Structural Stability of Mathematical Models of
National Economy
Abdykappar A. Ashimov, Bahyt T. Sultanov, Yuriy V. Borovskiy,
Zheksenbek M. Adilov, Askar A. Ashimov
Kazakh National Technical University named after K. Satpayev,
22 Satpayev Str, Almaty city, 050013, Kazakhstan
Abstract. In the paper we test robustness of particular dynamic systems in a compact regions of
a plane and a weak structural stability of one dynamic system of high order in a compact region
of its phase space. The test was carried out based on the fundamental theory of dynamical
systems on a plane and based on the conditions for weak structural stability of high order
dynamic systems. A numerical algorithm for testing the weak structural stability of high order
dynamic systems has been proposed. Based on this algorithm we assess the weak structural
stability of one computable general equilibrium model.
Keywords: Mathematical model, structural stability, parametrical regulation.
PACS: 05.45.-a
1 INTRODUCTION
Many dynamical systems, including the country's economic system, after some
transformations can be represented by systems of nonlinear ordinary differential
equations (with initial condition) of the following form [1], [2]
x (t )  f ( x(t )), u (t ),  ),
(1)
x(t 0 )  x0 .
(2)
Here t – time, t  [t 0 , t 0  T ] , T  0 , – a fixed number; x  x(t )  R m – function of the
(1), (2) system’s state; x 0  R m – initial state of the system, a deterministic vector;
u  u(t )  R q – vector of regulated parameters, it is assumed that functions u (t ) and
their derivatives are uniformly bounded;   A  R s – vector of unregulated
parameters, A – open connected set.
As it is known, the solution (evolution) of the researched system of ordinary
differential equations depends on the vector of initial values x0 , also on the values of
vectors of regulated (u) and unregulated (α) parameters. Therefore the results of
evolution (development) of the nonlinear dynamical system under given vector of
initial values x0 is determined by the values of regulated and unregulated parameters.
Also it is known [3], that in order to judge by the solutions of the system (1) about
the object it describes, the system must have the property of immutability of the
qualitative picture of the trajectories when there are small in some sense perturbations
of the right side of the system (1). In other words, the system (1) should have the
property of robustness, or structural stability.
Based on the above, in [4], [5], [6] we propose a theory of parametrical regulation
of market economy development, that is composed of eight components, two of them
consist the methods of assessment the conditions for robustness (weak structural
stability) of mathematical models of a country’s economic system from the library of
models without parametrical regulation and applying optimal laws of parametrical
regulation.
The paper presents the results of the research of robustness and weak structural
stability of the particular mathematical models of national.
2 METHODS OF STUDYING ROBUSTNESS (STRUCTURAL
STABILITY) OF A MATHEMATICAL MODEL OF ECONOMIC
SYSTEM
Methods of studying robustness (structural stability) of mathematical models of a
country’s economic system are based on the followings.
- Basic results of the theory of dynamic systems on the plane. It is known [7] that
the dynamic system defined in a neighborhood of the closed region G on the plane
with a boundary which is a simple closed curve is robust only if the region G: 1)
contains only robust singular points, 2) contains only robust limit cycles, 3) contains
no separatrixes going from a saddle to a saddle.
- Methods of testing the conditions of accessory of mathematical models to certain
classes of structurally stable systems, in particular, to systems with weak structural
stability.
Along with analytical opportunities of the research of structural stability of
particular mathematical models on the basis of stated results of the theory of
dynamical systems we can investigate approaches of structural stability research of
national economy mathematical models with the help of computational experiments.
In paper results of application of one computing algorithm [9] (of an estimation of
structural stability of considered mathematical models of the country economic system
on the basis of the Robinson's theorem [9, Theorem A]) are use. This theorem in
particular states that if a chain-recurrent set R( f , N ) of the flow f on some compact N
of its phase space is empty, then this flow f is weakly structurally stable in N. The
analogous statement is true for the case of a cascade that is determined by the
homeomorphism f. For specific mathematical model of economic system we can take,
for example, a parallelepiped of its phase space that includes all possible trajectories
of economic system evolution for the examined time segment, as a compact N.
If the studied discrete dynamical model, a priori, is a semi-cascade, checking the
invertibility of the mapping f, set on N should be a prerequisite for application of the
Robinson Theorem A for the assessment of its structural stability (as in this case the
semi-cascade, which is set by f, is a cascade).
3 EXAMPLES OF ASSESSMENT THE STRUCTURAL
STABILITY OF MATHEMATICAL MODELS
3.1 Investigation of the structural stability of the mathematical model
of the neoclassical theory of optimal growth
The mathematical model of economic growth [10] is presented by the following
system of the two ordinary differential equations, containing time (t) derivatives:
k  Ak   c  (n   )k ,

c

 1
c  1   (Ak  (  p)).

(3)
Here k(t) – capital (K) to labour (L) ratio, i.e. capital endowment of labour. In this
model there is no difference between population and the working force (labour); c(t) –
average per capita consumption; n – rate of population growth (or decrease):
L(t )  L0 e nt ; δ – rate of capital amortization,   0 ; p – discounting rate; e  pt –
function of discounting ( p  n ); α and A – parameters of the production function of
the type y   (k )  Ak  , where y – GDP to labour ratio, i.e. average labour
productivity ( 0    1, A  0 );  – parameter of the function of social utility, that
characterizes average welfare of population: U (c)  Bc  ( 0    1, B  0 ).
It is not difficult to check that the system (3) in the region R2 has a unique singular
1
1
 A 
(n   )(1   )  p  n 
 , c *  k * 
point: k *  
 . This point is a saddle point



  p 
of the system (3) under any values of input parameters of the model. The model (3)
does not have cyclical trajectories in R2 , and it is robust in any region N that belongs
to R 2 (if it is not the case when the singular point ( k * , c * ) lies on its boundary).
3.2 Investigation of the structural stability of the mathematical model
of Kondratiev cycle
The model is described by the following system of equations, that includes two
differential and one algebraic equations [11]
n(t )  Ay(t ) a ,

 x(t )  x(t )( x(t )  1)( y 0 n 0  y (t )n(t )),

 y (t )  n(t )(1  n(t )) y (t ) 2 ( x(t )  2    l 0 ),

n0 y 0

n 0  Ay 0  .
(4)
Endogenous variables: x(t) – effectiveness of innovations; y(t) – capital
productivity; n(t) – savings rate. Exogenous parameters of the model: y0 – capital
productivity, that corresponds to the equilibrium trajectory; n0 – savings rate, that
corresponds to the equilibrium trajectory; µ – rate of funds retirement; l0 –
employment growth, that corresponds to the equilibrium trajectory; A and α –
constants.
Preliminary parameter estimates of the model were carried out applying statistical
data for the Republic of Kazakhstan for 2001-2005, deviation of observed statistical
data from calculated values of endogenous variables does not exceed 1.9% for the
stated time interval.
As a result of applying the algorithm for assessing the chain-recurrent set for a
rectangular region N  [1.7; 2.3]  [0.066; 0.098] of the phase space Oxy of system (4),
we obtained the following assessment of the chain-recurrent set R(f, N) (see Fig.1).
Since the set R(f, N) is not empty, then based on the Robinson theorem we cannot
judge about structural stability of the Kondratiev cycle model in N. However, since
  l0
the nonhyperbolic singular point – center ( x0  2 
, y 0 ) [11] is located in N,
n0 y 0
then the system (4) is not weakly structurally stable in N.
FIGURE 1. The assessment of chain-recurrent set for the model of Kodratiev cycle.
3.3 Research of weak structural stability of the mathematical model
of a country’s economic system taking into account the effect of
government expenditures and interest rate on government debts on
the level of economic growth
The mathematical model of a country’s economic system for investigating the
effect of government expenditures rate out of gross domestic product and interest rate
on government debts on economic growth, that is proposed in [1], after appropriate
alterations, is represented as a system that has 13 algebraic and 4 differential
equations; from which we select the following:
dp
Q
 
p.
dt
M
(5)
The endogenous variables are: p(t) – price level, Q(t) – total stock of goods in the
market with respect to some equilibrium state; М(t) – total production capacity; LG(t) –
total government debt,   0 – a constant.
The following assertion is proved with use of Robinson's theorem A and the
formula (5).
Assertion. Let N – be the compact set that belongs to the domain
( M  0, Q  0, p  0) or ( M  0, Q  0, p  0) of the phase space of the system of
differential equations of the mathematical model, i.e. four-dimensional space of
variables ( M , Q, p, LG ) , the closure of N interior coincides with N. Then the flow f,
which is defined by the model is weakly structurally stable on N.
3.4 Research of weak structural stability of computable general
equilibrium model with knowledge sector
The considered model [12] is represented by 67 difference and 631 algebraic
equations. With the help of these equations we calculate values of 698 endogenous
variables. This model also contains 2045 estimated exogenous parameters. According
to the results of parametrical identification and model calculation for the period from
2000 to 2015 we obtained the trajectory L of the model in the phase space X and chose
a compact N  X that contained the trajectory.
Having applied the numerical algorithm for assessing weak structural stability of a
discrete dynamic system for the selected compact N, we evaluated the chain-recurrent
set R( f , N ) as an empty set.
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