Keywords: dynamical systems, systems modelling, parametrical regulation, task of variational calculation, structural stability Abdykappar A. ASHIMOV*, Kenzhegaly A. SAGADIYEV*, Yuriy V. BOROVSKIY**, Nurlan A. ISKAKOV*, Askar A. ASHIMOV* DEVELOPMENT AND USAGE OF THE MARKET ECONOMY PARAMETRICAL REGULATION THEORY ON THE BASIS OF ONE-CLASS MATHEMATICAL MODELS 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 [1] including economic systems of nations [2, 3], 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 ) 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 R l - a vector of (regulating) parametrical influences; W, Х are compact sets with non-empty interiors Int ( X ) Int (W ) and respectively; a vector of uncontrolled parameters (1 , 2 , , m ) R m , is an opened connected set; maps f ( x, u, ) : X W R n and f f f , , are continuous in X W , [t 0 , t 0 T ] - a fixed interval (time). x u ____________________________ * Laboratory of Systems Analysis and Control, Institute of Problems of Informatics and Control, 221 Bogenbay batyr str., Almaty, 050026, Republic of Kazakhstan, e-mail: [email protected] ** Kazakh-British Technical University, 59 Tole be str., Almaty, 050000, Republic of Kazakhstan, e-mail: [email protected] 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 (λ) 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 nonregulated vector λ 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 λ 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 for research of influence of the consumer charges of the state on development of economy [10]. 2. RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF A MATHEMATIC MODEL WITHOUT PARAMETRICAL REGULATION 2.1. MODEL DESCRIPTION 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: dM I M , dt pb (2) dQ Mf , dt p (3) dLG rG LG G n p n L sR L nO (d P d B ) , (4) dt dp Q p, dt M Rd RS L ds s d S max 0, , R min{ R , R } , S dt R Lp dp 1 1 (5) (6) LG , (7) r2 LG , (8) d B r2 LG , (9) 1 s x 1 1 p , (10) R d Mx , (11) O 0 pMf , (12) G pMf , (13) 1 I 1 1 f 1 1 x , (14) L (1 n L ) sR d , (15) 1 (1 )n p (1 n p ) G (16) n0 (d d ) n p nL (1 n L )n p sR ( rG ) L , B P O L * O G L I , R S P0A exp( p t ) 1 L , . 1 pP0 exp( p t ) p (17) (18) G L Here: M stands for total production capacity; Q – the general stock of the goods in the market; – total amount of a public debt; p – price level; s – average real wages; LP - volume of manufacture debts; d P and d b - accordingly owner’s and bank’s dividends; R d and R S accordingly demand and supply of labor; , - parameters of function f(x); x - solution of f x s p equation; L and O - accordingly consumer expenses of employees and owners; I - investment flow; G - consuming government spending; ξ - norm of reservation; β - the attitude of average rate of return from commercial activity to the rate of return of the investor; r2 – interest rate on deposits; rG - interest rate on government bonds; O - coefficient of owners disposition to consumption; π - state's consumer expenses share in gross domestic product; n p , n0 , n L - accordingly rates of taxes on cash flow, dividends and income of employees; b - rate of fund-capacitance of power unit; μ - factor of power unit leaving which is caused by degradation; * - norm of amortization; α - time constant; Δ - time constant, which is set of characteristic time scale of process relaxation of wage; P0 , P0A - initial number of workers and an aggregate number ablebodied accordingly; p >0 - appointed tempo of demographic growth; ω – per capita consumption in group of employees. The model parameters and initial conditions for the differential equations (2)-(6) were obtained on the basis of economic data of the Republic of Kazakhstan [17] at 1996-2000 ( r2 = 0.12; rG = 0.12; β = 2; n p = 0.08; nL = 0.12; s = 0.1; n0 = 0.5; μ = μ* = 0.012; Δ=1) or are assessed through the solution of the parametric identification task (ξ=0.1136; π=0.1348; δ=0.3; ν=34; O = 0.05; b = 3.08; α = 0.008; Q(0) = - 125000). 2.2. RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF THE MATHEMATIC MODEL Let us conduct statement of rigidness (structural stability) of a model under consideration in a closed field Ω, 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 C1 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 R d > R S . In this case differential equations (6) are replaced with the conditions of stability of variable s. Statement 1. Let N be a compact set in area ( M 0, Q 0, p 0) or ( M 0, Q 0, p 0) , of the phase area of differential equations system obtained from the (2-18), i.е. four-dimension area of variables ( M , Q, p, LG ) ; closure of interior of N coincides with the N. Then the flow f determined by the (2-18) is weakly structurally stable on the N. F.i., a parallelepiped could be chosen as the N, with boundaries p pmin , p pmax , LG LG min , LG LG max . Here M M min , M M max , Q Qmin , Q Qmax , 0 M min M max , Qmin Qmax 0 or 0 Qmin Qmax , 0 pmin pmax , LG min LG 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 dp Q p, that the variable p(t) for all the t 0 in the first case from equation (5) of the system dt M has got a derivative that is either more than some positive constant under the N ( M 0, Q 0, p 0) or less some negative constant under the N (M1 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 nonemptiness 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 3.1. THE TASK OF CHOICE OF EFFECTIVE PARAMETRICAL REGULATION LAW The possibility of a choice of the optimum law of parametrical regulation at a level of one of two parameters ξ (j = 1) and π (j = 2), and in the interval of time [t0 , t0 T ] were investigated in the environment of the following algorithms. 1)U 1 j (t ) k1 j M M0 const j , M0 3)U 3 j (t ) k 3 j p p0 const j , p0 2)U 2 j (t ) k 2 j M M0 const j , M0 4)U 4 j (t ) k 4 j p p0 const j . p0 (19) Here: U ij - is a i-law of regulation of j-parameter, i 1 4, j 1 2 ; t 0 – is the time of the start of regulation, t t 0 , t 0 T ; k ij is a tuned factor of the relevant law ( kij 0 i ); M 0 , p0 are initial values of the corresponding variables; const j is a constant that is equal to the assessment of the values of j-parameter by the results of parametric identification. Handling the law U ij under the fixed k ij in the system (2-18) means the substitution function U ij from (19) in equations of the system (2-18) instead of a parameter ξ or π. The task of selection of an optimal parametrical regulation law for the economic system at the level of one of the economic parameters (ξ, π) was set in the following form: to find on the basis of mathematical model (2–18) an optimal parametrical regulation law in an environment of the set of algorithms (19), i.е. to find an optimal law (and its factor k ij ) out of the set { U ij }, which would have provided a maximum for the criteria K 1 t0 T Y (t )dt , T t0 (20) where Y Mf is a gross domestic product. The closed set in the space of continuous vector-functions of target variables of the system (2) - (18) and regulating parametrical influences is defined by the following conditions pij (t ) p ** (t ) 0.09 p ** (t ), ( M (t ), Q(t ), LG (t ), p(t ), s (t )) X , 0 u j a j , i 1,4, j 1,2, t [t 0 , t 0 T ]. (21) Here a j - is the biggest possible value of j- parameter, pij (t ) are the values of price levels under the U ij regulation law; p ** (t ) are model (counting) values of price levels without parametrical regulation, X is a compact set of allowable values of the specified parameters. The formulated task is solved in two stages: - at the first stage optimal values of factors k ij for each law U ij are defined by way of their values sorting in relevant intervals (quantized with small step), which provide maximum K under the restrictions (21); - 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 will be chosen. The results of the numerical solution of the problem of choosing an optimum parametrical regulation law at a level of one of economic parameters for economic system of the state show that the best result K 177662 can be received via the usage of the following regulation law 0.095 M M0 0.1136 . M0 (22) Let us notice that the value of the criterion without usage of the parametrical regulation equals to K 170784 . 3.2. RESEARCH OF DEPENDENCE OF THE OPTIMUM LAW OF PARAMETRICAL REGULATION ON THE VALUES OF UNCONTROLLED PARAMETERS The calculating experiment [6] brought about the graphs of dependence of the optimal value of criterion K on parameter values (r2 , nO ) for each of the 8 probable rules U ij , i 1,4, j 1,2 . Figure 1 presents the given graphs for the rules U 21 and U 41 , which give the biggest value of a criterion in area Λ, the intersection of corresponding surfaces and the projection of this intersection upon the plane of values λ, consisting of the bifurcation points of this parameter. This projection divides the rectangle Λ into two parts, where the controlling rule U 21(t ) k 21 M M0 const1 M0 (23) is optimal for one part, and U 41(t ) k 41 p p0 const1 p0 (24) is optimal for the other; at the projection itself both rules are optimal. Fig. 1. Graphs of dependences of the criterion’s K optimal values on the parameters of interest rates on deposits r2 and rates of taxes on dividends nO. 3.3. RESULTS OF RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF THE MATHEMATICAL MODEL WITH PARAMETRICAL REGULATION Application of the found above optimal laws of parametrical regulation U 21 and U 41 , means replacement of parameter ξ by the relevant functions in the equations (7, 8, 16), 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 0, Q 0, p 0) or ( M 0, Q 0, p 0) , of the phase area of the differential equations system obtained from the (2-18), i.e. four-dimension area of variables ( M , Q, p, LG ) ; closure of interior of N coincides with the N. Then the flow f determined by the (2-18) and (23, 24) 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 for research of influence of the consumer charges of the state on development of economy [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. REFERENCES ANOSOV D.V., Rough systems, Proceedings of the USSR Academy of Sciences’ Institute of Mathematics, Vol. 169, 1985, pp. 59-96 (in Russian). [2] ASHIMOV A., BOROVSKIY YU., ASHIMOV AS., Parametrical Regulation Methods of the Market Economy Mechanisms, Systems Science, Vol. 35, 2005, No. 1, pp. 89-103. [3] ASHIMOV А., BOROVSKIY YU., ASHIMOV AS., VOLOBUEVA O., On the choice of the effective laws of parametrical regulation of market economy mechanisms, Automatics and Telemechanics, 2005, No 3, pp. 105112 (in Russian). [4] ASHIMOV А., SAGADIYEV К., BOROVSKIY YU., ISKAKOV N., ASHIMOV AS., 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] ASHIMOV А., SAGADIYEV К., BOROVSKIY YU., ISKAKOV N., ASHIMOV AS., 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] ASHIMOV А., SAGADIYEV К., BOROVSKIY YU., ISKAKOV N., ASHIMOV AS., On the market economy development parametrical regulation theory, Proceedings of the 16th International Conference on Systems Science, Wroclaw, Poland, 2007, pp. 493-502. [7] ASHIMOV А., SAGADIYEV К., BOROVSKIY YU., ASHIMOV AS., 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, Coventry, UK, 2006, pp. 15-19. [8] GUKENHEIMER J., CHOLMES P., Nonlinear fluctuations, dynamic systems and bifurcations of vector fields, Institute of Computer Researches, Moscow – Izhevsk, 2002 (in Russian). [9] MATROSOV V.М., CHRUSTALYOV М.М., ARNAUTOV О.V., 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., POSPELOV I., SHANANIN A., Experience of mathematical modeling of economy, Energoatomizdat, Moscow, 1996 (in Russian). [11] PONTRYAGIN A., The ordinary differential equations, Nauka, Moscow, 1970 (in Russian). [12] ROBINSON C., Structural Stability on Manifolds with Boundary, Journal of differential equations, 1980, No. 37, pp. 1-11. [1]

Download
# 319.50Kb - G