CHAOS AND CELLULAR AUTOMATA Claude Elysée Lbry Uiversité de Nice, Faculté des Scieces, parc Valrse, 06000 NICE, Frace. Keywrds: Chas, bifurcati, cellularautmata, cmputersimulatis, dyamical system, ifectius disease. Ctets 1. Chas 1.1. Itrducti 1.2. Oe Dimesial Discrete Chatic Systems 1. 3. Tw Dimesial Discrete Chatic System 1.4. The Cascade f Bifurcati 1.5. Ctiuus Chatic Systems 1.6. The Stry f Chas i Dyamical Systems 2. Cellular autmata 2.1. Itrducti 2.2. The S.E.I.R. Mdel fr Epidemics 2.3. Aalysis f the I.R.N autmat 2.3.1. Defiitis 2.3.2. Sme simulatis 2.4. A partial Thery fr the (I, R, N, G) Autmat 2.5. The Need fr a Thery whe e uses Cellular Autmata i Mdelig. 3. Cclusi Glssary Bibligraphy Bigraphical Sketch Summary I this chapter, we are ccered with Chas thery ad Cellular Autmata thery. These tw theries have i cmm the fact that they were very ppular sme time ag ad the fact that they are itimately cected t the develpmet f digital cmputig. The chapter is divided i tw parts the first e develped t chas, the secd e devted t cellular autmata. We d t try t make a survey f all aspects f these theries. We just ccetrate few aspects which seem t be relevat t mdelig ad give cautiary advises relative t these theries i the prcess f mdelig. 1. Chas 1.1. Itrducti Cmpare the tw sigals i Figure 1a:
Figure 1a: Tw sigals They lk very differet; the e belw is icely peridic sice the e abve has apparetly regularity. It lks radm. But it is t! Bth sigals are btaied by iterati f the very simple scheme: (1) U U ( U ) + 1 =λ 1 the first e with λ= 3.8 the secd e with λ = 3.5. This is a defiitely determiistic prcess! The first sigal is called chatic because it is differet frm a pure radm sigal. T make the differece betwee chas ad pure radm we cmpare the chatic sigal abve with the sigal btaied by plttig successively the result f a call t the radm fucti f a cmputer. Oe btais the fucti shw i Figure 1b: Figure 1b: A radm fucti This sigal lks similar t the chatic e but if we plt, t the value f the sigal agaist time, but the value at time + 1 agaist the value at time we btai the fucti i Figure 2: Figure 2: Chatic ad radm sigals This illustrates the essetial differece betwee chatic sigals ad radm sigals. The
ideal mathematical ccept f radmess requires that tw successive ccurrece f U must be idepedet, which meas that the prbability f appearace f ay value f 1 U + is t affected by the kwledge f U. The visual csequece f this is the apparetly uifrm repartiti the square f the ccurreces f the pits f crdiates ( U+ 1, U). Actually defiiti f perfect radmess is mre striget ad requires that all fiite sequeces are mutually idepedet, but we d t g further i this directi. Cmpletely differet is the chatic sigal. Sice the ext U + 1 is related t U by the fuctial relati: ( ) U + 1 = 3.8 U 1 U it is t surprisig at all that all the pit f crdiates ( U U ) graph f the mappig: ( ) U 3.8 U 1 U + 1, are ctaied i the Ntice that the radm prcess f ur cmputer is t able t realize a true mathematical pure radm prcess, but ly a apprximati f it by sme determiistic device. 1.2. Oe Dimesial Discrete Chatic Systems Csider a mappig f frm sme buded subset K f dyamical system defied by: + 1 = ( ) x f x ad dete by ( ) R it itself ad csider the x x the trajectry issued frm x (this meas the sequece issued frm x ad defied by iducti with the abve frmula. Defiiti : Sesitivity t iitial cditis Oe says that the system pssesses the prperty f sesitivity t iitial cditis sme subset A f K if there exist sme psitive umber M such that fr every x i A ad fr every (small) psitive ε e ca fid sme iitial cditi x ad a iteger such that: ( ) ( ) x x x x M We illustrate this by the fllwig experimets with the system (1) fr the value 0.8 f
the parameter. We have pltted the differece U ( U 0) U ( U 0 ) differig successively frm 6 9 12 10, 10, 10 i Figure 3. fr iitial cditis Figure 3: The fucti U ( U 0) U ( U 0 ) fr differet iitial cditis We see that fr few iterates (15,25,35) the tw trajectries are impssible t distiguish (up t the precisi f the pixel f the scree) ad the, abruptly, the differece becmes visible at upredictable. The differece grws expetially which meas that t add sme cstat umber t the umber f iterates where the trajectries are udistiguishable e has t divide by sme cstat factr the distace f the tw iitial cditis; i ur example e have t divide by 1000 t btai te mre iteratis. Defiiti : We say that system (2) is chatic if it has a trajectry which is dese a subset A which is a attractr, if it has the prperty f sesitivity t iitial cditi subset A ad, mrever, peridic rbits are dese i A. Because f the sesibility t iitial cditi, what is the meaig f ur umerical cmputatis is t clear. All what we kw is that ur cmputer simulati has thig t d with the actual trajectry! Frtuately there is a deep mathematical result which state (i a precise mathematical way) that every cmputer simulati is actually clse t the true trajectry issued frm a iitial cditi which is t the e we used i the cmputati but is clse t it. By the way, i a chatic system, it is impssible t predict the trajectry issued frm a iitial cditi but what we cmpute is a typical pssible utcme f the system. 1. 3. Tw Dimesial Discrete Chatic System S far we have csidered a e dimesial chatic system. A example f a famus chatic system i tw dimesis is the Hé mappig, prpsed i 1976. Csider the mappig: ( x, y) ( y + 1 ax 2, bx) ad the iteratis:
x = y + 1 ax + 1 y + 1 = bx Figure 4: Tw successive elargemets fr the iterates f the Hé mappig 7 fr the values a = 1, 4 ad b = 0.3. Figure 4 shws 10 iterates f this mappig startig frm sme iitial cditi. Each elargemet is a cpy f the previus picture. This is ather feature f chatic systems. The fractal ature f each trajectry. - - - Bibligraphy TO ACCESS ALL THE 19 PAGES OF THIS CHAPTER, Visit: http://www.elss.et/elss-sampleallchapter.aspx Alluche J-P, ad C. Reder (1984), Oscillatis spati-temprelles egedrées par u autmate cellulaire. Discrete Applied Mathematics 8 : 215-254. Cartwright M.L. ad J.E. Littlewd. (1945) O liear equatis f the secd rder; the equati 2 y k 1 y y + y = bλk cs λ t+α, k large. J; Ld Math. Sc. 20: 180-189 ( ) ( ) Checier A. (1989). Systèmes Dyamiques Différetiables. Ecycpedia Uiveralis. Emetrut G-B ad L. Edelstei-Keshet (1993), Cellular Autmata Apprach t Bilgical Mdelig. Jural f Theretical Bilgy 160 : 97-133. Feigebaum M. (1978), Quatitative uiversality fr a class f liear trasfrmatis. J. Stat. Physics. 19: 25-52. Lbry C. a H. Elmzi (2000). Cmbiatrial Prperties f sme Cellular Autmata Related t the Msaïc Cycle Ccept. Acta Bitheretica 48: 219-242. Lrez E. (1963). J. Atms. Sci. 20: 130-141.
May R. (1976). Simple mathematical mdels with very cmplicated dyamics, Nature 261, 459-467 Picaré H. (Euvres cmplètes. Gauthier- Vilars 1951 Smale S. (1976). Differetial Dyamical Systems. Bull. A.M.S. Wieer N., ad A. Rseblueth (1946). The mathematical frmulati f the prblem f cducti f impulses i a etwrk f cected excitable elemets. Archives f the Istitute f Cardilgy f Mexic 16 : 202-2065. Wlfram S. (1994) Cellular Autmata ad Cmplexity. Addis-Wesley Bigraphical Sketch Claude Lbry, was br i 1943 ad gt a thèse d état at the uiversity f Greble, i ptimal ctrl, i 1972. He was the appited as prfessr at the Uiversity f Brdeaux ad jied the Uiversity f Nice i 1981. His mai iterest is mathematics applied t autmatic ctrl ad atural systems where he became a specialist f ctrllability. Durig the eighties he used the tls f Nstadard Aalysis i sme prblems f sigular perturbatis f differetial systems.frm 1990 t w he has bee very active i creatig iterdiscipliary teams r etwrks with the Cetre Natial de la Recherche Scietifique, the Istitut Natial de Recherche Agrmique, the Istitut Natial de Recherche e Ifrmatique et Autmatique. Besides his scietific activities he has a strg ivlvemet i the prmti f mathematical research i develpig cutries, especially Africa.