set of r laws of control the following
functional (criterion) is considered:
K
t0 T
 G( x (t ))dt .
(4)
t0
Setting the task of a choice of a dynamic system parametric regulation set of laws in the
environment of the finite set of algorithms looks like the following.
Under the fixed    find the set of r laws U  U is js , s  1  r from the set of algorithms


(3), which provides the supremum of the values of criterion (4)
K  sup
(5)
U
under the fulfilled conditions (1,2) for the given time interval.
Theorem 2. There is a solution to the task of finding the supremum of criterion K under the
usage of any selected set of laws U from the set of algorithms (3) with the restrictions (1) and (2):
t0 T
 G( x (t ))dt 
t0

sup
.
(6)
( ki1 j1 , ki2 j2 , , kir jr )

Herewith, if the set of possible values ki1 j1 , ki2 j2 , ..., kir jr of factors of laws of the considered task is
limited, then the indicated supremum for the selected set of laws is reached. The task (1)-(5) has got
the solution for the finite set of algorithms (3).
The proof of the theorem 2 is present in the (Ashimov, 2006a).
3.3. Sufficient conditions for the existence of a bifurcation point of extremals of one
variational calculation task
Let us suggest the following definition, which characterizes the values of parameter  , under
which the replacement of one optimal law for another becomes possible.
Definition. Value *   would be called a bifurcation point of the task (1)-(5) extremal, if
under    there were, as minimum, two different optimal sets of the laws from (3), that would
differ at least by one law U ij , and if in each neighborhood of the point  there is such value
   , for which the task (1)-(5) would have a single solution.
6
The following theorem provides sufficient conditions for the existence of a bifurcation point
of the extremals for the considered variations calculus task at the choice of parametrical regulation
set of laws in the given finite set of algorithms.
Theorem 3 (about the existence of a bifurcation point). Let for the values of the parameter 1
and 2 , ( 1  2 , 1 , 2   ) task (1)-(5) has the relevant unique solutions for the two different
sets of r laws from (3) that would differ at least by one law U ij . Then there is at least one point of
bifurcation    .
The proof of the theorem 2 is present in the (Ashimov, 2006a).
4. DEVELOPMENT and USAGE of the THEORY on PARAMETRICAL REGULATION of
an ECONOMIC SYSTEM EVOLUTION BASED on a MODEL of OPTIMAL GROWTH
Development and application of this theory to definite tasks of market economy
development parametrical regulation involves choice of one or several mathematic models,
conforming to the main objectives of economic system developmental directions. This requires such
supplementary studies as assessment of rigidness (structural stability) of chosen mathematic
model(s), choice of definite parametric regulation laws and analysis of their dependence upon the
values of uncontrolled parameters  .
4.1. Optimal growth model description
The mathematic model of economic growth (Yanovsky, 2002) represented by a following
system of two ordinary differential equations, which contains time derivatives ( t ):
k   Ak   c  (n   )k ,

c

 1
c  1   (Ak  (  p)).

(7)
Here k indicates the ratio of capital ( K ) to labour ( L ), i.e. amount of capital per an
employee. This model does not differentiate between the population of a country and workforce
(labour);
c - average per capita consumption;
n - level of growth (or reduction) of population, L(t )  L0 e nt ;
 - level of capital depreciation,   0 ;
p - discounting level, e  pt - discounting function ( p  n );
A and  - parameters of a production function type y   (k )  Ak  , where y is the ratio of
GDP to labour, i.e. average labour efficiency ( 0    1, A  0 );
 is a parameter of social usefulness function which characterizes average population
welfare: U (c)  Bc ( 0    1, B  0 ).
The first equation of system (7) is Solou’s functional equation of economic growth theory.
The
second
equation
of
this
system
is
obtained
from
the
functional’s


0
0
 pt
 c  ( p  n )t
dt maximum, which characterizes total welfare of the whole
 U (c) L(t )e dt  BL 0  e
7
population at the time interval 0  t   . This functional is maximized under the limits of k (0)  k 0 ,
k   Ak   c  (n   )k , 0  c(t )   (k (t )) and constant values of parameters  , n , p , A , B , 
and  .
The solution to system (7) will be considered in some closed field  , with a boundary – a
simple closed curve, which belongs to the first quadrant of the phase plane R2  {k  0, c  0} .
k (0)  k 0 , c(0)  c 0 , (k 0 , c 0 )   .
4.2. Analysis of rigidness (structural stability) of mathematic model theory on optimal growth
without parametrical regulation
Let us conduct statement of rigidness (structural stability) of a model under consideration in
a closed field  , basing on the definition of rigidness and the theorem on necessary and sufficient
conditions of rigidness (Bautin, 1990). Let us prove the following assumptions prior to the
statement.
Lemma 1. System (1) in field R2 has a single singular point
1

1

k *   A   ,
  p 




 *
*  ( n   )(1   )  p  n 
.
c  k 




(8)
This point is a saddle point of the system (7).
The proof. If to equate the right parts of the system (7) equations to zeros, expressions (8)
would be obtained. It is obvious, that k   0, c   0 . Let us fix down the Jakobi’s matrix
determinant for the right parts of the equations (7) in the point ( k  , c  ):
 1

( p   )(( n   )(1   )  p  n) . As far as under all the given values of the parameters
(1   )
A,  ,  , p, n,  of the mathematical model the   0 is accomplished, the discovered singular
point ( k  , c  ) is a saddle point of system (7).
Тheorem 4. Should the right parts of system
 x   f1 ( x, y ),

 y   f 2 ( x, y )
(9)
be smooth functions in some area 1  R 2 and system (9) has in this area the only singular
saddle point ( x  , y  ), then system (9) would be rough in a closed area  (   1 ), containing
point ( x  , y  ).
The proof. Let us see if system (9) in area 1 does not have any cyclical trajectories. Let us
assume the opposite: in area 1 there is a cyclical trajectory. Then inside it there should be at least
one singular point, and a sum of Poincare’s indexes of singular points located inside this cycle
8
should be equal to 1 (Bautin, 1990, p. 117). But in area 1 there is only one saddle point with index
1. It is a discrepancy.
Let us see if the stable and unstable separatrixes of the saddle point ( x  , y  ) do not form a
trajectory in field 1 . Let us assume the opposite: the stable and unstable separatrixes of the saddle
point ( x  , y  ) make up one specific trajectory  located in the 1 . Then this trajectory (or, if
available, the second trajectory, composed of other stable and unstable separatrixes) together with
the saddle point should be the boundary of the limited cell  2 , placed in field 1 . Let us consider
the semi-trajectory L outbound from some point ( x1 , y1 ), where ( x1 , y1 ) is an inner point of  2 .
Then, due to the absence of cyclical trajectories and uniqueness of the singular point, accumulation
points of L could only be the boundary of cell  2 (point ( x1 , y1 ) cannot be the only accumulation
point of L , being a saddle point) (Bautin, 1990, p. 49). Let us consider now the semi-trajectory
L , outbound from the point ( x1 , y1 ) in the opposite direction from L . It is obvious that the
boundary of  2 cannot be viewed as accumulation points of L . There is a discrepancy due to the
absence of other singular points and singular trajectories in field  2 .
Consequence 1. System (7) is rough in closed field  (   R2 ), which contains inside
point ( k  , c  ) under any fixed values of parameters n, L0 ,  , p, A,  , B,  from the corresponding
fields of their tasks.
The fact of absence of bifurcations of a phase-plane portrait of system (7) in field  under
the change of the parameters mentioned above in their assignments fields follows from this
theorem.
4.3. The task of choice of effective parametrical regulation law
Let us now consider an opportunity of an effective state policy realization through the
choice of optimal regulation laws taking an economic parameter – level of capital depreciation (  )
as a sample.
The choice of optimal parametrical regulation laws is realized in the medium of the
following dependences set:
k (t )
k (t )
1)U 1 (t )  1
  * , 2)U 2 (t )   2
  *,
k ( 0)
k ( 0)
(10)
c(t )
c(t )
*
*
3)U 3 (t )  3
  , 4)U 4 (t )   4
 ,
c ( 0)
c ( 0)
Here Ui is an i law of regulating parameter  ( i  1,4 ); i –is an adjustable factor of the i regulation
law,
– a constant equal to the base value of parameter  ;
i  0 ;  *
k (t )  k i (t )  k (0), c(t )  ci (t )  c(0); ( k i (t ) , ci (t ) ) is a solution of system (2) with initial conditions
ki (0)  k0 , ci (0)  c0 under the usage of regulation law U i . Usage of regulation law U i is a
substitution of a function from the right parts (10) to system (7) instead of parameter  ; t  [0, T ] ,
t  0 is the time of regulation start.
The task of choice of optimal parametrical regulation law on the level of one of economic
parameters  can be formulated as follows: to find on the basis of mathematic model (7) an optimal
parametrical regulation law on the level of economic parameter  in the medium of the set of
9
algorithms (10), i.e., to find an optimal law out of multitude { U i }, which would provide the
maximum of criterion
T
K  BL0  e  ci (t )( p n)t dt  max
0
{U i , i }
(11)
under the limits
ki (t )  k (t )  0,09k (t ) , (ki (t ), ci (t ))   , t  [0, T ] .
(12)
Here (k (t ), c(t )) is a solution of system (1) without parametrical regulation.
The formulated task is solved in two stages:
- at the first stage optimal values of factors i for each law U i are defined by way of their
values sorting in relevant intervals (quantized with small step), which provide maximum K under
the limits (12);
- at the second stage an optimal law on a parametrical regulation of a parameter  based on
the first stage results for the maximum value of criterion K will be chosen.
The task under consideration was solved:
under the following values of parameters   0.5 ,   0.5 , A  1 , B  1 , k 0  4 , c0  0.8 ,
T  3 , L0  1 ;
under the following fixed values of uncontrolled parameters n  0.05 , p  0.1 ;
under based value of controlled parameter  *  0.2 .
The results of the numerical solution of a problem of a choosing optimum law of
parametrical regulation at a level of one of economic parameters for economic system of the state
show, that the best result K  1.95569 can be received with using of the following regulation law
k (t )
  0.19
 0. 2 .
(13)
4
Let's notice, that the value of the criterion without usage the parametrical regulation equals
to K  1.901038 .
4.4. Research of dependence of the optimum law of parametrical regulation from values of
uncontrolled parameters
Let us consider the dependence of the results of parametrical regulation laws’ choice on the
values of uncontrolled parameters (n, p) , whose values belong to certain field (a rectangular)  in
the plane. In other words, we will find possible bifurcation points for the variational task (7, 10, 11,
12) on the choice of optimal law on a parametrical regulation of the considered economic growth
model.
As a result of calculation experiment there were obtained graphs on dependences of the
value of an optimal criterion K on the values of parameters (n, p) for each out of the 4 possible
laws U i . Figure 1 presents the given graphs for the laws U1 and U 4 , which give the biggest value of
a criterion in area  , the intersection of corresponding surfaces and the projection of this
intersection upon the plane of values  , consisting of the bifurcation points of this parameter. This
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projection divides the rectangle  into two parts, where the controlling law U1 is optimal for one
part, and U 4 is optimal for the other; at the projection itself both laws are optimal.
Fig.1. Graphs of optimal values of a criterion.
Judging by the outcome of this research on a dependence of the results of solving the
considered variation calculations task on the values of unregulated parameters (n, p) , the problem
of choice of optimal parametrical regulation laws could be solved in a following way. If the values
of parameters (n, p) are located to the left of a bifurcation line in rectangular  (fig. 1), then the U1
law would be recommended as an optimal one, and if the values of parameters (n, p) are located to
the right of a bifurcation line in rectangular  , then the U 4 law would be recommended as optimal.
Should the values of parameters (n, p) be located on the line of bifurcation in rectangular  , then
either of U1 and U 4 laws would be recommended.
4.5. Analysis of rigidness (structural stability) of mathematic model theory on optimal growth
with parametrical regulation
Let us check the rigidness of system (7) using the parametrical regulation law U1 out of the
algorithms set (10) under any value of an adjustable coefficient 1  0 . For this, a corresponding
formula from (10) should be put in the right parts of system (7) equations which are equaled to zero,
thus there will be a system with unknown (k , c) (under the fixed other acceptable values of
variables and constants)
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k  k0
 
  0 )k  0,
 Ak  c  (n  1 k

0

 c (Ak  1  ( k  k 0   )  p)  0.
1
0
1  
k0
As far as a function from the right part of the second equation of system
function of a variable k is steadily decreasing and taking all the values under k
second equation has got a single solution - k * . For this value there the only solution
equation (14) would be found, in other words, system (14) has the only solution
(14)
(7) being the
 0 , then the
c* to the first
- ( k * , c * ) . If
(k * , c * )  R2 then it is obvious that system (7) with the regulation law U1 is structurally stable in
any closed area   R2 .
Now (k * , c * )  R2 . Let us find a Jacobi’s matrix determinant to functions f1 , f 2 - left parts
f1 * *
of corresponding equations of system (14) in this point. As far as
(k , c )  1 ,
c
f 2 * *
f 2 * *
c*
(k , c )  0 , then the determinant of this
(k , c ) 
( (  1) A(k * ) 2  1 )  0 ,
c
k
1 
matrix is   0 . Then, in this case, point ( k * , c * ) is the saddle point of system (7) with regulation
law U1 . Theorem 1 is followed by the fact of structural stability of the considered system in a
closed area   R2 containing point ( k * , c * ) .
In particular, system (7) remains structurally stable under the usage of optimal law (13)
found above.
The above methods could be used in order to test the rigidness of system (7) under optimal
law U 4 usage while finding the values of parameters (n, p) in the above mentioned area of
rectangular  (fig. 1).
5. CONCLUSIONS
1. The first version of the components of the market economy development parametrical
regulation theory is represented.
2. Main stages of handling the parametrical regulation theory for the purpose of elaboration
and realization of an effective state economic policy are pointed out.
3. The following results are provided in the framework of the market economy development
parametrical regulation theory under development:
3.1. The approach to parametrical regulation of a nonlinear dynamic system’s development in
the form of setting and solving the task of variational calculation on the choice of the optimum laws
of parametrical regulation within the given finite set of algorithms is offered.
3.2. An assumption about the existence of solution to the task of variational calculation on the
choice of the optimum laws of parametrical regulation within the given finite set of algorithms is
formulated and proved.
3.3. The definition of a bifurcation point of an extremal of a task of variational calculation on
the choice of the optimum laws of parametrical regulation within the given finite set of algorithms
is given.
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3.4. An assumption about the conditions sufficient for the existence of an extremal’s
bifurcation point of a task of variational calculation on the choice of the optimum laws of
parametrical regulation within the given finite set of algorithms is formulated and proved.
3.5. The example of finding the bifurcation points of the extremals of one task of variational
calculation on the choice of the optimum laws of parametrical regulation within the given finite set
of algorithms is provided on the basis of one optimal growth model.
3.6. Structural stability (rigidness) of an optimal growth model with and without usage of
parametrical regulation has been proved.
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