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Some Two-Steps Discrete-Time Anticipatory Models with ‘Boiling’ Multivaluedness


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AACIMP 2009 Summer School lecture by Alexander Makarenko. "Mathematical Modelling of Social Systems" course. 5th hour. Part 3.

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Some Two-Steps Discrete-Time Anticipatory Models with ‘Boiling’ Multivaluedness

  1. 1. Some Two- Steps Discrete- Time Anticipatory Models with ‘Boiling’ Multivaluedness. Alexander.S.Makarenko, Alexander S. Stashenko Institute of Applied System Analysis at National Technical University of Ukraine (KPI), 37 Pobedy Avenue, Kiev, 03056, Ukraine, E-mail: ; Abstract In this paper it is described and investigated some class of models and the concept which can make a universal methodological background for difficult social, economic and public systems concerning different spatial and time scales and hierarchical levels. These are nonlinear models of difficult processes with foresight expectations. In the review some existing models with foresight expectations is presented, and the new nonlinear model with the behavior similar to models of neural network is offered. The offered model has next differences from existing. At first the model is anticipatory, that is passing on two steps ahead, and secondly the function f (?) has a piecewise - linear character, and looks like the activation function of neurons. The condition for multivaludness had been found. Such multivaluedness is of special type of 'boiling tank' when the multiplicity had created at the restricted region of space. Suggested concept and principles allow developing some practical applications of models. Keywords: anticipatory element; multivaluedness; society models 1. Introduction It is known well, that the systems with anticipating have large prospects both in theoretical, and in the applied aspects [1, 2]. But for subsequent development of the theory of anticipatory systems a large number of concrete examples of such systems should be investigated. As it was indicated in the previous works of one of the authors (A.Makarenko) very prospect and interesting are the neuronets with anticipating elements, which corresponds to the models of society with accounting the mentality of individuals [3 - 5]. The mathematical investigation of complex hierarchical models with anticipatory property is the further research task. At present work we consider the example of the system from one basic element of anticipatory network. We had considered as the prototype the investigations of some economical models [6 - 8]. Remark that it allows considering some presumable economical applications at the end of our paper.
  2. 2. In the given work the offered two-step discrete model in time with anticipating had been proposed and investigated. The main result of analysis of the maps is the possibility of multy - valued transitions. In the paper all solutions of model are explored. Some explanation of choice between these branches of solutions is also proposed. Remark that earlier a one – dimensional model with anticipation had been used for money-economic processes [7, 8]. But the offered model has some principle differences. The first is two- steps in transitions, that is passing on two steps ahead, and second is that the function f(?) has a piece-vise linear character and belongs to the class of neuron responses function which is usual for neural networks [9]. Piece -vise linear character of function allowed making the thorough numerical - analytical analysis. A new type of significant behavior had been found, when the region of multy – valuedness in solutions is localized in space. At the end of work the example of multy – step model is proposed is of which is more complicated by the dimension of map. Because of presumable importance of results we also pose some discussion of its possible applications to economical problems. 2. Model description 2.1 Prototype from economics First of all we shortly describe the prototype model from the field of economics. Here we pose the description of economical terms for illustration the origin of prototype models. The nonlinear model of money - credit dynamics in continuous time consists of equating of the price adjusting [6- 8]: & p = α [m − p − f (π )], (1) & where p - logarithm of price level, m - logarithm of amount (here it is foreseen to be constant) of money, and p - the expected norm of inflation, that is: & π (t ) = Et [ p (t )], (2) In accordance with equality (1), the norm of change of price at the market of commodities relies on the surplus of demand for the real balances. The function f is logarithm of demand for the real balances of money. The discrete version of evolution equation of price (1) has such form: pt +1 = α ⋅ m + (1 − α ) pt − αf (π t ,t +1 ), (3) where now pt means logarithm of level of price in the moment of time of t. In equality (3) the function of demand of money in the moment of time of t relies on the norm of inflation of expected in a next period. π t , t +1 = Et ( pt +1 − pt ), (4)
  3. 3. That to complete a definite model in (3) it is needed to take into account the condition of passing ahead expectation. The hypothesis of passing ahead expectation results in the one-dimensional map in a form for pt+1: pt +1 = α ⋅ m + (1 − α ) pt − α ⋅ f ( pt +1 − pt ), (5) Remark that the equation (5) had been derived as economical model but it is (and other equations with anticipatory property) interesting mathematical object itself. 2.3 Two – step model of anticipatory element Here we introduce a new nonlinear model which substantially extends the one – dimensional equation (5). New features in proposed model are the next. The first is two- steps nature (that is passing on two steps ahead). The second is that the function f(?) has a piecewise - linear character, and looks like the transition function of neurons in neuronets [9]. Remark that piecewise character of nonlinearity usually allows developed mathematical investigations (see for example [10]). We will write down the offered model as follows:  pt +1 = α ⋅ m + (1 − α ) pt − α ⋅ f ( pt + 2 − pt +1 ) ( 6)   pt + 2 = α ⋅ m + (1 − α ) pt +1 − α ⋅ f ( pt + 2 − pt +1 ) (7 ) The function f(x) depends on to the parameter a and has the following expression:  f ( x) = 0, x≤0   f ( x) = α ⋅ x, x ∈ (0, 1α ] (8)  f ( x) = 1,  x > 1α 3 Model investigations 3.1 Inverse function representation We will rewrite equation (7) thus, that pt + 2 were found for the right side of equation. ( pt + 2 − pt +1 ) + α ⋅ f ( pt + 2 − pt +1 ) = α ⋅ m − α ⋅ pt +1 (9) We will put this in right part of equation (6). Converting into a comfortable form we will get the following equality: ( pt + 2 − pt +1 ) + α (1 − α ) f ( pt + 2 − pt +1 ) = α (1 − α )(m − pt ) (10) We will enter the following function: V ( x) = x + α (1 − α ) f ( x) (11)
  4. 4. Then we can write a next correlation V ( pt + 2 − pt +1 ) = α (1 − α )(m − pt ) , from which we can receive pt + 2 = pt +1 + V −1[α (1 − α )(m − pt )] ≡ F ( pt , pt +1 ) (12) V −1 in equation (12) means the inverse function V, when V is invertible or the proper function is definite by means one of inverting of function V, when V is not uniquely invertible. We will write down first derivative to the function V: V ′( x) = 1 + α 2 f ′( x) − α 3 f ′( x) (13) First derivative (13) to the function V at x < 0 that x > 1α has independent from a value, according to properties of function f, equal to 1 accordant (11). We are interested at the value of derivative function V at x ∈ (0,1/ α ] , so at exactly in this interval the function of f matters dependency upon x accordant (8): V ′( x) = 1 + α 2 − α 3 (14) From equation (14) we have next results α = 1/3 + 1/3{29/2 − 3(√ 93/2)}1/ 3 + 1/3{1/2{29 + 3 √ 93}}1/ 3 Approximate value is the next: a = 1.465... 2 3 Consequently subject to the condition 1 + α − α < 0 derivative to the function V has the negative value, that can mean that the points of maximum appear, and minimum of function V. And accordingly the ambiguousness of invertability of function V appears which is represented on Picture 1.
  5. 5. Picture 1. The graph of function V, subjecting to the condition of ambiguous invertability. Intervals (−∞,ν m ) and (VM , +∞) have one solution, interval (ν m , VM ) - three solutions. Dotted line represents the reverse transformation of function V. On Picture 1. we see a form of function V under a definite condition on a parameter. We have the maximum of function V - VM in the point π 2 ( VM = V( π 2 )) and minimum ν m in a point π 1 (ν m = V( π 1 )). There on graphic we see prototypes ? V( π 2 ) and V( π 1 ) in other points π 2 and π1 accordingly. It is visible therefore, that ? V −1 ( y ) has three separate points when vm < y <VM and one point otherwise. There is a question how to represent a reverse function because the three special cases. It is needed to remark that there is the fourth case, subject to the condition 1 + α 2 − α 3 = 0 for inverse function of V when there are not two points, but the whole interval of points, which causes the endless quantity of possible variants of the map F ( pt , pt +1 ) . −1 Consequently, a function V determines the behavior of the map F ( pt , pt +1 ) as follows. When V simply reversible, F ( pt , pt +1 ) is continuous, and non- continuous, when V is not uniquely reversible. 3.2 Map F ( pt , pt +1 ) We will write out equation for the map F ( pt , pt +1 ) as follows: F ( pt , pt +1 ) = pt +1 + V −1[α (1 − α )(m − pt )] (15) The map F ( pt , pt +1 ) is continuous, when V reversible and is discontinuous otherwise. The fixed point: V −1[α (1 − α )(m − pt )] = 0   ⇒ α (1 − α )(m − pt ) = α (1 − α ) f (0) V (0) = 0 + α (1 − α ) f (0)   * pt = m − f (0) = m (16) As soon as F ( pt , pt +1 ) = pt +1 is achieved a condition V −1[α (1 − α )(m − pt )] = 0 is executed, that is α (1 − α )(m − pt ) = V (0) , or specifying the last expression α (1 − α )(m − pt ) = α (1 − α ) f (0) . When
  6. 6. F ( pt , pt +1 ) has the points of break, there it can be nonexisting of the fixed point, however if exists, has a form in obedience to equality (16). We will consider the cut of the map F ( pt , pt +1 ) =0. How visible with Picture 2, right overhead part has the positive value, and the left lower part of cut accordingly negative value. That is getting in a positive region, value of the map F ( pt , pt +1 ) to be increased and increased to endlessness. Like there is the reduction of value of map, at the hit in the negative region of values. If to set the initial value from the region of ambiguousness close to the fixed point, on a next step, the map will get a few values, one of which, will translate the map in a positive region, the second value vice versa will translate the map in a negative region, that will result in rejection of the state in endlessness, one in positive, second in negative. And another solution will arise up, which will leave the state of map in the region of ambiguousness. Thus there will be the permanent troop landing from the region of ambiguousness in a plus and minus of endlessness. We can mark the region of multivaluedness origin as the ‘boiling’ multy - valuedness. The term ‘boiling’ had been introduced following visible analogy with boiling water tank when the molecules of water leave tank through the free surface of water. Some solutions cross the boundary of ‘rhomb’ and then tend to infinity. But some branches of solutions stay within the ‘rhomb’ to undergo to further multiplication. Picture 2. Cut of the map F ( pt , p t +1 ) = 0 at a parameter b>1.465.... Considered equations (6) – (8) have three solutions in the rhomb on the plane (with marked three vertexes m, 1* and 2*) and one solution in the rest of the plane. Consequently, we are interested in the case, when the state of map is constantly found in the region of ambiguousness. The most adjusted for such investigation is computer calculation. 3.3 Behavior of the map F ( pt , pt +1 ) in state space The software had been developed so that it is possible to visualize some branches of the states of map, but only those, that are not thrown out to infinity, that much facilitated understanding of processes, which take place in the region of
  7. 7. ambiguousness. Namely, on Picture 3 we can see two cycles of period 6, and the quantity of cycles is multiplied in course of time, that is we can see a tendency to phenomena which we may called „chaos”. Picture 3. The graph of forks of the map F ( pt , p t +1 ) . Parameters: a = 3.2, m = 0, p0 =m-0.15, p1 =m-0.2, limitation is maximal: m+2, minimum: m-2, without the forks, that are thrown out to infinity. The cycles are within the ‘rhomb’ with the multy - valuedness above. Picture 4. Map in state space. The sequence of values of ( pt , pt +1 ) is represented. With reduction of parameter a the period of cycles diminishes. Diminishing takes place until then, while a model can generate new cycles. As soon as a model loses such power, a process goes out slowly. That it is possible to see on Picture 5 (cycles of period 4).
  8. 8. Picture 5. The graph of forks of map F ( pt , p t +1 ) . Parameters: a = 2.5, m = 5, p0=m- 0.15, p1=m-0.2, limitation is maximal: m+1, minimum: m-1, without the forks, that are thrown out on endlessness. 3.4 Generalization to the N-step model with anticipating Further possible extension of the model consist in considering the possibilities of increasing the number of steps in the model For subsequent investigations it is possible to consider the model of such form:  pt +1 = α ⋅ m + (1 − α ) pt − α ⋅ f ( pt + N − pt + N −1 ) p  t + 2 = α ⋅ m + (1 − α ) pt +1 − α ⋅ f ( pt + N − pt + N −1 )  pt + 3 = α ⋅ m + (1 − α ) pt + 2 − α ⋅ f ( pt + N − pt + N −1 )   pt + 4 = α ⋅ m + (1 − α ) pt + 3 − α ⋅ f ( pt + N − pt + N −1 ) ...........................................   pt + N −1 = α ⋅ m + (1 − α ) pt + N − 2 − α ⋅ f ( pt + N − pt + N −1 )   pt + N = α ⋅ m + (1 − α ) pt + N −1 − α ⋅ f ( pt + N − pt + N −1 ) Research of nonlinear anticipating model with piece-vise linear functions confirmed basic conformities to the law in the offered nonlinear model. At first −1 accordance of behavior is confirmed by nature of reverse functions V ( pt ) which remember the two – dimensional case. A parameter a would turn out very influential on the behavior and nature of the map F ( pt , pt +1 ) , that was expected at construction of the given nonlinear two-step model with anticipating and with piece- vise linear functions. Critical value to the parameter which follows to bifurcation points appear at: α = 1/3 + 1/3{29/2 − 3(√ 93/2)}1/ 3 + 1/3{1/2{29 + 3 √ 93}}1/ 3
  9. 9. It is needed to remark that there is the fourth case, subject to the condition at the inverse to function V when there are not two points, but a whole interval of points, which causes the infinite number of possible variants of the map F ( pt , pt +1 ) . −1 A function V determines the conduct of the map F ( pt , pt +1 ) as follows. When V simply reversible, F ( pt , pt +1 ) is continuous, and otherwise when a function V is not uniquely reversible. But just in the preliminary investigations some new possibilities had been found. For examples for some parameters value we had found the possibilities of increasing the number of branches during time increasing. Summary So in proposed paper we have considered some examples of anticipatory models – namely discrete – time models of single element with two – step anticipation in time. Chosen form of nonlinearity (piecewise - linear) allowed considering in details the dynamical behavior of solutions, branching of solutions and possible ways for some type of complex behavior related with possible multy - valuedness. These results are interesting and new per se. But it may be supposed that such type of models may constitute one of the interesting fields of mathematical investigations of anticipatory system. Just many – step in time equations from Paragraph 3.4 are interesting objects. But much more interesting may be investigations of coupled systems of anticipated elements. One of the most important classes of such systems constitutes the multy – valued neuronal networks [5]. In case of the artificial neuronal networks usually some of the research problems are the architecture of networks, leading principles and investigations of their behavior. Remark that now we make some investigations on such networks. Other wide new class of research problem is the investigation of self – organization processes in the anticipating media, in particular in discrete chains, lattices, networks from anticipating elements. In such case the main problems are self – organization, emergent structures including dissipative, bifurcations, synchronization and chaotic behavior [11]. As it is seen from previous paragraphs such problems take new forms of presumable possible multy – valuedness in anticipatory systems. For example just definition of ‘chaos’ in such case should be reconsidered. Remark that such problems are new for recent theory. But currently already understanding of such phenomena possibilities may help in investigation and managing real systems. Especially important may be applications to social, economical etc. systems. Some outlines of possibilities were discussed in [3, 4]. The realizations of such research programs are the goals for further investigations. Here we pose only some discussion on possible applications of proposed models in economics. Recently the ideas of anticipatory nature of ‘homo economicus’ (participant of economical relation) and organizations explicitly (but sometimes only verbally) penetrate into the community of theoretic and practitioners in economy. Currently some explicit investigations of macro economical models with anticipation had been proposed [12, 13]. But these investigations are concentrated mainly on the stability problems. Described in present paper results extended to the new society models open the new possibilities for exploring economical behavior. The key is possible multy – valuedness in such systems and new understanding on decision – making role. As one
  10. 10. of possible topics for considerations we may foresee the investigation if uncertainty in such systems. Now one of the leading ideas is the the uncertainty in economical systems origins from dynamical chaos in it [14]. But as it follows from our investigations anticipation and multy – valuedness also may serve as the source of uncertainty in economical systems. Then presumable new tools for managing such uncertainty may follows from mathematical modeling of anticipatory economical systems. Thus in proposed paper we have discussed strict results on some mathematical models with anticipation and possible related issues, especially for economical systems. We hope that further investigation will follow to next new and interesting results. References 1. Rosen R. Anticipatory Systems. Pergamon Press. 1985. 2. Dubois D. Introduction to computing Anticipatory Systems.. International Journal of Computational Anticipatory .Systems, 1998. Vol. 2. pp. 3-14. 3. Makarenko A. Anticipating in modeling of large social systems - neuronets with internal structure and multivaluedness. International .Journal of .Computational Anticipatory Systems, Vol. 13., pp. 77 - 92. 2002. 4. Makarenko A. Anticipatory agents, scenarios approach in decision- making and some quantum – mechanical analogies. International Journal of Computational Anticipatory Systems, Vol. 15., pp.217 - 225. 2004. 5. Makarenko A. Multi- valued neuronets and their mathematical investigations problem // Abstract books5 th Int.Math. School: Liapunov functions method and applications. Simpheropol, Ukraine, Creamia, Tavria University, 2000. p. 116. 6. Sargent T.J, Wallice B. Stability of money and growth models with perfect foresees. Econometrica 1973, vol. 41. pp. 1043–1048. 7. Agliari A., Chiarella C., Gardini L. A stability analysis of the perfect foresight map in nonlinear models of monetary dynamics, Chaos, Solitons and Fractals, vol. 21 2004. pp.371 -386. 8. Mira C., Gardini L., Barugola A., Cathala J.C., Chaotic dynamics in two- dimensional noninvertible maps. Singapore: World Scientific, 1996. 9. Haykin S., Neural Networks: Comprehensive Foundations. MacMillan: N.Y., 1994. 10. Maistrenko Yu., Kapitaniak T., Szuminski P. Locally and globally basin in two coupled piecewise – linear maps. Phys.Rev.E , vol.E56, pp. 6393 – 6399, 1997. 11. Nicolis G., Prigogine I. Self – organization in nonequilibrium systems. N.Y., John Wiley & Sons, 1977. 12. Dubois D., Holmberg S. Modeling and simulation of management systems with retardation and anticipation. Abstract book of Int. Conf. CASYS’05, August 2005, Liege, p.5/5 13. Leydersdirf L. Hyper – incursion and the globalization of the knowledge – based economy. Abstract book of Int. Conf. CASYS’05, August 2005, Liege, p.8/8. 14. Dendrinos D. Chaos: challenges from and to socio- spatial form and policy. Discrete dynamics in nature and society. Vol.1, pp.9- 15, 1997.