Numerical simulations used to detect the chaotic evolution of the exchange rate described by a nonlinear determinist system

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1 Iue Volume 7 Numerical imulaion ued o deec he chaoic evoluion of he exchange rae decried y a nonlinear deermini yem Mirela-Carinel Voicu Arac In hi paper we preen a udy concerning he exchange rae evoluion governed y a hird-order nonlinear deermini dicree yem We preen ome reul concerning he unique fixed poin of he yem i ailiy and i aracion domain under cerain value of parameer and we alo preen he exience of period-wo cycle Given he nonlinear naure of he yem i mo complex ype of ehavior i he chaoic dynamic We canno deec hi ype of ehavior uing only analyical ool For hi reaon in order o deec he dynamic of he yem we will ue numerical imulaion I i known ha he Lyapunov exponen are a ool ued o ealih he ype of ehavior in nonlinear dynamic We will calculae heir value in order o ealih he ype of dynamic From our numerical imulaion we preen a cae in which he yem diplay a chaoic ehavior For hi paricular cae we alo conider a correponding yem of he econd order From he image of he figure in hi paper we can oerve a imilariy eween he image of aracor for each paricular order of he yem Keyword nonlinear yem numerical imulaion chao aracor A I INTRODUCTION CCORDING o [] a general equaion modeling he exchange rae evoluion i given y: X () In he aove equaion i he exchange rae a he momen ; X decrie he exogenou variale ha drive he i he expecaion exchange rae a he momen ; held a he momen in he marke aou he exchange rae a he momen ; i he dicoun facor ha peculaor ue o dicoun he fuure expeced exchange rae (<<) Thi model allow u o ake ino accoun wo componen for forecaing: a foreca made y he chari c a foreca made y he fundamenali f ( ): and Manucrip received Augu 7; Revied received epemer 7 The auhor i Profeor PhD of he Deparmen of conomic Informaic wihin he Faculy of conomic cience We Univeriy of Timioara Romania - hp://wwwfeuvro - and he i a memer of "Nichola Georgecu-Roegen" Inerdiciplinary Plaform - hp://wwwfeuvro/ngrpla/ ( mirelavoicu@feuvro) m m ( / ) / () c f / where m i he weigh given y he chari and m i he weigh given y he fundamenali a he momen The fundamenali aume he exience of an equilirium exchange rae If a he momen - he exchange rae i aove repecively elow he equilirium rae he fundamenali expec he fuure exchange rae o go down repecively increae wih he peed α More preciely if hey oerve a deviaion oday hen heir foreca i he following: f α α > The chari ue he pa value of he exchange rae o deec paern ha hey exrapolae in he fuure An equaion which give a general decripion of he differen model ued y chari i he following: c () f ( N ) (4) According o [] i i poile o pecify uch a rule in general erm a follow: C C C N c N (5) N The exac naure of hi rule i deermined y he coefficien c Thee can e poiive negaive or zero i The weigh m in equaion () given y chari i m β ( ) β > The parameer β meaure he preciion degree of he fundamenali' eimaion When he exchange rae i in he neighourhood of he equilirium rae chari' ehavior (6) INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC

2 Iue Volume 7 4 dominae When he exchange rae differ from he fundamenal rae hen he expecaion will e dominaed y he fundamenali In hi paper we conider he cae X (which mean ha ) and for chari we conider he expecaion: c (7) In equaion () we will ue he expecaion given y he equaion () and (7) In equaion () we will ue he expecaion given y equaion () In hi way we oain he following difference equaion: F ( x ' y ' z ' ' ' ' ) U for every poin ( x y z ) V ( α β ) and all > where F F F ime V U α o ha for every poin hen he fixed poin i U aympoically ale (aracing fixed poin) If here i a neighorhood ( β ) ( x x x ) ' ' ' F ( x y z ) ( x x x ) when ' ' ' ( x y z ) V ( α β ) α > and β > yem () ha a unique fixed poin and hi poin i () The fixed poin () i ale for ( 65) 65 Propoiion In he cae in which ( ) and unale for U α ( α ) β β In ecion we will preen ome analyical reul for equaion (8) and in ecion we will preen ome numerical imulaion II TH XCHANG RAT VOLUTION GOVRND BY A THIRD-ORDR NONLINAR DTRMINIT DICRT YTM If we denoe he form: wih β ( ) ( e ) (8) ln hen equaion (8) can e wrien in α ( ) α (9) β e R and Z We can rewrie equaion (9) in he following vecorial form: where : R R ( ) F( ) () F F ( x y z) ( F ( x y z) F ( x y z) F ( x y z) ) F x y z i defined in he following way: z F ( x y z) y and F ( x y z) ϕ ( z) z ψ ( z)x ( α ) ϕ z ( α )β and ψ ( z) z z β ( e ) β ( e ) wih A eady-ae exience uniciy and ailiy A fixed poin for yem () i a poin ( x x x ) for x x x F x x x We recall ha a fixed poin ( x x x ) i ale if for any U x x x here i a α β x x x uch ha which ufficienly mall neighourhood neighourhood V U The equilirium exchange rae mean ha money demand i equal o money upply When he dicoun facor i in he inerval ( 65) he fixed poin i ale In fac if i mall hi mean ha he exchange rae value i rongly influenced y he exogenou variale (which yield he equilirium value of he exchange rae) We have een ha in hi cae i i imporan o know he equilirium value ecaue he rader expec ha in a neighorhood of equilirium exchange rae he exchange rae will go ack o hi value for ( 65) For 65 we noice a ifurcaion Afer he ifurcaion he fixed poin i unale and i i urrounded y a limi cycle ha i ale (we oerve hi from imulaion) Wihin a neighourhood of he fixed poin all he ori aring ouide or inide he cloed invarian curve excep a he origin end oward he limi cycle under he ieraion of he funcion F Thi i a Neimark-acker ifurcaion α > he fixed poin () of yem () i gloally aracive Propoiion For ( ) β and ( ] B Period-wo cycle We hall now udy he exience of cycle of period wo if α and β > or if α and ( α ) β hen he yem () ha no cycle of ( α ) period wo ii) If ( ) α and α β α hen yem () ha only one cycle of period wo Thi cycle i {( ) ( )} where are he oluion of Propoiion i) Under he aumpion ( ) INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC

3 Iue Volume 7 5 ϕ( x) ϕ x he equaion ψ ( x) ϕ( x) which mean ha x ϕ( x) ψ ( x) ψ x ψ ( x) are he oluion of he equaion x ( α ) β ( e ) x β e α β e x ( α ) β ( e ) x x ( α ) β ( e ) β ( e ) x β e β e The numer verify he relaion < Le e he poiive numer If ( α ) β hen α α > ln and β α < ln β α If β hen ( α ) ln ln β α β α and ln ln β α β α I i poile o claify he differen aracor: aracing fixed poin aracing n-cycle quaiperiodic aracor and range aracor An aracor a an experimenal ojec give a gloal decripion of he aympoic ehavior of a dynamical yem When a deerminiic mechanim preen complex ehavior wih inermience we can conclude ha he erie evince chao under cerain condiion The eniive dependence on iniial condiion i one of he mo eenial apec o idenify chao We recall ha he eniive dependence on iniial condiion mean ha wo rajecorie aring very cloe ogeher will rapidly diverge from each oher The range aracor i aociaed wih a chaoic ae of ime evoluion and i characerized y he eniive dependence on iniial condiion A meaure of he average rae of exponenial divergence exhiied y a chaoic yem i given y he Lyapunov exponen of he yem; he poiiviy of one from hee exponen can ugge he preence of chao The Lyapunov exponen and are given y n { } n () e e e lim eigenvalue of J ( F( )) n where J ( F( )) repreen he Jacoian marix of he funcion F For a period-p poin he Lyapunov exponen and are given y From Propoiion and we oain he following propoiion: Propoiion 4 If ( 65) α and ( α ) β hen he fixed poin () of α yem () i only locally aracive III NUMRICAL IMULATION A Numerical imulaion for he yem () We now recall ome noion which will e ued in hi ecion We ay ha a e A i an aracing e wih he fundamenal neighourhood U if i verifie he following properie (ee [5]): ) araciviy: for every open e V A F U V for all ufficienly large ) invariance: F ( A) A for all ) A i minimal: here i no proper ue of A ha aifie condiion and The ain of aracion i he e of iniial poin x o ha F x i cloe o A when p { e e e } eigenvalue of J F( ) p () We recall now ha for an aracing period-p cycle he Lyapunov exponen are negaive; in cae of a ifurcaion poin a lea one Lyapunov exponen i zero; for a limi cycle one Lyapunov exponen i zero and he oher are negaive and for a chaoic ehavior he highe Lyapunov exponen i poiive while he um of he all Lyapunov exponen i negaive In order o compue he Lyapunov exponen when yem () diplay a chaoic ehavior we ue he mehod propoed in [] aed on he Houeholder QR facorizaion and he implemenaion mehod propoed in [8] We have made many numerical imulaion and we have found many iuaion in which he yem diplay a chaoic ehavior For he paricular cae where α 95 and he iniial where α we condiion ( ) ( ) ( ) ( ) α β ( e ) β ( e ) INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC

4 Iue Volume 7 6 inveigae he range of parameer β for which yem () preen a chaoic or a non chaoic ehavior We oerve differen inerval of value for β for which in general yem () diplay a chaoic ehavior Thee inerval are eparaed y an inerval of value of β which characerize a equence of period-douling ifurcaion for yem () In Figure - we preen he range aracor which characerize differen ype of inerval of value for β for which he yem diplay a chaoic ehavior (he fixed poin () i unale) he influence of peculaor increae more and more For cerain value of he parameer α and β and of he exchange rae he ehavior i expeced o e chaoic Thi mean ha he influence of peculaor increae and produce inailiy and he foreca of he exchange rae evoluion i difficul When ( 65) β 6 β 6 β 6 β 6 β 46 β β 46 β β 46 β β 46 β β 6 β β 6 β β 4 β 7 Fig Chaoic aracor in he cae 95 α ( ) ( ) in he pace ( ) In Tale we give he value of Lyapunov exponen in he cae of he range aracor preened in Figure - The image from hee figure eem o repreen he ame aracor which increae and i deformed In hi cae he fixed poin () i unale and yem () ha no cycle of period wo We oerve more value for β for which yem () diplay a chaoic ehavior β 4 β 7 Fig Chaoic aracor in he cae 95 α ( ) ( ) in he pace ( ) Parameer β alo influence he dynamic of he yem The peculaor have a high influence on he marke and creae inailiy We can ee how fa he aracor end oward he equilirium value when β decreae When β i high he evoluion of he exchange rae i around he equilirium value and he weigh given y he fundamenali end oward i maximum value When β INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC

5 Iue Volume 7 7 decreae hi weigh alo decreae and he value of he exchange rae and he value of equilirium are no cloe β 6 β 6 TABL The Lyapunov exponen in he cae 95 α β To calculae he Lyapunov exponen we have ued he implemenaion mehod propoed in [8] uing a VBA (Viual Baic for Applicaion) program in xcel and he image from Figure - are made uing Mahemaica β 46 β B Numerical imulaion for a yem of he econd order Now for chari we conider he expecaion: c () β 46 β In equaion () we will ue he expecaion given y he equaion () and () In equaion () we will ue he expecaion given y equaion () In hi way we oain he following difference equaion: α ( α ) β β (4) If we denoe ln hen equaion (4) can e wrien in he form: ( ) α ( ) α ( ) (5) ( ) β e β e β 6 β β 4 β 7 Fig Chaoic aracor in he cae 95 α ( ) ( ) in he pace wih R and Z We can rewrie equaion (5) in he following vecorial form: where : R R ( ) F( ) (6) F F ( x y) ( F ( x y) F ( x y) ) i defined in he following way: F ( x y) y F ( x y) ϕ ( y) y ψ ( y)x wih ( α ) ϕ y y β ( e ) and ψ ( y) y β ( e ) and ( α )β INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC

6 Iue Volume 7 8 For he paricular cae where α 95 c and he we inveigae he iniial condiion range of parameer β for which yem (6) preen a chaoic or a non chaoic ehavior β 6 β 6 β 46 β Lyapunov exponen in he cae of he range aracor preen in Figure 4 TABL Lyapunov exponen in he cae - 95 α β From Figure and 4 we can oerve a imilariy eween he image of aracor for each paricular order of he yem β 6 β 4 6 β β β 8 β 4 β β Fig 4 Chaoic aracor in he cae c - 95 α pace ( ) We oerve differen inerval of value for β for which yem (6) in general diplay a chaoic ehavior Thee inerval are [46 ) [68] [6] [446] [98] [78] very wo inerval preened here are eparaed y an inerval of value of β which characerize a equence of period-douling ifurcaion for yem (6) We preen in Figure 4 he range aracor which characerize differen ype of inerval of value for β for which he yem diplay a chaoic ehavior In Tale we give he value of IV CONCLUION Fixing he value of parameer and he iniial condiion uing numerical imulaion we can deec he dynamic diplayed y he nonlinear yem From he cae preened in he ecion we can oerve ha he dynamic of nonlinear yem can e very complicae In uch a udy he implemenaion mehod are very imporan If we ue a good implemenaion mehod we can quickly oerve many cae (for example fixing he parameer and iniial condiion and making only one parameer variale) In hi way we can conclude on he oained reul RFRNC [] ckmann J-P Ruelle D (985) - rgodic heory of chao and range aracor Review of Modern Phyic Vol57No Par I July [] De Grauwe P Dewacher H mrech M (99) - xchange rae heory: chaoic model of foreign exchange marke Blackwell Puliher [] V D Juncu M Rafiei-Naeini and P Dudek - Inegraed Circui Implemenaion of a Compac Dicree-Time Chao Generaor - Analog Inegraed Circui and ignal Proceing pringer Neherland IN: 95- Volume 46 Numer / March 6 Page 75-8 [4] Ulrike Feudel-Generalized model a a ool o udy he ailiy of nonlinear dynamical yem - Nonlinear Dynamic Chao and Applicaioni 6h Crimean chool and Workhop 5-9 May 6 [5] Ruelle D - "Chaoic evoluion and range aracor"- Camridge Univeriy Pre 989 [6] Voicu MC On he compuaional ool role in he nonlinear dynamic udy Proceeding of ICNPAA 4 5 h -4 June 4 Timişoara România Camridge cienific Puliher Ld UK pag [7] Voicu MC -Numerical imulaion ued o deec he chaoic evoluion of he exchange rae decried y a hird-order nonlinear deermini yem - The 7h WA Inernaional Conference on APPLID COMPUTR & APPLID COMPUTATIONAL CINC (ACACO '8) Hangzhou China April pag7-76 [8] Voicu MC Compuaional Implemenaion Mehod ued o Deec he Dynamic of Nonlinear Deermini Dicree yem- uropean INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC

7 Iue Volume 7 9 Compuing Conference Vouliagmeni Ahen Greece epemer pringer Verlag Proceeding [9] hp://wwwcuee/~ooma_/linalg/lin/node8hml INTRNATIONAL JOURNAL OF APPLID MATHMATIC AND INFORMATIC