Economy and Application of Chaos Theory

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1 Iteratioal Joural of Busiess, Humaities ad Techology Vol. 5, No. 5; October 2015 Ecoomy ad Applicatio of Chaos Theory Petr Dostál Bro Uiversity of Techology Faculty of Busiess ad Maagemet Istitute of Iformatics Kolejí 2906/4, Bro Czech Republic Imrich Rukovaský Bro Iteratioal Busiess School BIBS Lidická 81, , Bro Czech Republic Abstract The theory of chaos came ito beig i solutio of techical problems, where it describes the behaviour of oliear systems that have some hidde order, but still behave like systems cotrolled by chace. A liear model ca describe a real system oly if it is liear. If this is ot fulfilled, the such models ca simulate the real system oly uder ideal coditios, oly for a short time. If a system is a oliear dyamic, a determiistic system it ca geerate ot oly the permaet treds ad cycles, but it ca also iclude radom-lookig behaviour. Processes of such behaviour are preset i the ecoomy. Keywords: Chaos, attractor, logistic fuctio, Hurst expoet, fractals. 1) Itroductio The theory of chaos came ito beig i solutio of techical problems, where it describes the behaviour of oliear systems that have some hidde order, but still behave like systems cotrolled by chace. A liear model ca describe a real system oly if it is liear. If this is ot fulfilled, the such models ca simulate the real system oly uder ideal coditios, oly for a short time. If a system is a oliear dyamic, a determiistic system it ca geerate ot oly the permaet treds ad cycles, but it ca also iclude radom-lookig behaviour. Processes of such behaviour are preset i the ecoomy. Chaos i the fiacial market is discussed i may books (Trippy, 1995), (Dostal, 2013). The behaviour of ecoomic pheomea is a complex system (Rukovaský, 2005).I this respect we ca talk about two categories that are i oppositio to each other: order ad radomess. Chaos is somethig i betwee. It ivolves some degree of order. Some pheomea ca appear radom, but after the study of these pheomea some ier order ca be discovered that govers these pheomea. For example, the movemet of people at a railway statio ca appear accidetal, but i fact it is behaviour with order cotrolled by the arrival ad departure of trais. Also, the ecoomy ca exist i states of various degrees of order, chaos, ad radomess. The behaviour of a ecoomy is iflueced quite ofte by atural disasters, political chages, etc.i coectio with chaos it is possible to speak about a so-called attractor, which represets a equilibrium positio. A attractor is a coditio towards which a dyamical system evolves over time. That is, poits that get close eough to the attractor remai close eve if slightly disturbed. Geometrically, a attractor ca be a poit, a curve, or eve a complicated set with a fractal structure kow as a chaotic attractor. The geometrical explaatio of a attractor could be doe with the help of a pedulum. The attractor ca be: Poit - the stability is represeted geometrically by a poit. For example, whe a pedulum is displaced, its fial equilibrium positio is reached whe movemet ceases. Cycle - the balace is represeted geometrically by a limited cycle. For example, whe a pedulum is movig with costat eergy (potetial eergy plus kietic eergy = cost.), its equilibrium positio is reached whe movemet is cyclical. 18

ISSN 2162-1357 (Prit), 2162-1381 (Olie) Ceter for Promotig Ideas, USA www.ijbhtet.com Chaotic - it is represeted geometrically by order uderlyig the apparet chaos.

2 ISSN (Prit), (Olie) Ceter for Promotig Ideas, USA Chaotic - it is represeted geometrically by order uderlyig the apparet chaos. For example, whe the pedulum is drive by radom eergy, the the equilibrium positio will be represeted by a movemet i zoe (o poit or cycle). As we have said, the chaotic attractor ca be preseted geometrically i closed space, for example by a zoe iside of two ellipses i two-dimesioal space. See Figure 1. The area of equilibrium positio of a dyamic system could be explaied by a zoe betwee the outer ad ier ellipses. The zoe could represet for example the road for car racig. The trajectories of the cars are ot the same, but very similar, because of reactios of drivers o outer coditios. Eve if small accidets happe the cars stay o the road. If there will be a great accidet, the cars leaves the road. The same is it with equilibrium of ecoomy of states.see fig. 1. Figure 1: The presetatio of chaotic attractor 2) Logistic fuctio The complicated behaviour i ecoomy could be illustrated by the equatio (i literature called a logistic fuctiox i+1 = r. x i ( 1 x i ),where r is a costat ad x i is a variable. The equatio eables geeratio of behaviours that are cosidered to be poit, periodic, ad chaotic. Figure 2 presets the graph geerated by this equatio with iitial values x 0 =0.85 ad r = 3.5 or r = 3.7. The vertical axis represets the values x i+1 ad horizotal axis represets the values i. The curve i the graph represets chaotic behaviour, i that it has a hidde order that is described by the above-metioed logistic equatio, eve if this reality is hardly recogizable. x i i Figure 2: The graph of logistic fuctio with parameters x 0 =0.85 ad r = 3.5 or 3.7 Applicatio of logistic fuctio i ecoomy The complicated behaviour i ecoomy could be illustrated by the modified logistic fuctio i the form Pt Pt 1 ( Vt Pt ), where P t+1 simulates the market price of the share, σ t is the volatility of yield of share, 1 a t a isa coefficiet of market aversio agaist risk, V t is fudametal ier value of share i time t, V t P t is the respose of market o the chage of ier value. The iitial values (P 0, σ 0, V 0 ) could be differet i the same way as the coefficiets of market aversio agaist risk. If the value = 0 ad market is eutral to risk a = 0, the P t+1 = V t. 19

3 Iteratioal Joural of Busiess, Humaities ad Techology Vol. 5, No. 5; October 2015 Figure 3 presets graphically the course of P t+1 depedece o time with aversio to risk a = The course of the curve remids oe of real values of time series of shares, commodities, currecy ratios, ad values of idices o the stock market. P t t Figure 3: The graph of fuctio P t+1 with parameter a = ) Hurst expoet The calculatio of a Hurst expoet for a time series is doe by the algorithm _ 1 x( i1 t xi X ( t, ( xi x( )), i1, _ t 1,2..., ; S( 1 i1 _ ( xi x( ) 2, R( max X ( t, mi X ( t,, Where x = (x 1, x 2,..., x ) is the sequece of values of time series ad is the umber of values. The Hurst expoet is used for evaluatio of order or radomess of dyamic systems that geerate such time series. It could be for example for time series of records of GDP, uemploymet, etc. The Hurst expoet reaches values i the rage from 0 to 1. The value H aroud 0.5 meas, that there is a log-term cycle i the time series. If the value H gets ear to 0 or 1, it meas that there is a log-term cycle i the time series ad determiistic behavior is a part of time series. Hurst expoet (H) determies the rate of chaos. If the probability distributio of the system is ot ormal, or closes eough, a oparametric aalysis is eeded. Hurst expoet is such a method i the case of time series. Hurst expoet ca distiguish fractal from radom time series, or fid the log memory cycles. If the value of H = 0.5, the time series is ormally distributed, or has o log memory process. If (H i <H i+1 ), the time series is a ati-persistet or mea revertig time series. If the time series has bee up i the previous period, it is more likely to be goig dow i the ext period, ad vice versa. Whe (H i >H i+1 ), the time series is persistet or tred reiforcig time series. If the time series has bee up i the previous period, it is more likely to be goig up i the ext period, ad vice versa. Hurst expoet ca measure how jagged a time series is. The lower the H value is, the more jagged the time series is. The higher the H value is, the more apparet the tred is, ad the less jagged the time series is. The fractal dimesio is calculated as a value = 2.0 H. The costat c iflueces the value of H ad it was set to t 1t

4 ISSN (Prit), (Olie) Ceter for Promotig Ideas, USA Applicatio of Hurst expoet i ecoomy There are show some represetative results of calculatio of Hurst expoet cocerig the of Microsoft, Co. share, title Figure 4 ad NASDAQ idex Figure 5 of the America stock market. The preseted chart for t = t max ad graphs cofirm theory metioed above. The time series of Hurst expoet of MSFT share ad NASDAQ idex proves; that the more apparet the tred is, ad the less jagged the time series is. Figure 4: The graph MSFT - Hurst Expoet Figure 5: The graph NASDAQ - Hurst Expoet 4) Fractals The theory of chaos icludes fractal geometry. Fractal geometry as described by B. Madelbrot officially bega twety years ago. The study of the fluctuatio of market prices led him to foud fractal geometry. The character of fractals is the repetitio of a motif: we talk about so-called self-similarity ad self-relatioship (for example the structure of a whole tree has self-similarity as a structure of mai braches, small braches, twigs, etc.). Whe we elarge or reduce ay part of a fractal formatio, it will be similar to the origial oe. A fractal is a atural pheomeo or a mathematical set that exhibits a repeatig patter that displays at every scale. If the replicatio is exactly the same at every scale, it is called a self-similar patter. Fractals i stock market are metioed i may books (Peters, 1994). Applicatio of fractals i ecoomy This theory is icluded also i so-called Elliott waves. The waves have two phases, the first a impulse phase ad the secod a correctio phase. The impulse phase cosists of five breaks marked 1 5 ad a correctio phase cosists of three breaks marked a c. Figure 6a presets the situatio for a iitial icrease of a time series ad Figure 6b for a iitial decrease of a time series. These patters could be foud i time series with various sampligs; they are self-similar ad have self-relatioship. May time series are published i the mass media from the ecoomy brach ad the stock market. 21

5 Iteratioal Joural of Busiess, Humaities ad Techology Vol. 5, No. 5; October 2015 Figure 6a: Elliott wave - icrease Figure 6b: Elliott wave - decrease This search for waves is used for evaluatio ad predictio of time series such as shares, idices, commodities, ad currecy ratios o stock markets. The simulatio of chaos i ecoomy could be doe by Tamarichaotic attractor (Tamari, 2015) defied by equatios x' = (x-a.y).cos(z) b.y.si(z) y' = (x+ c.y).si(z) +d.y.cos(z) z' = e + f.z + g.ata{ [(1 - u).y] / [(1 - i).x] } for pricig - spiral versio z' = e + f.z + g.ata{ (1 - u) / (1 - i) x.y } for wealth - attractor versio. The drawig of Tamari ecoomic attractor ad its trajectory is preseted i Figure 7 where the depedece of z (pricig/wealth) o x(output)ad y (moey) is draw. The exoges are iertia, productivity, pritig, adaptatio, exchage, idexatio, elasticity, expectatios, uemploymet ad iterest. The values of exoges are: a- iertia, b - productivity, c -pritig, d -adaptatio, e - exchage, f -idexatio, g -elasticity/expectatios, u- uemploymet ad i- iterest. Figure 7: Tamari ecoomic attractor Tamari attractor eables the study of a coutry's ecoomy ad it is used for simulatio ad search of coutry's ecoomy, such as aalysis, plaig, predictio ad compariso with other atios' ecoomies. There are various methods of simulatio i ecoomy ad o the stock market (Dostal, 2014). 5) Coclusios The theory of chaos is still uder developmet; evertheless some outputs are becomig usable i busiess, such as the calculatio of Hurst expoets to evaluate the order or chaos of a busiess system. The properties of fractals are used for aalysis ad predictio of time series i stock markets. 22

6 ISSN (Prit), (Olie) Ceter for Promotig Ideas, USA Referece Dostál, P. (2014) Stock Market ad Vortex. Twelve Iteratioal Coferece o Soft Computig Applied i Computer ad Ecoomic Eviromet, Dostal, P. (2013). Advaced Decisio Makig i Busiess ad Public Services. Bro: CERM Publishig House. Peters, E.E. (1994) Fractal Market Aalysis, Joh Wiley & Sos Ltd., UK. Rukovaský, I. (2005) Evolutio of Complex Systems. 8th Joit Coferece o Iformatio Scieces. Salt Lake City, Utah, USA. Trippi, R.R. (1995) Chaos & Noliear Dyamic i the Fiacial Markets, USA Tamari, B. (2015).Attractors. [Olie] Available: (September 7, 2015) 23

Economy and Application of Chaos Theory

Economy and Application of Chaos Theory 1. Introduction The theory of chaos came into being in solution of technical problems, where it describes the behaviour of nonlinear systems that have some hidden