NONLINEAR PROBLEMS IN ECONOMIC DEVELOPMENT Laura Ungureanu Spiru Hare Universiy, Craiova, Romania ungureanu@lycos.com Ion Viorel Maei Absrac: Now, we aend o he beginning of a process of synhesis among he developmenal economic heories and he new heory o he complexiy. Mahemaics, he science of spaial forms and quaniaive relaions, is considered o be he basis of all oher sciences. In he analyses dynamics macroeconomic area we can observe a big variey of mehod and echniques for research flucuaes from economy and financial dae. Because a lo of economical models were elaboraed in las years, in his aricle we propose o presen some nonlinear echniques which can be used in economic analyses. For example, a complex way for evidence he economic cycles is o deermine limis cycles for he dynamical sysem which model he economic phenomenon. Keywords: complexiy, nonlinear heory, economic evoluion, equilibrium 1. THE ECONOMIC DEVELOPMENT PROBLEM The asks ha lay ahead he economis are so wide, conradicory and resrained ha hey reclaim more and more capious knowledge from oher fields as well, especially concerning he modelling of economic phenomena, he use of he mehods of mahemaical research. Economy, generally supposes consumers and producers, goods and services ha hey exchange, prices ha esablish he condiions of hese exchanges, saes of balance and fundamenal problems deermined by hem (he exisence of he balance, he uniqueness, he consancy or is esablishmen, he evoluion in ime). The use of mahemaics for solving such economic problems someimes deermines profound and unjusified reicence, because, on one hand he mahemaic device go complicaed, fac ha implies a major effor o know i, and on he oher hand, he discoveries useful o economy are raher recen as iming. The economic science has made grea progress in he laes years, bu his progress is raher unknown maybe because he mahemaic heory became exremely refined, requiring a coninuous effor o be undersood. A new approach of he economic developmen oulined in he years 70, when all counries were facing counless difficulies, alhough grea effors were made for he indusrializaion. The realisic specialiss in conemporary economic issues were forced o warn he public and he oher economiss abou he fac ha oo many hings were going badly, eiher in he sense ha here was oo much violence and oo lile economic reasoning (R. Kohari - 1974), eiher in he sense ha he developmen effors of many hundred million people are annihilaed due o cerain srucural violence (J. Galung - 1980), eiher in he sense ha he excessive division of work and he exaggeraed specializaion in cerain areas began o harden, even make inoperaive, he acual economic mechanisms (P. Hawken - 198). The economic dicionary defines he economic developmen as being a form of manifesaion of he macroeconomic dynamics which supposes, besides he economic growh of he counries, a se of quaniaive, srucural and qualiaive ransformaions, boh in economy, and in he scienific research, in people`s way of hinking and behaving. Developmen means, in he vision of J. Galung (1980), he close connecion beween he exisence ( o be ) and he maerial welfare ( o have ); in he vision of H.Chenery, developmen supposes muliple modificaions of he srucure, and in he vision of he Bariloche Repor, coordinaed by A.O. Herrera, i means he saisfacion of he fundamenal needs of all people, eliminaing he shocking inequaliies beween people abou heir forune and revenues. Numerous specialiss demonsraed ha he obsacles in he way o developmen, of he dynamic balance and of he harmonizaion of various ineress, are no only of maerial and echnical naure, bu also of social poliical naure.
They reached he conclusion ha if mankind desires o survive, so no only o be richer and happier, hen subsanial changes mus be made boh regarding he objecives pursued and he mehods used and he mechanisms hrough which hey are pu ino acion. The economic developmen, besides he quaniaive aspecs, surprises he disribuion of richness and of he revenue inside he sociey, he influence of he economic changes on he living level of he populaion, on he efficiency of using he economic resources, ec. The muli-dimensionaliy of he concep of economic developmen is revealed by he following aspecs: - he economic developmen implies economic growh, because i canno exis iself wihou a growh of he macroeconomic resuls on long erm; sill, in order o urn in economic developmen, he growh of he macroeconomic resuls mus be accompanied by qualiaively srucural ransformaions in economy, in he way of living and in he qualiy of people`s life; - he economic developmen denoes, unlike he economic growh, new echnical-economic economic social repors ha appear in he process of growh; - he economic developmen may be defined in a resrained way as he developmen of he producion facors, respecively of he main componens of economy, like: he developmen of he maerial infrasrucure, he opimizaion of he condiions of combining and using he producion facors, ec; - he noion of developmen refers no only o he changes of he repors beween he economic agens in he producion process, bu also o he change of heir behaviour oward environmen and sociey. These changes of concepion refer o he behaviour modificaions on producion, o he modificaion of he poins of view concerning i. The erm of economic growh resumes his process sin he economic hinking and preserves one of he mos passionae and conroversial issues of economic hinking of he pos war period. The economic developmen conains elemens of success and failure. In general, he economiss use he erm of economic developmen in a broader sense, which incorporaes insiuional and culural changes (Lewis). This way, he capialisic and he socialis economic developmen exiss. The classic economiss (Smih, Malhus, Ricardo, Marx, Mill) are ineresed in he developmen of he social sysem, in one word in he economic developmen which corresponds o he economic evoluion. In exchange, he economic growh corresponds o he expansion or balanced change of economy. The economic developmen is associaed o he insabiliies and sochasic behaviour, while he economic growh o he sabiliy and balances expansion. In conclusion, he economic developmen of a counry disinguishes he se of ransformaions of quaniaive and qualiaive naure, appeared in he economic social and scienific echnical srucures as well as in he psychology and behaviour of he sociey as a whole. The evoluion of he economic hinking from he saic microanalysis o he dynamic macro analysis represened a real progress in he economic science. The vision abou he economic movemen of sociey began o ge close more and more o he vision abou he movemen of he living organisms, he human sociey represening a complicaed organisms in which hey self propel.. NONLINEAR PROBLEMS IN ECONOMY.1 The complexiy sciences In he real economy, a he level of he economic agen, of he branch or of he naional economy, he aciviies can be followed, analyzed and opimized wih he help of he modern mehods approached in a uniary concep, sysemic and srongly absraced and mahemaised. The modern heories aim mainly a he economic balances, ever more complex and more inerdependen. The sysemaic research of he economic evoluions show ha, in he conemporary economy, he srucural changes and he oscillaions are he rule and no he excepion, and he consan saes generally become insable when cerain parameers vary. Subsequenly, economy evolves o economic cycles or o siuaions of chaos, imposing a complex sudy insead of he classic heories.
The appariion of some global, planeary problems, ha influence he process of growh, of he developmen on naional, zone or global scale, like he depleion of cerain non regeneraive naural sources, he srucure of he populaion, he deerioraion of he naural environmen, he developmen of he echnique, he problem of alimenaion, he excessive urbanizaion, he economic under developmen, he poliical economy led o he invesigaion of he muliple heoreic problems and especially pracical in he field of he evoluional economy. In general, he economic processes have by heir inerial naure, a inrinsic coninuiy, he jumps being excepions o he rule. Bu hese jumps exis, generaing disconinuiies, heir knowledge being necessary boh as an inrinsic phenomenon bu especially due o he propagaed effecs, since i is known ha small perurbaions, by amplificaion, may lead o big effecs, someimes caasrophic. In he economic processes we encouner a series of relaions of non linear ype: he curve of reques, he curve of offer, he average raio of consume, he relaions cos produced quaniy, P.I.B. is cyclic flucuaion, producion facors of producion, revenues a budge axing rae (Laffer curve), he relaive increase of unemploymen he rae of unemploymen (Phillps curve), he reques of a produc on he marke wih he passing of ime and he succession of he sages in he life of he produc (launching, growh, mauring, sagnaion or decline). From he economeric poin of view, he classic mehods, based on coninuiy, lineariy and sabiliy proved o be inadequae, in order o be able o represen economic phenomena and processes wih a higher degree of complexiy. The researchers are compelled o follow hese processes in a dynamic way, o sudy he changes of quaniaive order which inerfere beween he economic variables involved as well as he resuls obained wih heir help. Besides oher characerisics, he mahemaic models allow he inroducion of a isomorphism beween he real and ideal economic sysem, represened by he model. Wih heir help, he approach of he insable behaviours of he differen non linear economic sysems becomes possible his way, being underlined more ofen he fac ha he lineariy and sabiliy acually represen paricular cases of economic evoluion. If he radiional economic dynamic was based on he famous principle of correspondence of Samuelson, according o which small perurbaions of he parameers in he sysem deermine small changes of he variables, he new concepion, which is dominan in he acual dynamic, considers ha small changes of he parameers may lead o qualiaive modificaions of he dynamic behaviour. This way, he sysems may become from sable insable, from deerminisic, chaoic, from linear non linear. The non lineariy of he evoluion of a number of quie many phenomena from physics, biology, ecology and economy led o he ouline of cerain modern sciences, srucured in he las years, sciences ha ry o approach, o concepualize and hen o use a differen face of realiy, more flucuan, more dynamic. These sciences are he resul of he inegraion of cerain models, heories and echniques of solving he sysem of non linear differenial equaions, of a change of perspecive, from which ofen appear new saring poins in he aemp o beer undersand he phenomena sudied. If he models are adequae, hen from he knowledge of heir soluions we can deduce he behaviour of he modelled phenomena. Alhough i is said ha each non linear model has is own heory, hey also have common reas, unifying, and he behaviour no maer how srange of heir soluions having a corresponden in he aspec of he modelled phenomenon. The fac ha his behaviour was no ye signalled is due o he complexiy of he non linear problems, whose sysemaic sudy began only a few decades ago. The complexiy appears due o he large number of such elemen ha ineracs simulaneously. The complexiy appears in he organizaion of he whole under he pressure of he infinie combinaions in which hey may inerac. I is obvious ha he definiions evolve gradually in order o sugges ha, o ge he essence of complexiy, he classic mehodology of sudy canno be used, which supposes he fragmenaion of he whole and he sudy of he pars his way isolaed. From his perspecive, he science of complexiy is, firs of all, anoher way of raionally approach Realiy, anoher way o build an onological vision of he universe o be able o cach non linear phenomena, singulariies, synergies, evoluions.
I can be said ha he componen elemens are simple, and heir ineracion law is a he same ime simple. The complexiy appears due o he increased number of such elemens which inerac simulaneously. The complexiy appears in he organizaion of he whole under he pressure of he infinie combinaions in which hey can inerac. In order o undersand he behaviour of a complex sysem, we mus undersand no only he evoluion of he pars, bu also he way in which hey generae, by ineracing, he whole iself. I is however ineresing he definiion of he word hazard, which in he same dicionary is an even ha depends on is causes, so as an insignifican difference in causes may produce a considerable difference in effecs. Wih oher words, hazard defines a caegory of evens whose sensibiliy o he iniial condiions is big, fac ha makes he predicion o be difficul from he principle poin of view. Now maer how hard we ried o conrol he iniial condiions, somehing will exis: a small flucuaion, a drop, a fricion, a local ineracion and he non repeiive which will make impossible he idenical repeiion of an experimen. There are no pracically wo iniial siuaions idenical. Alhough i ges close o 0 years of exisence (1987 Sana Fe) he science of complexiy in no perceived a is real value, neiher as scienific imporance, nor as pragmaic opening. Why? A possible cause is he fac ha requires from hose who approach he field of complexiy a change of he perspecive from which realiy is looked a, a holis approach (global, conexual, inegral), differen from he reducionism one specific o he acual paradigm. In anoher way, i is no only he novely of informaion, of he objec iself (fracal geomery, he heory of chaos, synergeic, geneic algorihms, ec) bu also of he perspecive from which hey have o be looked a, inegraed and valorised. The organizaion a he academic level of cerain conceps, heories and new models, like: coevoluion, complex sysem, auo-organizaion, emergency, he heory of chaos, he heory of caasrophes, synergeic, fracal geomery, ec deermined he frame and defined he perspecive from which sysems ha evolve can be sudied, modify unpredicably in repor wih he flux of informaion, energy and maer ha crosses i: living sysems (ecology, sociology, economy) or arificial sysems based on arificial inelligence. The science of complexiy is from his perspecive a facor of iner-disciplinary coagulaion of an inellecual environmen capable o undersand and help in a concree manner he ransiion process from he indusrial sociey o he informaional one, respecively o he one based on knowledge (Knowledge Based Sociey Lisbon 000). Objecs like: econo-physics or juris-dynamics are examples of he cross-breeding of he classic objecs wih he new se of conceps, echniques and mehods delivered by he science of complexiy. The science of complexiy, like any oher science, has a well defined objec of sudy, for which i elaboraed mehods and specific echniques of approach, pursuing he comprehension and use of phenomena, processes, properies and insrucions ha derive from hese sudies. In a concree way, he science of complexiy deals wih he sudy of complex sysems and develops adequae echniques and models for he descripion of heir behaviour in ime and space. The appariion and developmen wihou preceden of he Inerne, he raise of he calculus power of he acual compuers which can ge o conain housands of processors, he very rapid developmen of elecommunicaion, he appariion and coninuous exension of he cyberspace and of he virual sysems, he passage o mehods based on inelligen agens and ohers, deermined a special ineres for cyberneics and he heory of sysems, he only ones capable o offer a sysemic vision, inegraive, on a world found in a dramaic process of complexificaion... Some aspecs of non-lineariy The scienific research of naure showed is complexiy. The problem of a science of complexiy was brough ou in July 1991 in Physics Today by Philip Anderson, professor a Princeon Universiy in he aricle Is Complexiy Physics? I is a Science? Wha is i?. In ime his science, of complexiy, began o ouline is objec hrough new mehods, ohers han he ones used so far. Among hese are: - The fracal geomery sudies forms wih irregular aspecs boh in space and in ime, wih properies of auo-similariy and measurable in space wih non enire dimension. Science begins where i can be measured, where a simple qualiaive approach is no longer saisfacory, where a
precise delimiaion is required, in he limi of an error, deermined and considered accepable. And here begins he role of he fracal dimension which can disinguish hrough a number, he srucure differences. Wha saisics canno do, he fracal dimension can do up o a cerain hreshold. And ogeher, he wo ways of esimaing such a complicaed profile allow special shadows. The fracal may easily represen similar forces ha ac a more levels, offering a socking mehod of images and daa much more compac han he linear vecors, he irregulariies becoming essenial pars of he model. If he linear equaions fail o build inrinsic, unpredicable and chaoic sysems, hen heir accomplishmen is possible wih he help of fracals. All hese advanages lead o he adopion of fracals in many fields like meeorology, seismology, cardiology bu also economy. Few people know ha Benoi Mandelbro, based his fracal geomery basing himself especially on he successful simulaion of he endency of he prices of he consume goods. This way, his simulaions from 195 on he price of he coon coninue o exacly predic he variaion quaniy from he price of he coon, boh monhly and yearly. Mandelbro proposes in 197 he modelling of he sock evoluions wih he help of a disribuion no used so far in economy: he sable pareian disribuion. I was proved ha his disribuion reflecs very well he real disribuion of daily variaions on periods of five years (1900-1905) and of foureen years (1944-1958), and of he monhly variaions for a period of 60 years. Due o his model, he managed o disinguish regulariy inside a irregulariy. This model allowed he racing of cerain ficive sock evoluions amazingly resembling o he real ones. They seem so real ha expers could no differeniae hem from he real ones. Some advenured o make, saring from ficive emissions, a series of complex commens and previsions []. The sudies of fracal geomery disinguished new properies of he naural objecs and marked he main differences beween hem and arefacs. Besides a beer modelling, he fracal approach allowed he idenificaion of he imporance of he recursive processes in he modelling of he naure phenomena and in generaing srucures wih an exremely complicaed aspec hough very simple mechanisms. This way, his analysis has pragmaic implicaions in fields like: elecommunicaion (fracal anenna, fracal compressions in he mobile elephony), biology (he quaniaive evaluaion of umours, he sudy of he processes of morphogenesis, he operaive evaluaion of he sae of healh) bu also in economy (he diagnosis of sabiliy a macroeconomic scale, he diagnosis of cerain economic processes, he fracal marke). - The chaos heory sudies he dynamic of complex sysems and inroduces a new mehodology of invesigaion and new conceps (ransiion sceneries o chaos, srange aracors, ec). Toward he end of he 60ies he mahemaician James Yorke gave for he firs ime he erm chaos a mahemaic saus and meaning. The heory of chaos addresses, like he heory of caasrophes, o he mahemaic conribuion of he science of dynamic sysems. There is a big number of applicaions of he heory of chaos in economy. We give as examples he models of Benhabib and Day (1981, 198), Benhabib and Nishimura (1985), Grandmon (1985, 1986), Day (198, 198), Suzer (1980), Deneckereand Pelican (1986), Baldrin and Monrocchio (198), Sacey (199). In his model, Sacey considers ha he dependence beween he curren profi π and he previous one π 1 is no linear because he raise of expenses generaes a raise of he profi only in cerain limis, aking ino consideraion he law of he decreasing efficaciousness, generally valid in economy. The dependency relaion beween he curren profi and he previous one has he formπ + 1 = Aπ Bπ. The non linear erm is inroduced o show ha if he profi raises oo much, is limiaion will appear, given by he negaive values generaed by Bπ. Sacey demonsraed ha for A < 1 he company will regiser a raising profi while is growing. For 1 < A < he model is sable, he rajecories ending o he balance poin π * = 1/ (aracor). In exchange, for A. 57 a chaoic rajecory is obained, any modificaion of he iniial condiion, even a sligh variaion of i, deermining an evoluion rajecory oally differen from he iniial one.
- The caasrophes heory approaches he sudy of he criical saes, of singulariies, hrough he consrucion of a model ha allows he comprehension and he analysis of he phenomena ha happen in naure. This heory may be regarded as one of he possible soluions for surpassing he dilemma conneced o he modelling of cerain non linear dynamics, non conform o he heory of he coninuous funcions and o oher heories based on coninuiy hypohesis, balance and sabiliy of he opimum soluions. I resors o his heory in order o solve problems ha canno be approached in a radiional manner. The heory of caasrophe was inroduced by Rene Thom in 197 and popularized by Zeeman in 1976. I supposes he esablishmen of he muliude of poins in which he considered funcions have criical poins, operaion ha akes place in he space of he phases and he esablishmen of he muliude of poins in which hey have singulariies, operaion ha akes place in he space of parameers, also known under he name of conrol space. In [4] is considered a company ha produces x household objecs, which i sells a he price p on a marke wih perfec compeiion. The oal cos is C ( x) = x + ax + bx + c and using he a change of coordinae y = x +, and marking = r α + ar + b, β = r + ar + br + c where r = a / can be wrien C( y) = y + α y + β. The oal revenue is py and he profi funcion has he expression of π ( y) = y αy β. As he maximizaion of he profi is no affeced by β, by resoring o a ranslaion he profi may be wrien as f ( x, p α) = π ( x) + β = x + ( p α) x. Supposing ha boh he echnical producion and he cos condiions remain unalered, ha is α and β are consan he only relevan parameer is he price p which raises or decreases as a response o he reques condiions, respecively offer of he considered economic branch, condiions on which he producing company does no hold conrol. This way, f can be considered a poenial funcion ha can be brough a he sandard canonical form from he heory of caasrophes. Equalizing wih zero he firs derivae we obain / f ( x) = x + p α = 0 which in he space of phases, bi-dimensionally, (coordinaes x and p α ) represens a parabola, is he muliude of he criical poins of his caasrophe. Furher on, equalizing wih zero he second derivae oo, // ( f = 6x = 0 ) and eliminaing he coordinae x beween he wo equaions, he equaion p α = 0 is obained, which in he uni-dimensional space of parameers (which has only one coordinae p α ) represens a single poin only, he origin, he only poin of ramificaion of he fold caasrophe. The p α criic oupu is x = ±. So, he equaion / f ( x) = x + p α = 0 does no admi real roos for p α < 0 and has wo real roos corresponding o he maximum and minimum for p α > 0 (figure 1). The maximizaion of he profi may end o he maximum in he branch when p varies. The price may decrease, according o he oupu x. The company may evenually lose bu coninue o produce unil i passes over he maximum poin, and a origin where he maximum and he minimum unify, a p α = 0 and x = 0, a jump akes place
which leads o a dropping of he price. According o he iniial condiions, his way an evenual bankrupcy of he company can be foreseen. So, he condiions for he maximizaion of he profi are simple: p α > 0, and he quaniy ha p has o be produced is x α * =. This issue was presened as a saic issue of opimizaion bu i can be made explici and dynamic by considering he gradien funcion x = f ( x,..) = x + ( p α) x ha is he oupu pursues he maximizaion direcion of he profi. - The bifurcaion heory was crysallized in he years 70 as a consequence of he accumulaion of a grea number of resuls in he non linear funcional analysis, on one hand, and in he differenial opology, on he oher. The unificaion of all hese resuls allowed he disinguishing of principles, of he fundamens of he heory of bifurcaion. Developed in more conexs, in he beginning preponderanly applicaive, concree, in he heory of bifurcaion many conceps and resuls were formulaed, someimes difficul o be harmonized and hus unified. Differen auhors use differen conceps of bifurcaion and his hing began o be fel also in he sudies of economic mahemaics. Mos noions in he heory of bifurcaion suppose he knowledge of a very advanced mahemaic device. The heory of bifurcaion sudies he opological and differenial changes, named bifurcaions, of he non linear applicaions in cerain singular poins named bifurcaion poins. I was imposed following he asceraining ha, due o non lineariy, he mahemaic models of paricular sciences, he presence of he bifurcaion is he rule and no he excepion. On he oher hand he exisence of more aracors for he same value of he parameer lead o he explicaion of many paradoxes of hese sciences, imposing anoher sense o he noions of solving and soluion of an equaion, conribuing o he formaion of he paradigm of complexiy. The classic sudies on he equaions ha used o model phenomena of he maerial world, referred o a cerain soluion of he equaion, corresponding o a single poin on he bifurcaion diagram, or only o sudied soluions, in ha diagram, on a single branch, sopping in fron of he bifurcaion poins. Through his, he phenomenon governed by ha equaion was no sudied in is oaliy, bu only for he physical siuaions corresponding o he posiioning in a poin or on a branch of he bifurcaion diagram. This lead many imes o conradicions beween heory and experimen. The heory of bifurcaion eliminaes hese shorcomings hrough a global sudy, complee and deailed of he bifurcaion diagram. In [5] we considered a model ha supposes a Cauchy problem for a sysem of wo ordinary differenial equaions of firs order in real field. I describes he evoluion of he capial of a company and of he work force implied. Le K be he capial a momen and L he volume of he work force (number of persons employed). Then he company has he business number y given by he producion funcion y = F( K, L ). The evoluion of he capial is according o he developmen poliic of he company, by he share par of revenues desined o invesmens, ( 1 δ ) π, where π is he ne profi realized in he year, profi ha may be allocaed enirely o developmen or only parially, and ha is he par remained afer covering he dividends o he shareholders of he company, in a par of δ. Subsequenly, δ π is he mass of dividends and ( 1 δ ) π is he remained volume for invesmens. Taking ino accoun he depreciaion of he capial wih he average coefficien and he revenues obained from he liquidaion of he damped acives, a he recovery cos of λ he mahemaic model of he developmen of a company is obained, by using he basic equaion of he evoluion of he capial and considering he producion funcion of he ype Cobb-Douglas wih raising efficaciousness: µ
K = Aγ (1 δ ) K L µ (1 λ ) K, L = α1k + α L α 0. In his sysem K and L :R R are unknown funcions which depend on he independen variable (ime). By ransformaions of coordinaes he considered sysem is equivalen o: x = cx y + bx, y = x + α y 1, In he equaions only remained hree parameers, b, c and α. This reducion has as economic consequence he disinguishing of cerain expressions, funcions of primary economic parameers, which inerfere in he evoluion of he capial and of he work force inside a company (figure ). This way, he same value of a new parameer corresponds o a grea diversiy of values of he old economic parameers, being formed in his manner classes of equivalen economic siuaions. In [6] i is shown ha for cerain values of he parameers, his model admis a limi cycle, so a periodic evoluion in ime disinguishes for he wo variables considered. The behaviour of he non linear sysem is given in he figure below, where i can be noiced boh he variaion of he work force repored o he capial, heir variaion in ime, as heir ridimensional evoluion. Fig.. Phase porrais for a) linear case b) non-linear case. I is demonsraed ha locally, he parameric porrai around he balance (1,0) for he non linear sysem is opologically equivalen o hose around he origin for he linear sysem, even if his balance is non hyperbolic. In [6] we demonsrae ha his balance poin in neiher of he ype Hopf nor of he ype Bauin, and ha he possible non resonan erms, which may lead o he opological non equivalence beween he linear case and he non linear one, are sricly of higher degree han six. From he economic poin of view i may be followed he variaion of he capial K and of he work force L in ime, saring from significan iniial daa corresponding o cerain poins from he space of parameers. This way, here are siuaions in which he considered sysem admis a periodical soluion which corresponds o a cyclic economic evoluion. I can be disinguished boh he negaive phenomena like he lowering of producion and he raise of unemploymen, as well as he posiive ones, characerized by he modernizaion of he producion capaciies which can deermine boh he increase of he reques of consume goods and he deerminaion of he degree of occupaion of he work force.. CONCLUSION The sudy of such sysems, sared wih sudies made by researchers in he field of mahemaics and naural sciences, lead o he developmen of cerain conceps and mehods fundamenally new. Alhough heir applicaion in he economic science is sill in he beginning, some remarkable resuls have already been obained of grea ineres for economiss. There are numerous fields and economic conexs in which he non linear echniques are exremely useful, like he behaviour of he capial markes and of he exchange rae, he problems of he exernal
deb, he economic depressions, he hyperinflaion and he banking risk, he esimaion of he naural rae of unemploymen, evoluions of capial or of he work force. So, here is heoreic-pracical finaliy only by using cerain modern and performan mehods, of analysis of he non linear dynamic sysems, capable o surprise he essence of he phenomena and economic processes researched, o realisically evaluae he dimensions and he endencies of heir evoluion in he fuure. We can say ha he paradigm of complexiy is able o offer he concepual frame in which hey can projec, evaluae and coordinae processes specific o he living world (biology and environmen, sociology and economy), respecively he processes governed by arificial inelligence (ingredien more and more used in informaics). We can sae ha his paradigm is essenially conneced and even condiions he accomplishmen of he objecives raced hrough he documen from Lisbon 000 regarding he srucuring in Europe of he economy based on Knowledge, form of organizaion superior o he Indusrial one (he indusrial sociey) or o he Informaional one (he Informaional sociey) and which may assure a durable developmen of man on Earh. BIBLIOGRAPHY [1] A.Bouo-Invenion of Forms, Nemira Publishing, Buchares, 1997. [] A.Georgescu- The Theory of Bifurcaion heory and applicaions, Universiăţii Publishing of Pieşi, Pieşi, 000. [] B. Mandelbro- Fracal Objecs, Nemira Publishing, Buchares, 1998. [4] P. N. V. Tu- Dynamical Sysems, Springer, 1994. [5] L.Ungureanu-Elemens of Economic Dynamics, Universiăţii Publishing of Pieşi,000. [6] L.Ungureanu-Sabiliy and Bifurcaion in Two Mahemaic Models of economic Dynamic Universiăţii Publishing of Pieşi, 004. [7] R.Voinea, I.Sroe- Inroducion in he Theory of Dynamic Sysems, Academia Româna Publishing, Buchares, 000. [8] W.B.Zhang-Economic Dynamics, Springer-Verlag, Berlin, 1990.