Mathematical Models to Determine Stable Behavior of Complex Systems

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS Matheatical Models to Deterine Stable Behavior of Coplex Systes To cite this article: V I Suin et al 08 J. Phys.: Conf. Ser View the article online for updates and enhanceents. Related content - Mechatronics: Dynaical systes approach B T Fijalkowski - A MATEMATICAL MODEL FOR PREDICTING NIGT-SKY Mark A. Yocke, enry ogo and Don enderson - Matheatical odeling of piezoresistive eleents M Gereias, R C Moreira, L A Rasia et al. This content was downloaded fro IP address on 7//08 at 00:05

2 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 Matheatical Models to Deterine Stable Behavior of Coplex Systes V I Suin, A V Dushkin, T E Solentseva Voronesz Institute of the Federal Service for the Execution of Sanctions of Russia Lipetsk State Technical University» Abstract. The paper analyzes a possibility to predict functioning of a coplex dynaic syste with a significant aount of circulating inforation and a large nuber of rando factors ipacting its functioning. Functioning of the coplex dynaic syste is described as a chaotic state, self-organized criticality and bifurcation. This proble ay be resolved by odeling such systes as dynaic ones, without applying stochastic odels and taking into account strange attractors. Key words: coplex dynaic syste, chaos, self-organized criticality, bifurcation, stochastic odels, strange attractor, stochastic integral, Ito integral, Stratonovich integral, fractal diension, Lyapunov theory, Fokker-Planck-Kologorov equation.. Introduction. The proble of the paper is defined as follows: analysis and synthesis of coplex systes (CSs) with considerations for atheatical odeling [] and application of non-linear systes is a coplex proble where no universal ethods of synthesis and odeling exist, thus, ipeding solution of a diverse range of applied probles, including those with possibility to control CS functioning. To ake it possible to control functioning of a CS, let us assue that it transfors input (control) signals U ( t) into output signals ( t), describing the CS's state at oent t with considerations of possible disturbances ξ ( t). Real-life CSs, including social ones, ay have ultiple inputs, outputs and disturbing influences. Constructing a foralized odel of social CSs is difficult due to the having certain peculiarities []: - stochasticity; - non-linearity; - tie-dependence; - deterinistic nature at a sall-scale tie interval; - non-stationary behavior; - ipossibility to give adequate description of a syste being studied; - indeterinateness. Control of social CSs for partial probles ay be described with non-linear systes. Thus, solution of partial probles linked to CS control with non-linear systes is a tiely task, because any odels of real-life CSs ay be reduced to a certain partial proble. Content fro this work ay be used under the ters of the Creative Coons Attribution 3.0 licence. Any further distribution of this work ust aintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336. Theoretical Part CS functioning with non-linear systes with rando inputs ay be described with differential equations in the following for [3]: d ( t) = f ( t, ( t) ) + σ ( t, ( t) ) G( t), () dt where is a state vector; f ( t, x), σ ( t, x) are a vector and a atrix function; G ( t) is a standard Gauss's white noise. For differential equation (), the initial state is described as a rando vector, whose values have a known probability density. The Ito integral is often eployed to find the stochastic integral; it is calculated as a liit of convergent integral sus in a ean-square sense: N σ ( τ, ( τ )[ W ( τ ) W ( τ )], () i i i + i i = where W ( t) is a standard Wiener rando process under an assuption that W ( t ) 0, [ ( )] 0 = M W t 0 for all t > t 0, while vector W ( t) has a noral distribution, assuing that the process is unifor and has independent increents. As an alternative to the Ito integral, a Stratonovich integral ay be used, as well as a generalized θ -integral. The proble of analysis, synthesis and odeling of a CS control syste is often reduced to a deterinistic proble with preset liits [,3]. Thus, the proble of CS output paraeter analysis is based on deterination of a given probability density p 0( x ), vector f ( t, x) and atrix a ( t, x ) = k σ ( t, x ) σ ij il jl ( t, x ), and in finding the law describing changes in probability l= density p ( t, x). Using the Fokker-Planck-Kologorov equation: ( t, x) n n [ f ( t, x) p( t, x) ] + a ( t, x) p( t, x) p n =. (3) t = i i = j = x x ij i x i i j Equation (3) is a parabolic partial-derivative equation; application of different nuerical or analytical ethods, one ay find the full inforation on the behavior of a dynaic CS. CS control with odeling on the basis of non-linear systes [,7] CS inputs is sets of functional uncertainties, deterined by a easured deterinistic disturbance and described with an equation in the for of: ( ) ( ) ( ) x = f x + G x ( u + w x, t + ω), (4) where x R is a state vector; u is a control signal; f ( x) and G ( x) are sooth vector functions and and atrix functions; w ( x, t) is an unknown vector function describing uncertain characteristics of CS's functioning. For CS in the for of (4), it is possible to use a control odel, where for a certain class of functional uncertainties and for soe liited external disturbances, an asyptotic stability ay be reached for soe variables of the CS as described with non-linear odels, which ay be put into a noral canonical for. The proble of state estiate of a hierarchic social syste takes an iportant place in the analysis of such systes [4]. For exaple, in [5], such estiate ay be ipleented by applying the least square ethod with extension of state space. owever, the least square estiation is perfored on the basis of a large nuber of calculations, thus, lowering the accuracy.

4 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 For any CS, deterining its paraeters is a coplex task due to a large volue of inforation circulating inside the CS, as well as a ultitude of rando factors influencing the functioning of the syste. Such behavior of a dynaic CS corresponds to a state of chaos, bifurcation, self-organized criticality [5]. CS functioning in the state of chaos ay be described if initial conditions are known, that is, the state of the CS largely depends on input paraeters, which ay be easured with a certain level of error. It is clear that prediction of CS functioning is not a siple task due to the fact that description of functioning of non-linear dynaic systes is challenging due to non-linearity and local instability of their dynaics; also, a sall initial error is leading to a change in CS paraeters which is exponential with respect to tie, coplicating long-ter prognosis. This issue ay be partially alleviated for solution of siple autonoic dynaic systes without use of stochastic odels [6] by applying strange attractors. Regularity of chaotic behavior of a CS correlates (at different tie scales) with fractal behavior (at different spacial scales). In both CS and fractals, characteristic uncertainty distributions behave following a power law and not the exponential one. It should be highlighted that CSs functioning as fractals [7] are changing at the edge of chaos and their functioning ay be described with deterinistic chaos, and in this case analysis on the basis of long-ter forecasting is possible. In absolutely chaotic CSs, forecasting is possible only over a tie interval depending on Kologorov's K-entropy: K = li li li P ln P l N N i i i i N i 0 0 i. (5) τ 0 0 τ N 0 N Study of dynaic CS behavior by studying fractal diensions (behavior of dynaic systes with strange attractors) is conducted by deterining systeic indicators of such CS [7], which is characterized with its rando chaotic behavior even for a certain deterinistic odel. Analysis of behavior of dynaic CSs with fractal diensions is based on the analysis of two principal states: oveent to an attractor and oveent along the attractor. Analysis of such CSs has shown that they will never reach a state of equilibriu while a transitional state fro one etastable state to another is possible in case of a sall-scale disturbance of the CS. A full description of CS dynaics with an attractor is possible only when the nuber of CS variables is the sae as the diensionality of the attractor; thus, fractal diension of the attractor which is a characteristic of instability is also a characteristic of CS etastable state. Fractal diension of CS or diension of its confority is deterined with a forula: loga N d =, (6) loga r where is a nuber of equal sub-objects; is a coefficient of siilarity. In applied probles, instead of (6) the following equation is used: log Na ( ε ) d = li, (7) ε 0 a log a ε where N ( ε ) is the inial nuber of radius spheres; ε is a radius necessary to cover set A. Lyapunov's theory is practical for prediction of change in CS paraeters with the help of nonlinear systes and ters of attractor dynaic diension on the basis of the following forulas: ) Kaplan-Yorke forula [7] 3

5 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 d L j λ = j + k k = λ j +, (8) where j are deterined in accordance with condition λ λ + + λ 0 and + j > λ + λ + + λ j + λ i + < 0 ; ) Young's forula λ d = + k, (9) L k = λ n where is the nuber of non-negative Lyapunov exponents; 3) Mori forula p λ + 3 k d = + k =, (0) L q λ k k = where is the nuber of non-negative; p is the nuber of positive; q is the nuber of negative Lyapunov exponents. Quantitative calculations of attractor diension [8] ay be conducted on the basis of tie series of observations. Analysis of CS functioning in fractal geoetry ters ay be conducted on the basis of the Fokker-Planck-Kologorov equation (3) with the right part which is linear with respect to the function being deterined [] and with the help of an instability criterion: 3 d 3 < d < f, d,5 d f < <, ln f d =,5, li S () f 0,5 j = f 0, d = d, 5 3d d,5 f < <, d 0 d f < <. Analysis of forula () allows one to conclude that a fractal with diension in intervals < d < 3;.5 < d < ; < d <.5;0 < d < is a flicker noise, and thus in CSs with such diension, a phenoenon of self-organized criticality appears. Studies of dynaic CSs are then conducted in such case when only the type of tie series of observations ay be deterined [9], while a foral description of functioning of such systes is 4

6 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 probleatic. In this case, attractor diension ay be deterined fro tie series for different paraeters describing the CS functioning using Lyapunov exponents [0] or the Kologorov's K- entropy. Diension of attractors ay be found with Kologorov's K-entropy fro the following hypotheses:. Let us assue that the dynaic syste ay be described as a tie series.. There is a global attractor in the dynaic syste. 3. The syste transits fro one state to another on the attractor during easureent. Fro the above-entioned, the following conclusion ay be ade. Applying ethods for deterination of fractal diension, it is possible to predict CS behavior using non-linear systes, deterining an instability criterion fro the fractal diension of the attractor, i.e., in this case the CS transits into a deterinistic chaos state, which allows deterining CS's behavior in a sustained stable state. For a strange attractor, the set of its trajectories is not very large, so Takens theore is inapplicable, but fractals with fractal ausdorf diension ay be enclosed in space, where d is significantly large. Let us assue that during the experients the variables describing oveent of the dynaic syste was deterined. In this case, correlation diension will be deterined in the following way. Let us select a coponent of vector x j ( t) out of x( t) = ( x ( t), x ( t),..., x ( t) ), which is a solution vector for a certain syste of non-linear equations and for vector ξ ( t) of a for, allowing one to enclose the strea generated by the syste in R +, where etrical properties of space { x ( t) } and { ξ ( t) } are the sae: ξ ( t) x ( t) x ( t + τ ),..., x ( t + τ ) d R (, ), > 0 = τ. () j j j During the experient, the attractor diension of the dynaic syste is not known, so, the correlation diension of the attractor shall be deterined step-by-step for =, = 3 etc., using the forula given below: ln C ( ε ) d = li (3) ε 0 lnε where N C ( ε ) = li Θε ξ ( t ) ξ t (4) N N i j = i j, is a generalized correlation integral. In forula (3), * certain = and d and keep getting deterined until d significantly changes around * d is the correlation diension of the attractor: * d at * d = d. (5) It is clear that the right part of the syste of non-linear equations was not involved in the calculation. Thus, sequences{ x j ( iτ) }, i =, N ay be described as a tie series of observations and the right part of the syste of non-linear equations ay not be linked to any syste of equations. Using such approach, the enclosing space ay be characterized, aking it possible to deterine the correlation diension of the attractor. 5

7 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 The prediction interval for paraeters on the attractor ay be deterined with the help of tie series of observations. The easure of predictability of paraeters on the attractor is the su of positive Lyapunov exponents, which deterine the quantitative easure of the syste's divergence rate [0]. 3. Practical Part The process of education is described with a atheatical odel of control, which is a social CS. The process of foration of prognostic paraeters of education allows assessent of duration of the education. It is held that portions of education and tie interval are equal. For each student during the education process their individual characteristics were deterined, i.e., fractal diensions in the for of assessent of the nuber of variables in the education quality function, providing a criterion for the education quality with possibility to deterine an aount of repeated study for a required portion of educational inforation in coparison with predicted values of the paraeters. The assessent will be perfored for different values of γ and γ (individual features of students). n экс is the ean duration of education, obtained experientally; n is duration of education, obtained fro forula (6): ( σ ) ln ln α tn n = N + (6) ln γ n is a factor showing by what-fold the duration of education is worse than the ean duration of n экс education of the studied process. * The dependence level was deterined fro the correlation coefficient between level Q of theoretical knowledge and the level of practical knowledge. Taking into account that с { 0,,, 3} has 4 values, while ranking correlation coefficient takes values on interval [0, ], let us divide the interval into four parts and assign rank q to each of the:, if 0.8 < Q, if 0.5 < Q 0.8 q = (7) 3, if 0. < Q 0.5 4, if 0 < Q 0. In accordance with forula (7), tables (), () are obtained: 6

8 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 Table. VSU: Nuber of point with rank [q, с]. q с q Table. LSTU: Nuber of point with rank [q, с]. с Fro the data in Tables and, a link between grasp of theoretical aterial and the level of practical knowledge is deterined with the help of Spearan's rank correlation and then these two variables are plotted against each other. Taking into account that there are related ranks, it is possible to average the with the Spearan coefficient, using the forula: 6 d ρ = i, (8) в n 3 n where n is saple size. Ordered assessents for each group for all instructors and lecturers, total points, GPA, dispersion and standard deviation are given in Tables 3 and 4. Table 3. Statistical assessent of a review work. (Voronezh State University). Group (exposure) Group (control) ( ) ( ) ( ) ( ) j j j j GPA s s

9 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 Table 4. Statistical assessent of a review work. (Lipetsk State Technical University). Group (exposure) Group (control) ( ) ( ) ( ) ( ) j j GPA s s The table uses the following notation: i k n k ( i ) = is the GPA in the i-th group for the k -th instructor; n is the nuber of i i ij n j = students in the i-th group; ( k ) is the score of the j-th student fro the i-th group given by the k-th () instructor; =.; n = n = 0; j =.; ij i ( k ) i j s is an unbiased estiate of dispersion for the i-th group and the k-th instructor. Unifority of the epirical data obtained was checked with standard ethods []. Fro the experient conducted, it was proven that the two saples fro the sae group are saples fro the sae population, i.e., in each group corresponding saples ay be united into a saple of a larger size and two collectives (exposure and control) ay then be copared. Testing of the statistical hypothesis has shown that both collectives are saples of different populations. It is worth highlighting, that the average values of the saples differ fro each other in exposure group = 3.4 and = in the control = 3. and =. 9, that is, the average value in the exposure group is higher than in the control one, eaning that the process of education is ore efficient in the exposure group than in the control group.[,7,]. Let us copare the result of testing on a three-point scale (0, and 0.5 points ) in the two groups. In total, S = 50 arks were given. On the assuption that the saples ay be consolidated, let us deterine the average grade for different instructors in both groups and find relative W. The results are given in Tables 5 and 6. Assessent of changes in the quality of knowledge [,] will be conducted with the help of a coefficient, giving a quantitative characteristic to different educational practices. K + K K = 0 (9) i j 8

10 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 W where K = 0 is an indicator characterizing reduction of unfailiarity; 0 W 0 indicator characterizing an increase in knowledge. W K = is an W Group Table 5. Nuber of, 0.5 and 0 scores in exposure and control groups. (Voronezh State University). Instructor Score GPA Relative score GPA Relative score Group Table 6. Nuber of, 0.5 and 0 scores in exposure and control groups. (Lipetsk State Technical University). Instructor Score GPA Relative score GPA 0 9 Relative score

11 International Conference Inforation Technologies in Business and Industry 08 IOP Conf. Series: Journal of Physics: Conf. Series (08) 0336 doi :0.088/ /05/3/0336 The calculations gave the following results for the VSU: K = =.05; K = =.7. (0) The level of long-ter knowledge has the value of K =. : LSTU: K = =,; K = =. 08. () The level of long-ter knowledge has the value of K =. 5. Thus, use of the thesis research ethod led to an increase of the quality of education by the factor of. in the VSU and by the factor of.5 in the LSTU. The results of the experient have shown that the nuber of Excellent arks has grown, while the nuber of Failed arks has fallen. 4. Conclusion Dynaics of changes in a CS with initial uncertainty were studied. With the application of deterinistic chaos, a stochastic behavior of dynaic syste trajectories in the phase space was deterined together with a possibility for a deterinistic chaos to arise due to unstable dynaics. A ethod was developed to deterine whether certain state paraeters of a CS fall into the required region, using tie series of observations without need to use non-linear systes. Dynaics of changes in educational behavior with initial uncertainty was studied experientally. A ethod was developed to deterine whether the state paraeters of a single given student fall into a required region using tie series of observations and without need to use any non-linear systes; a portion of educational inforation necessary for a given student was deterined. References [] Feigenbau M 983 Universality in Behavior of Non-Linear Systes Advances in Physical Sciences [] Goldberger E, Rigney D R 990 Chaos and Fractals in uan Physiology V Mire Nauki (Scientific Aerican, Russian Edition) [3] Tarasenko V V 000 Fractal Geoetry of Nature: Socio-Cultural Diension/ Synergetic Paradig. Multitude of Searches and Approaches. (Progress-Traditsia) [4] Duran B, Odel P 997 Cluster Analysis (Statistika) [5] Zhukovsky V I, Molostvov V S 988 Multi-criterial Decision-Making In Uncertainty Conditions (International Manageent Research Institute) [6] Zayatina O M 009 Syste Modeling (TPU Press) [7] Peters E 004 Fractal Analysis of Financial Markets: Application of Chaos Theory to Investents and Econoics (Internet-trading) [8] Vagin V N, Golovina Ye Yu 004 Certain and Reasonable Induction in Intellectual Systes (Fizatlit) [9] Trubetskov D N 998 Turbulence and Deterinistic Chaos Sarov Educational Journal [0] Grinchenko V T, Matsypura A A 007 Snarskiy Introduction to Non-Linear Dynaics. Chaos and Fractals (Moscow: LKI Publishing) [] Suin V I, Solentseva T E 04 «Modeling Education with Tie Series of Observations» (Voronezh: Nauchnaya Kniga Publishing Center) [] Suin V I, Solentseva T E 008 Foralization of the Social Syste Control Proble. Conflict Theory and Its Applications: All-Russian Scientific and Technical Conference. Part. (Voronezh: Voronezh Institute of igh Technologies) 0