SOCIAL AND POLITICAL EVENTS AND CHAOS THEORY- THE DROP OF HONEY EFFECT

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1 Academic book J.A. Filipe, M.A.M. Ferreira. Social and poliical evens and chaos heory - he "drop of honey effec". Emerging Issues in he Naural and Applied Sciences 203; 3(), DOI: 0.783/einas.203/3-/0 SOCIAL AND POLITICAL EVENTS AND CHAOS THEORY- THE DROP OF HONEY EFFECT José Anónio Filipe, Manuel Albero M. Ferreira Insiuo Universiário de Lisboa (ISCTE-IUL), BRU-IUL, Lisboa (PORTUGAL) s: jose.filipe@isce.p, manuel.ferreira@isce.p DOI: 0.783/einas.203/3-/0 ABSTRACT The applicaions of chaos heory began firs by naural sciences and hen follow o he humaniies and social sciences. Since he beginning go rapidly a developing field. Chaos is exremely complex and difficul o be idenified in he real life. Bu using he available informaion, i is possible, anyway, up o a cerain exen, o find specific mahemaical relaionships for problems o be solved. In his work some issues are presened in order o consider his subjec in he humaniies and social sciences fields Key words: Chaos heory, dynamical sysems, poliics, buerfly effec, drop of honey effec. INTRODUCTION The mos of he progress in he area of chaos heory was eviden afer he 960s. Chaos, in he sense i is sudied, is consequenly ye a no very well known field and is sill disan from being compleely undersood or deermined. Chaos is exremely complex 26

2 EMERGING ISSUES IN THE NATURAL AND APPLIED SCIENCES and difficul o be idenified in he real world, using he available informaion. However, up o a cerain exen, i is possible o presen specific mahemaical relaionships for problems o be solved in several areas. As soon as he idea of nonlineariy was inroduced ino heoreical models, chaos go a lile more obvious. A very complex srucure is observed in he field daa and jus simple paerns can be found and approximaed heoreically. Complex paerns o be go hrough models are anoher maer. In any even, we canno jus grab a nice lile se of daa, apply a simple es or wo, and declare "chaos" or "no chaos." (Williams, 997). Chaos occurs as far as known in deerminisic, nonlinear, dynamical sysems. The word chaos, in common sense, presumes he exisence of urbulence and disorder. The predisposiion o a profound change in he direcion of a phenomenon generaes an own force, undersood as a deep change ha resuls from small changes in heir iniial condiions. The chaos is - from his poin of view - somehing exremely sensiive o he iniial condiions. The sensiive dependence on iniial condiions shows how a small change a one place or momen in a nonlinear sysem can resul in large differences o a laer sae in he sysem. The undersanding of inherenly nonlinear phenomena presen in social siuaions shows ha i is possible o use mahemaical models in he social environmen and social issues analysis. Moreover, when his does no happen, some kind of qualiaive analysis is ye possible o be performed by following he ideas of chaos heory. The deerminisic chaos presen in many nonlinear sysems can impose fundamenal limiaions on he human abiliy of predicing behaviors. Addiionally, he analysis of a big number of condiions by a single deerminisic resul may creae he possibiliy of having a prospecive oucome in erms of adapaion and evoluion. In he conex of arificial life models his has led o he noion of "life a he edge of chaos" expressing he principle ha a delicae balance of chaos and order is opimal for successful evoluion (Campbell and Nonlinear means ha oupu isn direcly proporional o inpu. Or ha a change in one variable doesn produce a proporional change or reacion in he relaed variable(s). 27

3 Academic book Mayer-Kress, 997). Neverheless, he essence of life may conduc o specific siuaions ha someimes bring new siuaions creaing a new order even considering exremely difficul siuaions. A subsanial par of his work is in (Filipe and Ferreira, 203). 2. SOCIO-POLITICAL SITUATIONS AND CHAOS In social phenomena, chaos may be evidenced in many siuaions. Hisorically, simple facs wih no visible significan consequences have regisered considerable impacs ha could no be predicable a he iniial momen. Nowadays, such a kind of siuaions goes on occurring in many social conexs around he world. The recen Arabian Spring is an example of he way he buerfly effec can be found when causing a wide spread regional poliical reform in he poliical regimes of some counries in ha geographical area. The flapping of he buerfly wings may be represened by he immolaion by fire of a Tunisian salesman ha was he saring poin for he regime change in Tunisia firs and hen he conagion o he neighbors Egyp and Libya. The consequences would be seen as well in Syria where a civil war is sill in course. The well-known buerfly effec could also be named as he drop of honey effec 2, expression go from he wonderful ale wrien by he Armenian poe Hovanés Tumanian ( ). Presening anoher example and considering he poliical siuaion in Greece here is a new sage o be sudied for Greek, European and World economy. The poliical saus quo has been broken in Greece: a new pary has aken an advanage ha i has never had. The emergen crisis in Greece, which was fel severely afer he Greece-Troika agreemen. Throughou his Program, Greece has o respec an auseriy program in order o pu naional budgeing a accepable levels and is complied o obey he agreemen ha conduced Greek people o severe self well being sacrificing. This siuaion made Greeks o voe in favor of a new siuaion. This enire new siuaion has imposed a new socioeconomic condiion o European Union and o he World ha has 2 I is abou how an apparenly insignifican and inoffensive drop of honey provokes a war. 28

4 EMERGING ISSUES IN THE NATURAL AND APPLIED SCIENCES hreaened he world economic sabiliy. The possible bankrupcy in Greece has ormened world leaders and a new saus quo is being redefined for Europe wih considerable implicaions for he world. 3. CHAOS IN MATHEMATICAL TERMS As Williams (997) says, phenomena happen over ime as a discree, separae or disinc, inervals 3 or as coninuously. Discree inervals can be spaced evenly in ime or irregularly in ime. Coninuous phenomena migh be measured coninuously. However, we can measure hem a discree inervals 5. Special ypes of equaions apply o each of hose wo ways in which phenomena happen over ime. Equaions for discree ime changes are difference equaions and are solved by ieraion, he mos of he imes, or analyically. In conras, equaions based on a coninuous change (coninuous measuremens) are differenial equaions. The erm "flow" ofen is associaed o differenial equaions 6. I follows a mahemaical model ha works he conceps of chaos heory and conribues o explain he possible presence of some effecs based on he idea of chaos. So, in Berliner (992) i is referred ha non-inveribiliy is required o observe chaos for one-dimensional dynamic sysems. Addiionally i is said everywhere inverible maps in wo or more dimensions can exhibi chaoic behavior. The sudy of srange aracors shows ha in he long erm, as ime proceeds, he sysems rajecories may become rapped in cerain bounded regions of he saes space. The model presened in Berliner (992) is an example in wo dimensions of he Hénon map, displaying he propery of having a srange aracor. The Hénon map appears represened by he equaions: 3 Examples are he occurrence of earhquakes, rainsorms or volcanic erupions. Examples are air emperaure and humidiy or he flow of waer in perennial rivers. 5 For example, we may measure air emperaure only once per hour, over many days or years. 6 For some auhors (see Bergé and Pomeau, 98), a flow is a sysem of differenial equaions. For ohers (see Rasband, 990), a flow is he soluion of differenial equaions. Noe ha for he Navier Sokes equaions, ha describe he moion of fluid subsances, surprisingly, given heir wide range of pracical uses, mahemaicians have no ye proven ha in hree dimensions soluions always exis, or ha if hey do exis, hen hey do no conain any singulariy. 29

5 Academic book x y ax 2 () and y bx, (2) for fixed values of a and b wih 0,, This inverible map possesses srange aracors and simulaneously has srong sensiiviy o iniial condiions. The Hénon map, represening a ransformaion from R o R, which Jacobian is b. If 0 b, he Hénon map conracs he domains o which i is applied. These maps are said o be dissipaive. On he conrary, maps ha mainain he applicaion domain are said o be conservaive.. MODELLING MATHEMATICALLY THE DISSIPATIVE EFFECT ON POLITICS Considering he model in Berliner (992), i is possible now o sugges a model on his basis for economics poliics in he area of fisheries 7. So, if a general siuaion is considered, he following equaions may represen a sysem in which fish socks, a ime, are given by x and caches by y. The model is as described below: x F ( x ) y and y bx. (3) I is a generalizaion of Hénon model. The Jacobian is equal o b. As is a porion of x, 0 b. So, i is a dissipaive model and y he values of x are resriced o a bounded domain. Considering he paricular case: x x y, and y bx. () 7 And also, evidenly, in he area of oher reproducing and harvesing naural resources. 30

6 EMERGING ISSUES IN THE NATURAL AND APPLIED SCIENCES so, x x y and x 2 x bx 0 (5) 2 Now, solving he characerisic equaion associaed o he difference equaion (see Ferreira and Menezes, 992) i is obained: b b k or k ; calling 2 2 b and being 0 b, comes ha 3. So, 0 if 0 b and 3 0 if b, being 0 when b. Consequenly for 0 b, b b x A A2 2 2 (6) And for b, x A A2 2 Finally, for b x b A cos arccos A2 sen arccos 2 b 2 b (7) (8) In hese soluions, A and A2 are real consans. Noe ha he bases of powers are always beween 0 and. So, lim x 0 and whaever he value of b, he dissipaive effec is real, even leading o he exincion of his species. Of course, his is 3

7 Academic book eviden according o he hypoheses of his paricular siuaion for he model. Concluding his approach: -The general model does no allow obaining in general such explici soluions. -Bu, of course, wih simple compuaional ools i is possible o obain recursively concree ime series soluions afer esablishing he iniial value and o check he dissipaive effec. x 0 Anoher example may be presened for poliics, in general, considering he poliical credibiliy. Call x he poliical credibiliy, of a poliician or of a pary measured, for insance, in number of voes, or in he number of chamber s members, or even in money, in he year ; and consider b he credibiliy rae, b. I is admissible ha in he year +, x x bx, ha is: in a cerain year he poliical credibiliy is he one of he former year plus, or minus, a par of i. So: x ( b) x 0 (9) Solving his difference equaion (see Ferreira and Menezes, 992) i is obained 8 : ( ) x, 0 x0 b b and x x, 0 0 b. (0) Then, according o his model, if he credibiliy rae is null he poliical credibiliy is kep unchanged, assuming always he iniial value. If 0 b, he poliical credibiliy follows an increasing exponenial pah. If b 0, he poliical credibiliy follows a decreasing exponenial pah converging o 0. Finally, if b, x is permanenly null. Evidenly, values like b 0 define poliical credibiliy pahs ha may lead o people s chaoic behaviors. 8 Evidenly, his is he compound ineres capializaion formula, a ineres rae b used for financial purposes. 32

8 EMERGING ISSUES IN THE NATURAL AND APPLIED SCIENCES 5. CONCLUSIONS Along his sudy i was shown ha in poliics, he possible exisence of chaos may be evidenced for innumerous siuaions. Hisorically, here are a lo of simple facs, hough insignifican in he momen for he consequences hey had, ha in a compleely unexpeced way gave raise o huge impacs ha could no be prediced, or even guessed, a he iniial momen of is occurrence. Definiively, hey are siuaions for which he oupu is no direcly proporional o he inpu. Nowadays, such a kind of facs goes on happening in many socio-poliical conexs around he world. I is a leas srange ha he simple despie he greaness of he personal sacrifice immolaion by fire of a Tunisian salesman was he saring poin for he regime change in Tunisia firs and hen he conagion o Egyp and Libya. The consequences would hen be seen as well in Syria where a bloody civil war is sill in course, despie he acual and relaively promising peace enaive. Two mahemaical models wih difference equaions were presened in his paper. They conribue o idenify possible chaoic siuaions, in poliics, hrough he models parameers values. The more accurae is hese parameers evaluaion, he more is he usefulness of each of he models. REFERENCES. Berliner L.M. (992), Saisics, Probabiliy and Chaos. Saisical Science, 7 (), Bjorndal T. (987), Producion economics and opimal sock size in a Norh Alanic fishery. Scandinavian Journal of Economics, 89 (2), Bjorndal T. and Conrad J. (987), The dynamics of an open access fishery. Canadian Journal of Economics, 20(), Campbell D.K., Mayer-Kress G. (997), Chaos and poliics: Applicaions of nonlinear dynamics o socio-poliical issues. 33

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12 EMERGING ISSUES IN THE NATURAL AND APPLIED SCIENCES 38. Prigogine I. (993), Les Lois du chaos. Paris: Flammarion. 39. Prigogine I., Nicolis G. (989), Exploring complexiy : an inroducion. New York: W.H. Freeman and Company. 0. Prigogine I., Senglers I. (98), Order ou of chaos. Boulder (CO, USA): New Science Library.. Radu M. (2000), Fesina Lene: Unied Saes and Cuba afer Casro. Wha he experience in Easern Europe suggess. Probable realiies and recommendaions, Sudies in Comparaive Inernaional Developmen, 3 (), Winer. 2. Rasband N.S. (990), Chaoic dynamics of nonlinear sysems. New York: John Wiley. 3. Scones I. (999), New ecology and he social sciences: wha prospecs for a fruiful engagemen?, Annual Review of Anhropology, 28, Thrif N. (2008), Non-Represenaional Theory, Rouledge, New York, USA. 5. Williams G. P. (997), Chaos Theory Tamed. Washingon, D. C.: Joseph Henry Press. 37