On the development and usage of the market economy parametrical
regulation theory on the basis of one-class mathematical models
Abdykappar A. Ashimov*, Nurlan A. Iskakov*, Yuriy V. Borovskiy**, Bahyt T. Sultanov*, Askar
A. Ashimov*
* Laboratory of Systems Analysis and Control, Institute of Problems of Informatics and Control
221 Bogenbay batyr str., Almaty, 050026, Republic of Kazakhstan
** Kazakh-British Technical University, 59 Tole be str., Almaty, 050000, Republic of
Kazakhstan
E-mail: * ashimov@ipic.kz, ** yuborovskiy@topmail.kz
Abstract
The paper offers further development and
applications of the theory of a parametrical regulation
of market economy evolution. This theory consists of
the following sections: formation of a library of
economic systems’ mathematical models; of rigidness
(structural stability) of mathematical models;
development of parametrical regulation laws etc. The
work contains new results of the considered one class
models’ rigidness research with and without
parametrical regulation.
1. Introduction
Many dynamic systems [8] including economic
systems of nations [9, 10], after certain transformations
can be presented by the systems of non-linear ordinary
differential equations of the following type:
dx
 f ( x, u ,  ), x(t0 )  x0 ,
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  Rl - a
vector of (regulating) parametrical influences; W, Х are
compact sets with non-empty interiors - Int (W ) and
Int ( X )
respectively;
a
vector
of
uncontrolled
parameters   ( ,  ,,  )    R ,  is an
opened connected set; maps f ( x, u,  ) : X W    R n
1
and
f
,
x
f
,
u
f

2
m
m
are continuous in X W   ,
[t 0 , t 0  T ] - a fixed interval (time).
As is known [4], 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 (development) of a nonlinear
dynamic system under the given vector of initial values
x0 is defined by the values of both controlled and
uncontrolled parameters.
Main components of market economy parametrical
regulation theory have been developed and offered in
[4, 5, 6] based on the fact of a possible description of a
country’s economic system with the help of a
mathematic model of type (1). Thus, in [3, 2] an
approach is offered to finding the vector u values in the
form of extremals of variation calculation task on the
choice of parametrical regulation laws from the given
set of parameters’ dependences on these or those
significant endogenic indicators of economic system’s
evolution. These laws are deduced from the conditions
of economic processes optimal evolution, when certain
parameters (e. g., price level) are preserved within the
given scales. In [7] a dependence of the deduced
optimal regulation laws on a non-regulated vector 
change has been investigated. An assumption has been
proved on the existence of solutions to the above
mentioned variation calculation tasks. A definition is
provided and an assumption is proved on the sufficient
terms of existence of extremals bifurcation points of
the above-mentioned variation calculation tasks under
the parametrical perturbations [4].
Development and usage of this theory in solving of
certain tasks of parametrical regulation of market
economy development depend on the choice of
mathematic models meeting the requirements of
economic system evolution. This requires additional
study of roughness (structural stability) of the models
chosen [1, 12]. The results of this research allow
considering an adequate nature of mathematic models
as well as a structural stability of the economic system
described by this model. Development and application
of a theory for the chosen models require that definite
parametrical regulation laws should be chosen, and the
dependence of the chosen laws on the values of the
non-controlled parameter  should be studied as well.
The paper presents latest results of development and
application of the parametrical regulation theory to a
mathematical model of an economic system with the
account of foreign commerce [3].

 i   s i i 

xi 
1  
1   i   p i 
1 i
i

Rid
(11)
1
 1 i
 1  i
f i  1  1 
x 
 i i 

;
(13)
G
i   i pi M i f i ;
(14)
 iL

 iI 
(1  n L i ) si Rid
;
(15)
1

1  n pi
2.1. Model description
n0 i (d iB  d iP )  n p i  O
i 
dM i

 i M i ;
dt
pi bi
(2)
dQi

 M i fi  i ;
dt
pi
(3)
dLGi

dt
rG i LGi
(4)
 G
i
 n p i  i  n L i s i RiL
 nO i (d iP
dpi
Q
  i i pi ;
dt
Mi
 R d  R S 
dsi
s
 i max 0, i S i ,
dt i

Ri

RiL

LPi 
 d iB );
(5)
LGi ;
k qi M i f i
i
 (1  n pi ) G
i 

(16)

 n L i  (1  n L i )n p i s i RiL 
n pi ( ji   i  ij )   i L pi  rG i LPi };
RiS  P0Ai exp(  p it )
i 
(1  CiL i
 iL
pj
pi
1
1   ii
,
j  3i
;
;
) P0 i ( pit )
(27)
p2
O p2
С1
p1
p1
L
 12 
1 
 1O ;(18)
L p2
O p2
1  C1 
1  C1 
p1
p1
p
p
С 2L 1
С 2O 1
p2
p
2
 21 
 2L 
 O2 ;(19)
L 1 p1
O 1 p1
1  С2
1  C2
 p2
 p2
С1L
1  1I  1L  1O  1G  21  12 ; (20)
(6)
1
 2   2I   2L   O2   G2  12   21 . (21)

min{ Rid , RiS };
1  i
(12)
O
i   0 i pi M i f i ;
{
 iI
(10)
 M i xi ;
2. Research of rigidness (structural
stability) of a mathematic model without
parametrical regulation
The considered task is solved on the basis of the
parametrical regulation theory and on the sample of the
following mathematical model [10], which presents the
phase restrictions and restrictions in the permitted form
of the researched variational calculus task at the choice
of parametric regulations laws by the following
relationships:


;


i
1  i
d iP 
 i r2 i LGi ;
i
(7)
d iB   i r2 i LG
i ;
(9)
(8)
Here: i = 1, 2 is for a number of the state; t is for
time; M i – total production capacity, Qi – the general
stock of the goods in the market relatively some
balance; LGi – total amount of a public debt; pi –
price levels; si – average real wages; LPi -volume of
manufacture debts; d iP and d iB - enterprise and bank
dividends respectively; Rid and RiS – a supply and
demand of a labor respectively;  i ,  i - parameters of
function
f i( xi ) 
fi ;
si
pi
xi
– a solution to the equation
;  iL and  O
i -consumer spending of
workers and proprietors; iI - investment flows;  G
i consumption government spending;  ij – consumer
spending of the i-th country on an imported product
from the j-th country; θ - exchange rate of the first
country in relation to the exchange rate of the second
country ( 1   ,  2  1/  ); CiL (CiO ) - volume of
imported product units consumed by the workers
(proprietors) of the i-th country on the domestic
product unit;  i – norm of reservation;  i - ratio of
the average profit rate from commercial activities to
the rate of return of the investor; r2i - deposits rate ;
r1i - credit rate; rGi – government bonds rate; Oi factor of proprietors’ propensity to consume;  i -share
of consumption government spending from the gross
domestic product; nPi , nOi , nLi - rates of the tax to flow
payments, dividends tax and surtax accordingly; bi norm of a capital intensity of a production capacity
unit;  i -factor of power unit leaving which is caused
by degradation;  i - norm of amortization; ai - time
constant;  i - time constant setting the characteristic
time-scale of wages process relaxation;
P0i , P0Ai
- initial
values of amount of workers and total amount ablebodied population respectively; i - per capita
consumption in group of employees;  Pi  0 appointed tempo of demographic growth; k qi - share
of the gross domestic products reserved in gold.
This system reduces to a system of the eight
ordinary differential equations for the variables, si are
constant.
The model parameters and initial conditions for the
differential equations (2)-(6) were accepted on the
basis of economic data of the Republic of Kazakhstan
and the Russian Federation at 1996-2000
or
(bi , r1i , r2i , rGi , nPi , nLi ,  i , si , Oi , i ,  i* ,  i )
are assessed through the solution of the parametric
identification task ( i ,  i ,  i ,  i , bi ,  i , Qi (0)) .
2.2. Research of rigidness (structural stability)
of a mathematic model
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 sufficient conditions of rigidness [12]. The
conditions are as follows.
Let N  be some set and N be such a compact subset

N that the closure of interior of N would be N. Let
some vector field be given in the area of set N in N  ,
then this field would determine the C 1 flow f in this
area. The chain recurrent set of the flow f on the N is
marked as R( f , N ) .
Let the R( f , N ) be contained inside the N and have
a hyperbolical structure; besides this, the f on the
R( f , N ) would satisfy the transversality conditions on
stable and unstable manifolds. Then the flow f on the N
is weakly structurally stable. Particularly, if the
R( f , N ) is an empty subset, then the flow f weakly
structurally stable on the N.
We will assume further that Rid > RiS . In this case
differential equations (6) are replaced with the
conditions of stability of variables si .
Statement 1. Let N be a compact set in area
(M 1  0, Q1  0, p1  0) or (M 1  0, Q1  0, p1  0) ,
of the phase area of differential equations system
obtained from the (2-21), i.е. eight-dimension area of
variables ( M i , Qi , pi , LG i ) , i  1, 2 ; closure of
interior of N coincides with the N. Then the flow f
determined by the (2-21) is weakly structurally stable
on the N.
F.i., a parallelepiped could be chosen as the N, with
boundaries
M i  M i min , M i  M i max , Qi  Qi min , Qi  Qi max ,
pi  pi min , pi  pi max , LG i  LG i min , LG i  LG i max
Here
0  M i min  M i max ,
0  Qi min  Qi max ,
Qi min  Qi max  0
.
or
0  pi min  pi max ,
LG i min  LG i max .
The proof. To start with, let us make sure that the
half-trajectory of the flow f beginning in any point of
the set N under a certain value of t (t>0) comes out
from the N. Let us consider any half-trajectory starting
in the N. There are two cases possible for it if t  0 : all
the half-trajectory points are left in the N, or for a
certain t a point of the trajectory does not belong to the
N. It follows in the first case from equation (5) of the
system dp1   Q1 p1, that the variable p1(t) for all the
dt
M1
t  0 has got a derivative that is either more than some
positive
constant
under
the
N  (M 1  0, Q1  0, p1  0) or less some negative
constant under the N  (M 1  0, Q1  0, p1  0) , that
is, the p(t) unlimitedly rises or tends to zero under the
unlimited increase of the t. This is why, the first case is
impossible, the orbit of any point from the N comes out
from the N.
As far as any chain recurrent set R( f , N ) is an
invariant set of this flow, in case of its non-emptiness it
should consist of whole orbits. Consequently, in our
case the R( f , N ) is empty. The assumption has been
proved.
3. Research of rigidness (structural
stability) of the mathematic model with
parametrical regulation
Handling the law U i ,  under the fixed ki ,  in the
system (2-21) means the substitution function U i , 
from (22) in equations of the system (2-21) instead of a
parameter  i ,  i or  .
The task of selection of an optimal parametrical
regulation law for the economic system of the icountry at the level of one of the economic parameters
( i ,  i , ) was set in the following form: to find on the
basis of mathematical model (2–21) an optimal
parametrical regulation law in an environment of the
set of algorithms (22), i.е. to find an optimal law (and
its factor ki ,  ) out of the set { U i ,  }, which would
have provided a maximum for the criteria
Ki 
3.1 The task of choice of effective parametrical
regulation law
The possibility of the choice of an optimal set of
laws of type (3) of parametrical regulations was
researched: at the level of one of the 3 parameters
 i (   1) ,  i (   2) ,  (  3) ; at the time interval
[t 0 , t 0  T ] and in an environment of the following
algorithms.
1) U1i,   k1i, 
2)
U 2i , 

M i (t )
 consti ;
M i (t0 )
k2i , 
3) U 3i ,   k3i , 
M i (t )
 consti ;
M i (t0 )
pi (t )
 consti ;
pi (t0 )
4) U 4i ,   k4i , 
1 t0 T
 Yi (t )dt ,
T t0
(23)
where Yi  M i f i is a gross domestic product of the istate. The calculation experiments studied the impact
of parametrical regulation of the first country (i=1).
The closed set  in the space of continuous vectorfunctions of discharge variables of the system (2-21)
and regulating parametrical influences is defined with
the following relationships:
1
p
(t )  p1* (t )  0.09 p1* (t ),
( M i (t ), Qi (t ), LGi (t ), p i (t ), s i (t ))  X ,
(22)
pi (t )
 consti
pi (t0 )
Here: U i ,  - is a α-law of regulation of βof the i-state,   1  4,  1  3 ,
Mi (t )  M ,  ,i (t )  Mi (t0 ), pi (t )  p ,  ,i (t )  pi (t0 ), t 0 – is
parameter
the time of the start of regulation, t  t0 , t0  T  ,
M  ,  ,i (t ) , p ,  ,i (t ) are values of the production
capacity and price levels of the i–state respectively
under the U i ,  -regulation law. ki ,  is a tuned factor
of the relevant law ( ki ,   0 i ); consti is a constant
that is equal to the assessment of the values of βparameter by the results of parametric identification.
(24)
0  u   a  ,   1, 4,   1, 3, t  [t 0 , t 0  T ]
Here a  is the biggest possible value of a  i
(t ) are values of price levels under the
parameter, p
i
U 
(t )
law
of
regulation;
pi* (t )
are
model
(accounting) values of price levels in i-state without
the parametrical regulation, X is a compact set of
possible values of the given parameters.
The formulated task for the first country is solved in
two stages:
- at the first stage optimal values of factors k1 ,  for
each law U 1 ,  are defined by way of their values
sorting in relevant intervals (quantized with small
step), which provide maximum K 1 under the
restrictions (24);
- at the second stage an optimal parametrical
regulation law of a parameter based on the first stage
results for the maximum value of criterion K 1 will be
chosen.
is optimal for the other, both laws are optimal for the
projection line itself.
3.2. Research of dependence of the optimum
law of parametrical regulation on the values of
uncontrolled parameters
3.3. Results of research of rigidness (structural
stability) of a mathematical model with
parametrical regulation.
The given task of variational calculation considered
its dependence on a two-dimensional factor
  (r2,1,  ) of the mathematical model, whose
Application of the found above optimal laws of
parametrical regulation U 21, 2 and U 41, 2 means
possible values belong to some area (rectangle)  on
the plane.
As a result of calculation experiment dependence
graphs of the optimal value of criterion K 1 on the
values of parameters (r2,1 ,  ) were obtained for each
of the 12 possible laws U 1 ,  ,   1,4,   1,3 . Figure 1
demonstrates the given graphs for two laws, U 21, 2 and
U 41, 2 , which give the biggest value of the criterion in
area  , an intersection line of corresponding surfaces
and a projection of this intersection line upon the plane
of values  , which contains the bifurcation points of
this two-dimensional parameter. This projection
divides the rectangle  into two parts; the regulating
law
Figure. 1. Graphs of dependences of the criterion’s
K 1 optimal values on the parameters of interest rates
on deposits r2,1 and currency exchange rate θ
M 1 (t )
 const12
M 1 (t 0 )
is optimal in one of these parts, and the law
p (t )
U 41, 2  k 41,2 1  const12
p1 (t 0 )
U 21, 2  k 21, 2
(25)
(26)
replacement of parameter  1 by the relevant functions
in the equation (14), the rest equations of the model
remain unchanged. The proof of the weak structural
stability of the mathematical model given in i. 2.2. and
basing on equation (5), allows to find the following.
Statement 2. Let N be a compact set in area
(M 1  0, Q1  0, p1  0) or (M 1  0, Q1  0, p1  0) ,
of the phase area of the differential equations system
obtained from the (2-21), i.e. eight-dimension area of
variables ( M i , Qi , pi , LG i ) , i  1, 2 ; closure of
interior of N coincides with the N. Then the flow f
determined by the (2-21) and (25, 26) is weakly
structurally stable on the N.
4. Conclusions
This paper gives the following latest results of the
development and usage of the parametrical regulation
theory for a mathematical model of an economic
system with the account of foreign commerce [10]:
- the model’s weak structural stability for compact
areas of its phase space has been proved;
- the task of choice of optimal parametrical
regulation laws has been solved for maximizing an
average VAT of a country under some limitations upon
price level increase;
- a dependence of the found optimal laws on the
values of non-controlled parameters has been defined
and a set of optimal laws bifurcation points has been
found;
- a weak structural stability of a model preserved
under the usage of the chosen parametrical regulation
laws influenced by the non-controlled parameters of
the model has been proved.
5. References
[1] D.V. Anosov, “Rough systems”, Proceedings of the
USSR Academy of Sciences’ Institute of Mathematics, 1985,
Vol. 169, pp. 59-96 (in Russian).
[2] A. Ashimov, Yu. Borovskiy, and As. Ashimov,
“Parametrical Regulation Methods of the Market Economy
Mechanisms”, Systems Science, 2005 Vol. 35, No. 1, pp. 89103.
[3] А. Ashimov, Yu. Borovskiy, As. Ashimov, and O.
Volobueva, “On the choice of the effective laws of
parametrical regulation of market economy mechanisms”,
Automatics and Telemechanics, 2005, No 3, pp. 105-112 (in
Russian).
[4] А. Ashimov, К. Sagadiyev, Yu. Borovskiy, N. Iskakov,
and As. Ashimov, ’’Parametrical regulation of nonlinear
dynamic systems development”, Proceedings of the 26th
IASTED
International
Conference
on
Modelling,
Identification and Control, Innsbruck, Austria, 2007, pp.
212-217.
[5] А. Ashimov, К. Sagadiyev, Yu. Borovskiy, N. Iskakov,
and As. Ashimov, (2007): “Elements of the market economy
development parametrical regulation theory”, Proceedings of
the Ninth IASTED International Conference on Control and
Applications, Montreal, Quebec, Canada, 2007, pp. 296-301.
[6] А. Ashimov, К. Sagadiyev, Yu. Borovskiy, N. Iskakov,
and As. Ashimov, “On the market economy development
parametrical regulation theory”, Proceedings of the 16th
International Conference on Systems Science, Wroclaw,
Poland, 2007, pp. 493-502.
[7] А. Ashimov, К. Sagadiyev, Yu. Borovskiy, and As.
Ashimov, “On Bifurcation of Extremals of one Class of
Variational Calculus Tasks at the Choice of the Optimum
Law of Parametrical Regulation of Dynamic Systems”,
Proceedings of Eighteenth International Conf. On Systems
Engineering, Coventry University, 2006, pp. 15-19.
[8] Gukenheimer, J., P. Cholmes, Nonlinear fluctuations,
dynamic systems and bifurcations of vector fields, Institute of
Computer Researches, Moscow – Izhevsk, 2002 (in Russian).
[9] V.М. Matrosov, М.М. Chrustalyov, О.V. Arnautov, and
Krotov V.F., “On the highly aggregate model of development
of Russia”, The Proceedings of the 2nd International
conference “Analysis of instability development based on
mathematical modeling”, Moscow, 1992, pp. 182-243 (in
Russian).
[10] Petrov, A., I. Pospelov, and A. Shananin, Experience of
mathematical modeling of economy, Energoatomizdat,
Moscow, 1996 (in Russian).
[11] Pontryagin, A., The ordinary differential equations,
Nauka, Moscow, 1970 (in Russian).
[12] C. Robinson, “Structural Stability on Manifolds with
Boundary”, Journal of differential equations, No. 37, 1980,
pp. 1-11.