Determination of the periodicity and synchronization of anticipative agent based supply-demand model

Transcription

1 IOP Coferece Serie: Material Sciece a Egieerig PAPER OPEN ACCESS Determiatio of the perioicity a ychroizatio of aticipative aget bae upply-ema moel To cite thi article: A Šraba et al 07 IOP Cof. Ser.: Mater. Sci. Eg Relate cotet - ON THE IN (I + Z) PERIODICITY IN QSO REDSHIFTS D. Will - Supply-ema 3D yamic moel i water reource evaluatio: taig Lebao a a example Hog Fag a Zhimi Hou - O the ychroizatio of two metroome a their relate yamic J C Carraza, M J Brea a B Tag View the article olie for upate a ehacemet. Thi cotet wa owloae from IP are o 8//07 at 0:55

2 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 Iteratioal Coferece o Recet Tre i Phyic 06 (ICRTP06) Joural of Phyic: Coferece Serie 755 (06) 000 IOP Publihig oi:0.088/ /755//000 Determiatio of the perioicity a ychroizatio of aticipative aget bae upply-ema moel A Šraba, M Bre, D Kofjač Uiverity of Maribor, Faculty of Orgaizatioal Sciece, Cyberetic & Deciio Support Sytem Laboratory, Kiriceva ceta 55a, 4000 Kraj, Sloveia Uiverity of Maribor, Faculty of Crimial Jutice a Security, Kotiova ulica 8, 000 Ljubljaa, Sloveia arej.raba@fov.ui-mb.i Abtract. The paper preet the traformatio of cobweb moel by icluig the aticipatio about the upply a ema. Develope traformatio lea to ocillatory behaviour. The perioic coitio of the moel have bee aalytically etermie by the applicatio of z-traform. Perioic olutio of the ytem are preete i the form of a ivere Farey tree, where the Gole Ratio path coul be oberve. The table of perioic coitio i give up to perio 8. The aget-bae ytem wa evelope i orer to how the poibility of cotrollig the ytem by varyig the ey parameter, which etermie the frequecy repoe of aget a their iteractio. A ote o applicatio i the toc maret ha bee provie.. Itrouctio Cotrollig upply a ema i proper balace become more challegig i the ew, upreictable ecoomy [4], [8], [5]. I the claical cobweb moel the fuctio of ema Q a upply Q are pecifie i the followig form: Q = a + bp () Q = c + P () where a, b, c a are parameter of upply a ema pecific to iiviual maret []. The price P a upply Q houl be retricte to the poitive value. I the cobweb moel it i aume that i ay oe time perio proucer upply a give amout (etermie by the previou time perio price) a the the price ajut o that all the prouct upplie are bought by cutomer. Iitial cobweb equatio coul be retate by itroucig aitioal time tep a factor A, B, C a D : bb c + a P + = ( A ( )) b (3) b Q + = ( C a ( D c)) b (4) Cotet from thi wor may be ue uer the term of the Creative Commo Attributio 3.0 licece. Ay further itributio of thi wor mut maitai attributio to the author() a the title of the wor, joural citatio a DOI. Publihe uer licece by IOP Publihig Lt

3 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 with iitial coitio: p a P = + (5) b bp + + a c P = (6) Q + = p (7) b Q = a + ( Q + c) (8) Term A a B i Eq. (3) coul be replace by the term P + or P, imilarly C a D i Eq. (4) by Q + or Q. Thi yiel 6 ifferet ytem combiatio efie by Eq. (3) a Eq. (4) that houl be tuie. The ytem combiatio further examie will have the followig term: A = P +, B = P, C = Q + a D = Q. Propoe replacemet chage the ifferece P a Q to tate equatio for P a Q. Thi yiel the followig et of equatio: bp c + a P + = ( P + ) b (9) b Q + = ( Q + ( a + ( Q c))) b (0) I the performe moificatio we are coierig the yamic which i epeat o two ifferet time tate value i Eq. (9) rather tha oe. If we reformulate Eq. (9) a Eq. (0), the epeecy of the future-preet-pat evet coul be oberve: bp + a c b P = + P + () b b bc Q = Q + + Q + a () Eq. () a Eq. () tate that the value of the preet i epeet o the pat a well a o the future. I thi cae we aticipate future upply a ema.. Perioicity of the aticipative cobweb moel Differet moe of cyclic behaviour repoe coul be oberve a the ytem yamic repoe, whe parameter i varie. Sychroizatio patter are ame by the hape of the Poicaré firtretur map repreetig the value of P, P +. The vertice of the olutio coverge to the ege poit of the petagram. Poit o the vertice form the lie at the perioic coitio value for parameter. The ytem i i traitio to the ext full polygo ychroizatio, which i etimate a the qua ychroizatio, where parameter i ear 0 (or hexago, epeig o the irectio of parameter variatio). The aalytical etermiatio of perioicity coitio i provie by the applicatio of z - traform, which i the bai of a effective metho for the olutio of liear cotat-coefficiet ifferece equatio. It eetially automate the proce of etermiig the coefficiet of the variou geometric equece that comprie a olutio [6]. The applicatio of z -traform o Eq. (9) a Eq. (0) with iitial coitio tate by Eq. (5) (8) give:

4 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 yz + y0z y0z Y ( z) = + z z Ivere z -traform yiel the followig olutio: (3) Y + + ( z) = y y 0 y ( ) ( 4 + ) ( 4 + ) + y ( ) + y ( ) y0 ( ) (4) I orer to gai coitio for the perioic repoe of the ytem the followig equatio houl be olve: Y ( z) = y 0 (5) Let u compute a umerical example of perioic olutio applyig the z -traform. The perio examie will be the perio of 9, i.e. = 9. I Eq. (5) oe houl put the coitio = 9. Oe of the poible olutio for the iitial coitio worth examiig i the followig: = + ( ( + i 3)) 3 ( ( + i 3)) 3 (6) 3 The term ( + i 3) (let u eote the term a ifferet imagiary value i polar form: * z ) coul be expree i the followig way by three π π z * = 3 (co + i i ) 9 9 (7) 8π 8π z * = 3 (co + i i ) 9 9 (8) 4π 4π z * = 3 3 (co + i i ) 9 9 (9) By puttig Eq. (7), (8) a (9) ito Eq. (6) a performig trigoometric reuctio, oe get the followig olutio: π 4π 8π = co = co 3 = co (0) By ipectig Eq. (6) a coierig the equatio for the root of complex umber [3]: 3

5 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 z = θ + π θ + π r(co + i i ) () the geeral form of the olutio for parameter coul therefore be efie a: πm = co () where i the perio a m =,,3,...,. A imilar proceure coul be performe for the arbitrary perio. A more geeral olutio, which applie parameter b, which wa fixe for the purpoe of etermiig olutio, i: πm = bco (3) The olutio coul i ome cae be expree i a alterative algebraic or trigoometric form. Table how the olutio for parameter up to the perio = 8. Alterative olutio coul be expree a the root of the polyomial. Table icorporate the Shape ymbol, which are importat i the tuy of the repoe of yamical ytem. Thi i epecially the cae i the examiatio of complex oliear yamical ytem [7]. Oe of the importat coitio gaie by the propoe ipectio i the value of the perio = 0, which correlate cloely with the perio = 5. The value of parameter i = (+ 5) with umerical value Thi olutio repreet the Gole Ratio (uually eote with Φ ) [9]. Some of the ifferet repreetatio of the olutio for the parameter value at perio = 0 are: π 0 = Φ = co = (+ 5) = (4) 5 The firt olutio of parameter at perio = 0 coect the coiere icrete ytem with the Fiboacci umber give by the ifiite erie: u+ ( ) 0 = Φ = + u= Fu F (5) u+ The fact, that the perioicity coitio of the examie icrete ytem icorporate the Gole Ratio umber Φ coul be oberve i other tuie of complex oliear expaio of the baic cobweb ytem, e.g. [3] Almot Homocliic Tagecy Lemma. Oe houl expect that the ymmetric repoe i -mappig to follow the patter with a match at a certai poit to the olutio of ychroizatio value, for example, Φ. The value that ievitably emerge with Gole Ratio Φ i 5 i our cae preet at perio = 5, where the value for parameter = = , ofte calle the Gole Ratio Cojugate (uually eote with φ ). Table i imilar to the gaie parameter value for the omai of -D yamic attractio by [4]. The metioe et of parameter i augmete with the value for the perio 8, which i ot tate i [4]. Table. Sychroizatio parameter value of perioicity coitio up to perio 8. 4

6 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 I Table ( * ) iicate that ice perio i o the bouary of the olutio iterval, the perioic repoe of the ytem epe o the iitial coitio. A example of umerical value of the perio- repoe i: a =, b =, c =, =, p =. The value for the tetrago mare with ( ** ) i Table i tae i the limit ice the ytem of equatio retur the uefie value whe = 0. Therefore oe houl coier the tetrago perio coitio a the value approachig zero, i.e I thi cae, the ytem repoe i uetermie ( ) at it critical poit. 0 Numerical value of the olutio for parameter cofirm the fiig of [5, 4] characterize by the followig Prop. about the omai of attractio for -D yamic by -imeioal liear bifurcatio aalyi. Determiatio of the perioicity coitio for the aticipative ytem of cobweb liear ifferece equatio lea to the followig propoitio, which provie the itercoectio betwee the perioicity cycle i the coiere aticipative cobweb moel a geeral bifurcatio coitio for the oliear icrete yamical ytem: πm Propoitio Perioicity coitio of the aticipative cobweb moel = bco equal bifurcatio coitio o the flutter bouary etermiig q -perioic fixe poit of the oliear icrete yamical ytem trj = coπω. p Here, Ω repreet a ratioal fractio Ω = ; b =. Prop. provie yamical iterpretatio of q bifurcatio perioicity coitio, which are importat i the aalyi of o-liear ytem where maifetatio of chao a turbulece might occur [5]. There i a igificat effort beig mae to further etermie the coceptual framewor of bifurcatio aalyi for which the cetral problem i bifurcatio cotrol. Prop. provie a poible implificatio of bifurcatio coitio repreetatio. Geometrical viualizatio of uch coitio i a importat factor i the aalyi of oliear icrete yamical ytem [7]. The emergece of ytem perioic tability i the hape of a -ie polygo coul be oberve ot oly i ecoomic ytem [0]; the -ie polygo a the Farey tree orgaizatio of the equilibrium coul be oberve i techical ytem [6, 5] a, for example, i laer cotrol a the paraigm of the chaotic ytem []. Data aalyi i the fiel of ecoomic by the Poicaré firtretur map, which exercie the perioic character, i preete i []. Accorig to Table gaie by the z-traform, the claificatio of the perioic olutio coul be raw, a how i Fig.. 5

7 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 Figure. Perioicity of the icurive cobweb moel i the form of a ivere Farey tree, where path of Gole Ratio coul be oberve (greye fractio) The tructure i Fig. how the ivere Farey tree which correpo to the value of the perioic coitio gaie by the z-traform. Oe of the quetio that aroe i the aalyi of imilar -D ytem i the quetio cocerig the rule a emerget orgaizatioal propertie that etermie perioicity [3]. I our cae, the chage of parameter caue the ytem to witch betwee ifferet perioic equilibrium. The orerig of the equilibrium i etermie by the geeral Eq. (3). The ratioal fractio m, which i i our cae traforme by Eq. (3) to the value of parameter, correpo to the Farey equece, which coul be repreete by the ivere Farey tree i Fig.. The greye fractio i the Farey tree poit to the equece of Fiboacci umber [0] { F i / F i+, i ℵ} a { F i / F i+, i ℵ}. A perioic parameter pace regio of ytem repoe i etermie by the coitio et > 0 a perioicity by et < 0. The claificatio at < 0 pecifie the agle, which are etermie by the three poit i the -D map i our cae, α < π ; i the cae where > 0, the agle of the map are α > π. The troget perioicity poit are etermie by the polygo tructure i -D mappig tartig with igo, triagle, etc. Other perioicity i the ubet of the mai ectio, which i etermie by the α a the Farey tree. The regio of o-perioicity etermie the repoe of the moel that i equal to the covetioal cobweb moel. Oe of the propertie of the hypericurive, oliear a chaotic ytem, which are of pecial iteret i the aalyi of complex ecoomic ytem [, 6, 7, ], i that they have cycle of every legth. It i importat to be aware of the orgaizatioal tructure of the perioic olutio provie by the emerget Farey tree icorporatig the aticipative cobweb moel a accorig to Prop. alo for the bifurcatio coitio of oliear ytem. 3. Equilibrium aalyi Equilibrium coitio of the tuie ytem correpo to the perioic olutio a are therefore of primary importace for the ecriptio of ytem repoe. The equilibrium coitio for the P 6

8 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 egmet of the ytem provie a iteretig repoe of the Q egmet, which exhibit a hexagoal hape i the two imeioal Poicaré firt-retur map. Sice hexagoal topology ha a pecial meaig i the aalyi of icrete yamical ytem [0], the equilibrium at = 3 a = 6 i examie. Other coitio are tate by the previouly tate perioicity coitio, which are ummarize i Table. The equilibrium coitio for the P egmet of the aticipative cobweb ytem coul be tate a: p c b c + a p a p a p c ( ) = = (6) b b b The equilibrium value of the parameter for the P egmet of the ytem are: b = 0 a, i thi cae P = 0. The equilibrium coitio for the a = c = p a Q egmet of the aticipative cobweb moel coul be tate a: b b b ( p a ( a + ( p c) c)) = p = a + ( p c) (7) b The Q egmet of the ytem ha o olutio. Whe the equilibrium coitio for the P egmet of the ytem are coiere i fact i all the cae, the Q egmet of the ytem coul ot be i a table tate. A graphic preetatio of the equilibrium coitio a = c = p a b = i how i Fig.. The hexagoal hape i ow a the poible optimum hape i the cotext of patial ecoomic [0]. The exitece of hexagoal hape i ow i pace ecoomy a explaie by the tructural tability [0]. Figure : Repoe of the Q egmet of the ytem while the P egmet i i equilibrium Propoitio The equilibrium coitio for the P egmet of the aticipative cobweb ytem efie by the equatio from Eq. (5) to Eq. (0) i: a = c = p a b =. Uer thee coitio, the Q egmet of the ytem ha the repoe of a hexagoal hape with vertice {( a,0),( a, a),(0, a),( a,0),( a, a),(0, a)} i Q, Q + mappig. While the repoe of the ytem for the P egmet i i equilibrium, the Q egmet of the ytem ha a hexagoal-lie hape with a igificat imeio of parameter a value a ege imeio of a a a. 7

9 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 Propoitio 3 Triagular ( ), i.e. three-perio repoe i -D mappig i etermie by the coitio b =. I orer to gai the term for the perio i the P value, oe houl apply perioic coitio. The value for time,,4 houl be ymbolically expree. By iertig Eq. (5) a Eq. (6) ito Eq. (9) the followig term i gaie: P b( p c) a c + p a = ( + ) (8) b b By repetitio of a imilar proceure for the equatio for P + 3, coierig the perio 4 coitio, i.e. P + 3 =, oe houl get the followig equatio: P b b( p c) a c + p a ( ( b b ) p c ) p c = 0 (9) with olutio b = R/{0}. Propoitio 4 Hexago ( ), i.e. ix-perio repoe i -D mappig i etermie by the coitio b =. The value for time,,6 houl be ymbolically expree. For example, etermiatio of P + i bae o expreio for P + a P, etermiatio of P + 3 i bae o expreio for P + a P +, etc. Perioic coitio i expree a P + 6 = P with olutio b = R/{0}. 4. Aget-bae aticipative cobweb moel Thi ectio ecribe the iteractio betwee everal ecoomic etitie moelle a aget. Aget iteractio repreet the alterative cotrol mechaim, which houl provie taig ocillatio a global equilibrium-eeig behaviour fou i real worl cae. The iitial feebac-aticipative cobweb moel ocillate oly for a certai parameter et, which repreet a thi borerlie i parameter pace. If the ytem parameter are ot o thi borer, the moel either ecay or prouce exploive behaviour [9]. There are everal approache to fixig the metioe problem. Oe of the metho for olvig thi problem i applicatio of the floor-roof priciple, which houl limit capital toc value a well a raw material, etc. [9]. I a evelope aget-bae ytem we move from imple liear yamic to icorporatio of the oliear rule a the elaye a aticipate iformatio cocept. The aticipate repoe of the ytem i therefore complex i behaviour ue to the coevolvig ature of the ytem [7]. I the propoe aget-bae formulatio oe houl aume that each aget ha a iiviual, local view of the problem poe, which i aree by the feebac-aticipative priciple. Real ecoomic ytem are etermie by their perioic repoe. Let u aume that there are may ifferet ecoomic ytem i our eviromet which iteract. Here the ecoomic ytem will be repreete a aget A iteractig with other aget with the goal of reachig geeral ytemic equilibrium by ifferetiatig parameter, which etermie the frequecy repoe of the ytem. Here, the quetio arie: I it poible to cotrol the ecoomic ytem by chagig the cotrol parameter, which actually alter the frequecy repoe of the ytem? The eviatio i price coul be uertoo a the chage of the ytem frequecy repoe. I other wor, if we chage the price, the frequecy of the ytem alter. Therefore, oe houl try to cotrol the ytem by chagig it frequecy repoe. 8

10 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 Coier the followig aget-bae aticipative cobweb moel of price P yamic erive from equatio Eq. (): bp + a c b P (30) = + P + Iitial coitio for Eq. (30) houl be tate i matrix form by Eq. (5, 6). I the above equatio matrix aotatio repreet colum vector, which have the ame arbitrary imeio etermie by the umber of aget. For the computatio of the ew value of price P, hift operator ρ o equece P i applie, which hift equece p P oe tep to the left: ρ ( P ) = p+ (3) proviig the forwar hifte value for P a P i Eq. (30). The eciio of chage i parameter will be epeat o the um of two price value at time + a time. Here, the relative value of the price by taig the rage of ytem repoe i the eomiator will be coiere: ξ = + + ξ e (3) ξ ξ I Eq. (3) ξ repreet the etimatio chai for r time tep compute i a imilar maer to P i Eq. (30) except for the iitial coitio, which are tate i matrix form a i Eq. (5, 6) for time 0, while for ξ (0) hift operator ρ i applie forcig the aticipatio priciple a ξ = f ( P + ). Beie the otatio for abolute value i the eomiator, the roof a floor operator are applie. I orer to perform the cotrol by variatio of parameter, where aget are preet, the followig tate equatio with the aaptive rule for i itrouce: where + + = (33) etermie the chage i cotrol parameter : β if e = = β if e = e e (34) I the above efiitio of the aget rule, the floor a ceilig fuctio over a vector of relative price e coier oly a fiite umber of lag. Oe houl otice that the metioe floor-roof operator are applie o vector rather tha o vector P, which woul mea the trict, covetioal implemetatio of the floor-roof priciple [9]. Parameter β i the iteity of aget reactio to the maret iequilibrium; β (0,). Iitializatio of vector i etermie by raom value r i [,], which fall withi the iterval of perioic olutio for the aticipative cobweb ytem. Certaily, oe coul alo aig a arbitrary value for a thi will alo be coiere. The iea capture i the above efiitio coier a ituatio where a ecoomic ytem, of which the maret price i the pat a etimate future i the highet, houl be cotrolle by icreaig the value of cotrol parameter, thu chagig the frequecy repoe of the ytem. The cae at the lower e of the maret price i ivere. 9

11 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73// A ote o applicatio Poible applicatio of the evelope aget-bae moel i i the fiel of toc exchage aalyi a cotrol where everal ecoomic ytem iteract. Here it i our itetio to how the maifetatio of perioicity a cotrol rule a ecribe i previou ectio o the real cae. Te top rae toc of the Ljubljaa Stoc Exchage [4] will be coiere, amely: AELG, GRVG, HDOG, IEKG, ITBG, KRKG, LKPG, MELR, PETG a SAVA (Fig. 3). Figure 3: Normalize toc value (upper) a average probability of floor-ceilig rule (lower) The perio of the obervatio will be from to (33 traig ay) icorporatig the critical fall i value from 008 owar; everal toc came i to maret after the iitial ate i.e. the lat toc oberve etere the mare o all of which preet 30,059 ata poit. The ormalize value of metioe toc are how i the upper part of the Fig. 3; here equity rate are coiere. The propoe floor-ceilig rule tate by Eq. (3-34) provie a meaigful iteractio betwee everal ecoomic ytem which occurece i Sloveia top te toc i how by Average probability i the bottom of Fig. 3; here the movig average of oe year wa coiere. It i urpriig, that the rule tate by Eq. (3-34) for aget bae moel occur with p = 0.46; σ = The value are how from 00 owar ice everal toc were ot yet lite o the toc exchage maret. Regarig the perioicity a the moel tate by Eq. () oe coul a, why i the etermiatio of price at time t epeat o price at t a t +? I the toc exchage maret thi woul be the cae, whe the price o the curret ay ( t ) coul ot be etermie. The curret price i ot ow before the cloig o the e of the ame ay. Therefore for time t, oly etimate price coul exit, which i mae o previou price ( t ) a expecte price ( t + ). There are metho to ehace the etimatio of curret price to exte the umber of time-lag coiere [8] which i the topic of ratioal expectatio a itertemporal equilibrium. A the example regarig the perioicity of aget, the maifetatio of perio ix i Sloveia toc 0

12 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 exchage maret i how i Fig. 4; here the movig average of oe year i tae. O y-axi the umber of perio ix occurece are how o average i oe year for te coiere toc, with x = 34 ; σ =. That woul mea, that the perio ix occur o average every two wee i oe of the te toc value. Perio ix i how here a oe poible maifetatio of equilibrium a how by Prop.. From Fig. 4 oe coul oberve, that the Number of perio ix occurece i fallig from the year 004 owar which might be oe of the iicator of ogoig itability i the toc exchage maret. Figure 4: Number of perio ix occurece i blue chip The importace of the quetio, how to cotrol everal iteractig aget i apparet i the cae of the global crii which i iicate i 008. It i a matter of further reearch to ivetigate the cotrol trategie o the real cae which woul provie global equilibrium or at leat compeate for the global iturbace. 6. Dicuio Uertaig the emergece of perioicity i a prerequiite for ecoomic ytem cotrol. The baic theory evelope i thi fiel coier the aalyi of eige vector a eige value [4]. The equality of the gaie olutio etermie by Prop. for the aticipative cobweb moel a geeral perioic olutio for icrete oliear ytem [4] provie a view of the Farey tree tructure, which i [4] uiveral i oliear expaio of the coiere ytem. Several aticipative cobweb ytem were itercoecte i orer to form a aget-bae moel where iteractio wa etermie by the icrete rule. By evelopig a aget-bae moel, the methoological platform i provie for a further examiatio of the iteractio of everal feebacaticipative ytem. The evelope ytem provie promiig reult obeyig the rule of iteractio a elf-ychroizatio fou i real worl ecoomic ytem. Aother importat fact i that aget with more iteractio provie better coitio for geeral ytem tability. A etaile aalyi of the aget-bae moel provie proof of ytem tability, which i oe of the ey coitio that houl be meet by aget-bae moel imulatig complex ecoomic ytem. The experimetal examiatio of the two aget moel iicate, that the etire moel coul be et i the global equilibrium moe. All the tate characteritic of the aget-bae moel a well a the repoe of the ytem for eight aget provie a promiig methoological platform for tuyig the iteractio betwee everal ecoomic etitie uch a aget. The propoe moel provie the mea for aalyzig iteractio, feebac, aticipatio, frequecy repoe, ychroizatio, taig ocillatio a ytem equilibrium. A itrouctio of feebac-aticipative ytem itercoectio a cotrol by varyig the parameter, which ifluece ytem frequecy repoe, repreet a ew perpective for the

13 V Iteratioal Worhop o Mathematical Moel a their Applicatio 06 IOP Publihig IOP Cof. Serie: Material Sciece a Egieerig 73 (07) 008 oi:0.088/ x/73//008 aalyi of complex ytem. The ey value i the Farey tree hol for the aget-bae moel a have provie a poible tartig-poit for the earch of aalytical olutio i the parameter pace. The preet tuy provie itercoectio of Kalor cobweb moel paraigm with a aticipative approach a aget-bae paraigm. The commo eomiator a a threa of the paper i the evolutioary emergece priciple hie i the form of the Farey tree icorporatig the Gole Ratio path. The evelope ytem cofirm the applicability of the cotrol rule where ytem repoe i cotrolle by the variatio of parameter, which ifluece the frequecy repoe of a particular aget. By examiig the evelope aget-bae ytem, it coul be coclue that iteractio of icrete ocillatory aget by itelf reult i table worig coitio. I the preet cae the rule of iteractio provie ytem tability a a ychroizatio property which i fou i real worl ytem [] a igificatly etermie iteractig ecoomic ytem. Acowlegmet Thi reearch wa upporte by the Sloveia Reearch Agecy (Programme No. P5-008 & Proj. No. RU/ ). Referece [] Berry B a Kim H 000 Log wave Chao Theory i the Social Sciece: Fouatio a Applicatio pp 5 36 [] Bichi G, Dieci R, Roao G a Saltari E 00 Joural of Evolutioary Ecoomic [3] Broc W a Homme C 997 Ecoometrica [4] Cavalli F, Naimzaa A a Pariio L 05 A cobweb moel for electricity maret th Iteratioal Coferece o the Europea Eergy Maret (EEM) pp 5 [5] Chechi G a Ryabov D 004 Phyical Review E [6] Galla J a Nue H 996 Joural Of Ecoomic Behavior & Orgaizatio [7] Hogg T a Huberma B 99 IEEE Traactio o Sytem Ma a Cyberetic [8] Homme C 998 Joural Of Ecoomic Behavior & Orgaizatio [9] Kott R 006 Fiboacci umber a the gole ectio (//006) [0] Kocic L a Stefaova L 004 Facta Uiveritati, Serie: Mechaic, Automatic Cotrol a Robotic [] Kociuba G, Heceberg N a White A 00 Phyical Review E [] Kopel M 997 Joural of Evolutioary Ecoomic [3] Kreyzig E 993 Avace Egieerig Mathematic (Hoboe, NJ: Joh Wiley & So) [4] LSE 009 Ljubljaa toc exchage (0//009) [5] Gori L, Guerrii L a Soii M 04 Dicrete Dyamic i Nature a Society [6] Lueberger D 979 trouctio to Dyamic Sytem: Theory, Moel a Applicatio [7] Matumoto A 997 Dicrete Dyamic i Nature a Society [8] Naimzaa A a Pecora N 06 Joural of Differece Equatio a Applicatio [9] Puu T a Suho I 004 Chao, Solito a Fractal [0] Puu T 005 Networ a Spatial Ecoomic [] Roe R 985 Aticipatory Sytem: Philoophical, Mathematical a Methoological Fouatio [] Roeblum M a Piovy A 003 Cotemporary Phyic [3] Sche K E 005 Joural of Evolutioary Ecoomic [4] Soi M 999 Dicrete Dyamic i Nature a Society [5] Soi M 996 Dicrete Dyamic i Nature a Society [6] Sprott J 994 Phyical Review E 50 R647 R650 [7] Strogatz S 994 Noliear yamic a chao: with applicatio to phyic, biology, chemitry a egieerig