THE FRACTAL TECHNIQUES APPLIED IN NUMERICAL ITERATIVE METHODS FRAME
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1 6 th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE THE FRACTAL TECHNIQUES APPLIED IN NUMERICAL ITERATIVE METHODS FRAME George MAHALU, Radu PENTIUC Uiversity of Suceava - mahalu@eed.usv.ro Abstract: May methods i the umeric calculus resolvig oliear algebraic or differetial equatio but igore some behavior of these problems. Same aspects ca cojure to the fractal iterative techiques. I this paper work is performed a critical aalysis about iterative fractal techiques which ca coduct to various implemeted applicatios. Study of the oliear equatio, treated ito iterative techiques, makes the subject of this paper. It cosists i a short revue of the most importat priciples of the fractal calculus ad complexity applicatios i fudametal scieces ad techologies. If tryig to solve the equatio z 4 -=0 i the complex pla, we ca obtai the Newto's fractal. Ad this is ot the sigle case whe a umeric method for solvig oliear algebraic equatios has same strage behavior. If we modify the fractal logistic equatio with the goal to perform a appropriate model for some physical applicatios, we ca obtai some iterestig ad amazig results. Key words: fractal, attractor, map, set, iterative, complexity, form.. INTRODUCTION May methods i the umeric calculus resolvig oliear algebraic or differetial equatio but igore some behaviour of these problems. Sice fractal aalysis developig were idetified a lot of strage features i the solutio of same equatios. A very iterestig pheomeo occurs i the solutio of the followig set of oliear differetial equatios called the Lorez system [5]: dx dy dz 8 = 0( x y ) = xz + rx y = xy z () 3
2 This system arises from problems related to fluid covectio ad to weather forecastig. Whe the r parameter lies i the 24.7<r<45 iterval, the solutio does ot coverge to a fixed poit i the t limit, or is there a limit cycle, but the solutio keeps movig aroud i a fiite area. The limit set of the orbit at t is geerally called the attractor. It has bee cofirmed umerically that the Lorez attractor system has ifiitely may foldig. Other strage attractors have bee foud i may systems with few degrees of freedom. The followig system, called the Rössler system [5], is famous for showig that chaos ca be produces with oly oe oliear term: dx dy dz = ( y + z ) = x y = z + xz (2) Attractors of ordiary differetial equatios with the degree of freedom less tha 2 are limited to either a fixed poit or a limit cycle, ad have proved ot to be strage. However, eve i system with oly two variables, chaos ca be foud if the system evolves discretely. A good example i this sese is the strage attractor of the Heo map. The equatios system i this case is: 2 + = ax + by y+ x = x Strage attractors i systems of ordiary differetial equatios also usually have fractal properties. By imagiig a plae i the phase space ad observig oly the poits where the orbits pass through the plae, the dyamical systems ca be reduced to a discrete map called the Poicaré map. The Poicaré map of the Rössler system, like the Heo map, is selfsimilar ad the Rössler attractor is also fractal. (3) 2. STRANGE ATTRACTORS Let us cosider a simple oliear map called logistic map or bifurcatio map: x+ = rx( x ), 0 r 4 (4) This is a example of iterative method applicatio o oliear fuctio. I the first regard it is a classical ad very kowledge path to resolvig without problems same umeric aalysis applicatios. However, we ca observe that the asymptotic behaviour of x depeds strogly o r parameter: for 0 r <, x decrease as ad x approach 0; for r 2, x mootoically approaches - /r; for 2< r 3, x approaches -/r with oscillatios; for 3< r 3.449, x is gradually approaches period motio of period 2; for 3.449< r 4, the system become ucotrolled.
3 The set of attractors of x is show i figure. X 0 3 r c 4 Fig. Bifurcatio diagram of logistic map for 3 r 4 r Historically, the logistic map was obtaied from the logistic equatio, which describes the growth of a populatio i a closed area: du = ( ε hu )u (5) If we put this equatio ito a differece equatio forms: u u Δt + = ( ε hu we obtai the logistic map if we chage the variables as: )u hδt r = + εδt x = u (7) + εδt The solutio of (5) ca be obtaied aalytically for ay iitial coditio u(0)>0. It mootoically approaches a fixed poit ε/h. By cotrast, the differece equatio for large iterval Δt ad the logistic map behave quite differetly, producig chaos. This kid of discrepacy betwee the solutio of a differetial equatio ad that of its differece equatio appears i ay oliear system if the differece iterval is sufficietly large []. Hece we have to be careful whe we umerically solve a differetial equatio by usig a differece equatio. If we modify the logistic equatio i the form: x+ = rx /( x ), 0 r 4 (8) we ca observe a iterestig result about the map equatio (figure 2). This result makes subject of some origial studies focused o the umeric methods i complexity calculus. If tryig to solve the equatio z 4 -=0 i the complex pla, we ca obtai the Newto's fractal [4] which show like i figure 3. Ad this is ot the sigle case whe a umeric method for solvig oliear algebraic equatios has same strage behaviour. (6)
4 r X Fig. 2 Bifurcatio diagram of modified logistic map for 0.0 r 4 3. FRACTALS BY MAPS For a give map [2]: x + = f ( x ) (9) the set of iitial poits {x o } whose iterated poits ever diverge ( x < for ay ) is called Julia set. For may maps, the Julia sets are kow to be fractals. A good example is the followig complex logistic map: I the same way, equatio: f ( z ) az( z ) =, where z=x+jy, j=. (0) g( z ) = z 2 b () coducts to other fractal. To set of complex parameters b such that successive iterates of z=0 uder g(z) do ot ted to is amed the Madelbrot set. This set has a fractal border. Whe we solve a algebraic equatio umerically by Newto's method, we have to iterate a map similar to (). If the equatio has several solutios, a iitial value for the iteratio will be attracted to oe of the solutio. The boudary of the set of poits that fially coverge to oe of the solutio becomes a fractal. Two iitial poits that are arbitrarily close ca approach distict solutios, if they start close to this boudary. Aother simple method to costruct fractals is provided by cotractio maps. It is trivial that the ivariat set of a sigle cotractio map is a poit. However, for two or more cotractio maps the ivariat set is the set X which satisfies: x 2 = f ( x ) f ( x )... f ( x ) (2) this is a fractal. For exa mple, i the case =2, the followig maps produce the Cator set i the [0,] iterval. x 2 + x ( x ) =, f ( x ) = (3) 3 3 f 2
5 I the complex plae we have the Koch curve if the mappigs are: 3 ( z ) = α z, f ( z ) = ( α )z + α, α = j (4) 2 6 f 2 + where z deotes the complex cojugate of z. Thus all regular (o-radom) fractals ca be expressed i this formalism, which because its simplicity is expected to become more importat i future. 4. RANDOM CLUSTERS Cosider a 2 or 3 dimesioal lattice ad distribute poits radomly o it with p probability. If eighbourig sites are occupied by poits, they are regarded as coected. By chagig the probability p of the occupatio of sites we ca estimate the critical probability p c ad fractal dimesio of clusters. The fractal dimesio of clusters is calculated i the followig way. We defie the mea radius of clusters of size s as: s 2 r =< i / 2 R s ( ) > (5) s i= where r i deotes the distace betwee the cetre of mass ad the i-th poit, ad < > idicates the average over all s-clusters. Whe R is proportioal to a power of s, the clusters are statistical fractals with dimesio D which satisfies the relatio: s / D R s ~ S (6) The result of simulatios show that (6) holds at p=p c ad the fractal dimesios are estimated as.9 (2 dimesioal lattice) ad 2.5 (3 dimesioal lattice) [3]. This value i the 2 dimesioal cases agrees with the experimetal value. 5. CLUSTERS IN SPIN SYSTEMS The best-kow model of magetic material is the Isig model [4]. I this model, spis which ca take oly the value + or - are arraged o a lattice. The total eergy (or Hamiltoia) E of the system is give by the equatio: + SiS j H Si, Si = (7) i E = J
6 Here, ΣΣ deotes summatio over earest eighbour sites. J is the couplig costat ad H is the exteral field. I thermal equilibrium, the probability of occurrece of the state with total eergy E is give by: W ~ e E / kbt where k B is Boltzma's costat ad T deotes temperature. I both 2 ad 3 dimesioal space, the Isig model is kow to show a phase trasitio at a critical temperature, T c. For T<T c, symmetry is spotaeously broke ad most spis take the same value, which idicates that the system is ferromagetic. O the other had whe T>T c, each spi takes the value + or - early idepedet of eighbourig spis ad the average of spi vaishes, which shows that the system is demagetised. At the critical poit T=T c, the characteristic size of clusters of the same spi diverges ad distributio of the clusters becomes fractal. The fractal dimesios of the clusters are estimated to be.88 i 2 dimesioal space ad 2.43 i 3 dimesios. (8) 6. CONCLUSION I the complexity theory is otable ivolvig of the iterative fuctios i behaviour of fractal patter. Study of the oliear equatio, treated ito iterative techiques, makes the subject of this paper. It cosists i a short revue of the most importat priciples of the fractal calculus ad complexity applicatios i fudametal scieces ad techologies. Were bee preseted also some ew ideas of aalysis to iterative relatios like as amed the modified logistic equatio. O this relatio ca be performig some studies with valuable results i umeric aalysis area. I bifurcatio diagram of logistic map (figure ) are preseted results of the ow program ru, that is writte uder C++ programmig laguage. 7. REFERENCES. D. Ioa, ş.a., Metode umerice î igieria electrică, Editura MATRIX ROM, Bucureşti, G. Mahalu, A. Graur, Iterative Fractal Algorithms, Advaces i Electrical ad Computer Egieerig, Faculty of Electrical Egieerig, Ştefa cel Mare Uiversity of Suceava, vol.4 (), o.2 (22), ISSN , pp.64-69, G. Mahalu, Fractal Techiques i Patter Recogitio ad Software Compressio, The Embedded Sigal Processig Solutios Evet, GSPx2004, October , Sata Clara Covetio Ceter, Sata Clara CA-USA. 4. G. Mahalu, C. Ciufudea, Iterative Fractal Forms Techiques, Third Europea Coferece o Itelliget Systems ad Techologies ECIT 2004, Techical Uiversity Gh. Asachi Iaşi, July , Iaşi, Romaia. 5. H. Takayasu, Fractals i the physical scieces, Machester Uiv. Press, Machester ad New York, 990.
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