78 Economy Informatics, no. 1/2001

Complexity and chaos in economic cybernetic systems. Chaotic models of economic systems functionality (Profit Non-linear Feedback Mechanism)

Prof. Emil SCARLAT PhD. Senior Lect. Virginia MARACINE PhD. Economic Cybernetics Department, Academy of Economic Studies Bucharest

Abstract: In over 50 years from the cybernetics appearance, the view on cybernetic systems has a continue growth. The specialists talk about three big stages in this evolution: the first class cybernetics, in which the cybernetic system haven’t a mathematical formalization; the second class cybernetics in which different formal models of the cybernetic systems appear and, finally, the third class cybernetics which assimilates and extends different new theories strong related with the general theory of the cybernetic systems. The mainly characteristic of this new direction is the assimilation among cybernetics methods of the new theories such as the dynamics of the non-linear systems, chaos theory, complexity theory, synergetic, and others. Even if, in appearance, it’s independent, those new approaches of the real systems appeals in facts many methods and theories that are components of cybernetics. In this situations, the attempt to assimilates and integrates its among the others cybernetics conceptions it’s natural. Key words: Cybernetic systems; Dynamic systems; Chaos; Chaotic systems; Determinist chaos; Stochastic chaos; Attractor fixed point; Chaotic trajectory.

From the determinist to the chaotic CST’s “father”. In 1960, he makes an 1systems experiment in which he simulates different Until recently, cybernetics studies the scenarios on weather evolution using a systems as determinists ones, meanly as twelve-equation system solve on the systems that could be describes in computer. After a period of time he review equational form, in which case the a sequence of calculus and, for time solutions obtained represents with a great saving, he start the experiment from the deal of certainty the real system acts. But middle of the sequence by introducing in often that it isn’t the true: the real system’s the computer the number 0.506 instead of behaviour is influenced by to many factors 0.506127 used in his first experiment. for could be completed described by the After one hour, he realise that his new mathematical instruments. The great result are completely different from the complexity of the systems has important first, even if the changes could be influences on the behaviour of the appreciate as insignificant at that time. cybernetic systems. Because he use a number very closed to But not only that: it was observed that even the first, he expect that the differences in for very simple systems the behaviour the results being directly related with the could be very complicated. This original error, namely a small ones. But his observation has conducted to the concepts expectations seem to be false. In this of chaotic system and chaotic systems experiment Lorentz has discovered one of theory (CST). the fundamental properties of the chaotic systems, the sensitive dependence on the 2. Chaos theory’ origins. Principles of initial conditions: even an imperceptible CST change in those conditions could generates Today, Edward Lorenz, an American drastic change in the long term behaviour weather forecaster, is considered as the of a system. Economy Informatics, no. 1/2001 79

Starting from here, Lorentz realised that proprieties which gives the chaos theory a it’s impossible to make an exact forecast great generality and comprehensiveness. on the weather and, in the same time his Even this theory it’s now in the middle of discover conduct at the chaos theory its evolution, we could enounce those appearance. In this way he make a new proprieties that characterise all dynamic, step on the road opened by the great non-linear and complex system, meaning French mathematician Henry Poincaré all potential chaotic systems. which, in 1892, in his essay “Science and a). The principle of the inherent Method” has enounced the above complexity mentioned principle for the first time, even Even the simpler dynamic determinist if he doesn’t realise it’s huge importance system could reach an extremely for the science. complicated behaviour when one of it’s Lorentz great merit consists in his attempts included process its repeated again and to discover other systems from nature for again until it becomes unstable therefore which this propriety works. Using his chaotic. twelve equations weather model, he has b). Lorentz’s principle (‘dandelion effect’) created a three-dimensional differential Little perturbations in the initial conditions system named ‘Lorentz’ equations’: of a system could generates major ìx& = a(y - x) behavioural changes on the long run, fact ï that makes impossible the long run y& = r × x - y - xz. í prediction over the system evolution. ï îz& = xy-bz c). The ‘order in disorder’ principle For certain values of the a, r, and b On the short term the system’s behaviour parameters, the solution of this system it’s predictable because there is a gap generates a chaotic trajectory in the three- between the moment of a little change and dimensional space called ‘Lorentz’s it’s effect apparition. This gap could be butterfly’. Specific to this trajectory is the used in order to make predictions over the fact that it never cross with him self, system behaviour. meaning that the system never repeat d). The inherent instability principle exactly. Starting from this observation, Chaotic systems are functions far from Lorentz makes a huge step proving that the equilibrium into a state of instability. A real systems which are dynamic, complex new equilibrium state appears in the and non-linear, never reproduces them past system only if the instability was entirely evolutions. Because of that, the chaotic traversed. systems appear as untidy, even random. e). The spontaneous self-organisation But they aren’t like that. The random principle behaviour implies a certain order, and the The pass to a new order state will be made real random systems aren’t chaotic, through a series of choices in critical because its could be represented using an points, which are not predictable. In those appropriate probability distribution. points, the system’ components co- For this reason chaos couldn’t be operates, reaches a consent and organise represented as an random move. It them self into a new structure. This new represents a third type of behaviour after structure it’s dissipating, being in fact the the determinist and the random ones. The beginning of a new transition to the chaos. chaotic movement appears both in the All those principles are valid for any simpler deterministic systems (‘determinist natural systems, economic systems chaos’), and into the extremely complex included. The most developed application stochastic systems (‘stochastic chaos’). of the chaos theory in the economic field is No mater what kind of chaos we find in the represented by the Capital Market study. natural systems, they have some common It’s admitted the fact that the Capital 80 Economy Informatics, no. 1/2001

Market, the stock exchange especially, Expenses’ represents a non-linear system, and chaos structure theory study precisely those kind of systems. Close by the models for the

Capital Market, recently it was elaborates a Pt Pt-1 series of models which describes economic cycles and fluctuations, economic growth process, inflation, economic politics Decision regarding the effects. We will try bellow to illustrate profit’ sharing those principles with a non-linear chaotic Figure 1 model of profit. The dependence between actual profit, Pt

and the previous, Pt-1 is not a linear one 3. Chaotic Models of Economic Systems because the expenses’ growth generates Functionality the future profit growth only to a certain (Profit Non-linear Feedback Mechanism) limit, beyond which rinsing the expenses Each component of the cybernetic firm’s will determine the reducing of the actual model includes one or several feedback profit (see the advertising expenses mechanisms for control and regulation of example). In order to illustrate this thing the material and informational flows we will use a non-linear relation as: between sub-systems or between those and P = AP - BP 2 . external environment. We will show how 1 0 0 such a mechanism (i.e. non-linear profit’ (1) mechanism - Stacey, 1992), its incorporate If B<role over the decision regarding P 2 = AP 1 - BP 1 the structure of the expenses for this year. P = AP - BP 2 In fact, these structure its dependent of the 3 2 2 ………………… decision of profit’ distribution to the finish 2 of the last year. P t+1 = AP t - BP 0t . In the follows we will consider that the Suppose that there is a maximum possible expenses made by the firm in current profit’ level, Pmax. We observe then, in period will generate a certain profit at the order that Pt+1 >0, is necessary that end of the period. From a profound Pmax0, and if Pt >A/B determines the adoption of a certain then Pt <0. decision regarding the distribution of the In this case we could introduce a new profit obtained in the previous year. variable: The figure no. 1 shows the feedback P p = t , relation between the two levels of the t P max profit, from the current and from the which express the profit obtained by the previous period. firm in year t as a fraction from the Economy Informatics, no. 1/2001 81 maximum possible profit. Obvious that A value of p, noted p*, for which we have 0interval [0,1]. If A>1 both fixed P t max A BP t = B max P = Bp t = Ap t . points are included into the domain [0,1], P B so its are interested. Finally, we have: In order to see the importance of those p = Ap - Ap 2 = A(1-p )p . 1 t t t t fixed points we give the graphic The relation representation of the function y= fA(p), for p t+1 = Ap t (1-p t ) = f A (p t ) , many values of A, with 0parameter A has is constant, p should y=p have well defined values. More then that, we could expect that the profit will change y gradual if a will change gradual. The calculus could be made as follow: A=0,6 We start with a certain given value p0; then fA(p) we calculates p1, p2, ... and so on. p 1 = f A(p 0 ); p 2 = f A(p 1 ); p 3 = f A(p 2 )... 0 p p p p 1 3 2 1 0 p In this way we obtain a sequence of Figure 2 iterations, which generates a sequence of The evolution of the profit in this case is values for the profit p called trajectory or obtained as follow: orbit. We consider an initial value of the profit, Of course, the first values of this trajectory p , on the p axis, and we draw a vertical depend on initial value of p. What isn’t 0 line to the fA curve. Then, we project the obvious in this moment is the fact that the intersection point on the y-axis. The trajectory’ possible behaviour its the same intersection of this projection with the for all most all the initial values p 0 main diagonal y=p determines the co- between 0 and 1 for a given value of A. ordinates for the next point p on the p- Yet, same initial values are different from 1 axis. Again, the vertical line from p the others. Thus, if we get p =0 we 1 0 intersects the f curve into a point which’ observe immediately that f (p )=0 and the A A 0 projection on the y-axis determines, trajectory is constant at p=0 for all the next through it’s intersection with the main iterations. 82 Economy Informatics, no. 1/2001 diagonal y=p, the co-ordinates for the next figure 4 is represented the profit’ trajectory point p2, and so on. in case A=3, in figure 5 is represented the In this case, we observe that the set of profit’ trajectory in case A=3,5, and in values p0, p1, p2,.., obtained for A<1 tends figure 6 is represented the profit’ trajectory to 0, that is the fixed point of the in case A = 3,54. application fA(p). For this reason, p*=0 is 1 also called attractor fixed point, or just A=3 y=p attractor for the iterative application y y=fA(p). The interval 0£p£1 is called attraction field for the respectively attractor, because each trajectory which have a point included into the interval [0,1] as initial value, will tends to p*=0 with each iteration. In the terms of the considered economic example, we concludes that, if A<1, the firm will registered a smaller profit (pt®0) when t rise. 0 p 1 Let consider now the case in which Figure 4 1initial condition p0=0,1 y=p and proceeding as above, we observe that y the trajectory tends to the fixed point p2*=1/3 (figure 3).

1

y y=p

0 p 1 Figure 5 1

0 p0 p1 p2 p3 1/3 1 p

Figure 3 By changing the initial condition for any y=p other p0Î(0,1), the obtained trajectory y tends to p=1/3. So this fixed point is attractor for any value p from the attraction field (0,1). We will obtain the same thing 0 p 1 for any function f (p) for which A takes A Figure 6 values between 1 and 3. We observe that, when A rise the trajectory Instead, if 3£A£4, even if no other fixed becomes more complicated, sifting from points appear the trajectory sift to the ‘cobweb’ form to the periodical something else but a fixed point. Thus, in alternation between two values. In this Economy Informatics, no. 1/2001 83 final case we obtained a two periods cycle The non-linear dynamic economic systems (figure 6). approach through chaotic systems’ theory If A continue to change, we will obtain is very well developed today. Thus they double periods sequence, so we obtain are created the chaotic versions for one of cycles with four periods, eight periods and the classic deterministic models such as so on. Samuelson, Hicks, Kaldor s.a. Also it was Finally, for A³3,57 we obtain a chaotic created original models for economic trajectory (figure 7). systems which incorporate the chaos, and 1 which reproduces a very wide spectrum of possible behaviours into an economic

medium extremely volatile.

References y 1. Baker, G., Gollub, J., “Chaotic Dynamics. An Introduction”, Cambridge University Press, New York, 1990; 2. Froyland, J., “Introduction to Chaos y=p and Coherence”, C. P., Oxford, 1990; 3. Gleick, J., “Chaos, Making a New 0 1 Science”, Viking, New York, 1987; p 4. Hall, N., “The New Scientist Guide to Figure 7 We observe now that, no mater the initial Chaos”, Penguin Books, 1991; condition is, after t iterations the profit 5. Hilborn, R. C., “Chaos and Non- could rich any point into the interval [0,1]. linear Dynamics: Introduction for Also, any modification in the initial Scientist and Engineers”, Oxford condition, even a smaller one, will offer a University Press, New York, 1994; trajectory total different from the initial 6. Mandelbrot, B., “Obiectele fractale: ² one. In this case we say that the dynamics form , hazard, dimensiune”, Nemira of the non-linear feedback profit Publishing, Bucharest, 1998; mechanism its a chaotic one when 7. Mullin, T., “The Nature of Chaos”, 3,57Economics. Time and Change in Non- possible anytime, the analysis of the linear Economics”, Springer-Verlag, condition of appearance of the chaotic Berlin, 1991. behaviour in the economic systems it’s totally justified.