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<<A then, for small values of P0, the into the sub-system profitability-costs. We second term from the right side of the all ready know that the profit and the relation its insignificant. The minus shows profitability are the mainly criteria of that the second term tends to decrease the performance for every company. As usual, profit. Using the relation (1), we could the profit gained last year will have a very writes for the next periods: 2 important 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 Pmax<A/B. Indeed, from analysis of this relation, we observe that 2 P t+1 = AP t - BP t , the profit at the end of current period, Pt, and, with Pt =A/B we obtain: is influenced by the structure of the 2 2 2 expenses in this period, and it influence the A A A A P t+1 = A× - B× 2 = - = 0 (2) profit in the past period, Pt -1, because the B B B B desire for a large profit this year If Pt <A/B then Pt+1>0, and if Pt >A/B determines the adoption of a certain then Pt <0.