Convergence Analysis of Transcendental Fractals

Transcription

1 International Journal Computer Appliations ( ) Convergene Analysis Transendental Fratals Shafali Agarwal Researh Sholar, Singhania University, Rajasthan, India Ashish Negi Phd, Dept. Computer Siene, G.B. Pant Engg. College, Pauri Garwal, Uttarakhand, India ABSTRACT Mandelbrot set with transendental funtion presents an out the ordinary field researh beause its magnifiene and geometrial omplexities. Earlier many researhers used feedbak system to study it and reveal new onepts unexplored the geometry transendental Mandelbrot set with new iterative approah. The objetive the paper is to analyze the work done by several researhers to visualize beautiful fratal graphis Mandelbar set with sine, osine funtions using Mann iteration method. Also the rate onvergene funtion in the presene transendental funtion and onjugay with Mandelbrot set and related omplex struture superior Julia set. General Terms Computer Graphis, Fratal Keywords Transendental superior Julia set, AntiJulia set, Fixed Point, Mann Iteration, Esape Time Fratal. 1. INTRODUCTION Frenh mathematiian Gaston Julia in 1918 investigated the iteration proess omplex funtion and attained a Julia set and gave a diretion to the fratal world [4]. Later Benoit Mandelbrot in 1979 studied a very omplex & perturbed struture whih is known as Mandelbrot set. A lot researhes arried out to unveil the geometrial and funtional ambiguity both the sets [14]. We are analyzing fratal images that have been generated from iterating polynomials type z d + with transendental funtion defined in the omplex plan. Here is a omplex parameter, d is an integer, bar sign denotes omplex onjugation & transendental funtion an be sine, osine or tangent. It has been observed that after applying muh iteration to these polynomials, omplex number z either move away from the origin and esape towards infinity or bounded to origin. Points with bounded orbit under iteration make up the filled in Julia set, whih we all K for the holomorphi and d K *d for the antiholomorphi ase. The topologial boundaries these bounded sets are Julia sets [3]. Points bounded to origin onverge to a fixed point. Fixed points an be lassified in three ategories: Asymptotially stable: Means all nearby solutions onverge to it. Stable: Means all nearby solutions stay nearby. Unstable: Means mostly all nearby solutions diverge away from the fixed point. Authors disussed about the singular values that plays an important role in determining the dynamis omplex system. Aording to them any attrating fixed point must have a singular value in its immediate basin attration [15]. A Transendental entire funtion possesses ritial and asymptoti values. Here a ritial value funtion is suh as there exists a z 0 belongs to and f(z 0 )=1 & f (z 0 )=0. In this ase, z 0 is alled a ritial point f and an asymptoti value f is suh as there exists a urve any symbol tending to infinity and f(z) -> a as z -> along any symbol [5]. The fratal image superior Mandelbrot set onsists small mini Mandelbrot set like images found in the sattered surrounding it is known as the midgets [1]. The study midgets in the Mandelbrot set is given by Philip [13] and Romera [16]. Fratal with onjugate z referred to as Mandelbar set, name is given by Crowe et al. beause its formal analogy with Mandelbrot set and also brought its features bifurations along axes rather than at points. The ritial point Mandelbar set is zero as in Mandelbrot set [20]. The study onnetedness lous for antiholomorphi polynomials defined as Triorn, oined by Milnor, plays intermediate role between quadrati and ubi polynomials [6]. Reently Shizuo [11] has quoted the Multiorn as the generalized Triorn or the Triorn higher order and presented beautiful images and their various properties. This set is partiularly interesting beause its topologial properties are different from the ones believed to be true for the Mandelbrot set. Shafali et al. have applied the onept onjugay to transendental Mandelbrot set and generated fratals [see 17, 18, 19]. They have iterated the funtion to ompute a preise value for the fixed point and hene the effiieny funtion depends on the rate onvergene the polynomial. Our paper fouses on the relative analysis all those funtions and to asertain that whih funtion onverges faster to fixed point than other. We will also see the beautiful Julia set images those funtions. 2. PRELIMINARIES & DEFINITIONS The formation fratal images is ahieved by using feedbak system. There are basially two types feedbak mahines i.e. one-step mahine and two-step mahine. Both types mahines an be haraterized by iterative proedures. Iteration methods are one way to ahieve the self-similarity exhibited by fratals [2]. One-step feedbak mahines are haraterized by Peano Piard iterations represented by the formula x n+1 =f(x n ) where f(x n ) an be any funtion. It is very useful mathematial tool and has been developed in partiular for the numerial solution omplex problems. In two-step feedbak mahine, the output is omputed by the formula x n+1 =f(x n, x n-1 ), whih requires two numbers as input and returns a new number as output[12]. Other related definitions whih will be required for subsequent analysis: (Superior iterates) Let X be a non-empty set real numbers and f: X->X. For x 0 belongs to X, onstrut a sequene {x n } in the following manner [8, 9]: x 1 =β 1 f(x 0 )+(1- β 1 )x 0 x 2 = β 2 f(x 1 )+(1- β 2 )x 1... x n = β n f(x n - 1 )+(1- β n )x n - 1 where 0 < β n <=1 and {β n } is onvergent away from 0. 24

2 International Journal Computer Appliations ( ) The sequene {x n } onstruted this way is alled a superior sequene iterates, denoted by SO(f,x 0, β n ) At β n = 1, SO(f,x 0, β n ) redues to O(f,x 0 ). This proedure was essentially given by Mann, first to study it for β n in 1955 [21]. Sine the results obtained in fratal modelling via Mann iterates are the super set their orresponding fratal models in the Piard orbit. (Superior Orbit) The sequene x n onstruted above is alled Mann sequene iteration or superior sequene iterates. We denote it by SO(x 0, s, t). Esape-time fratals (or superior orbit fratals) Many Julia and Mandelbrot sets have been omputed for polynomials in superior orbits. In the literature these are alled superior Julia sets and superior Mandelbrot sets respetively and are defined by Rani and Kumar [8, 9] as follows: (Superior Julia sets) The set omplex points SK whose orbits are bounded under superior iteration a funtion Q is alled the filled superior Julia set. A superior Julia set SJ Q is the boundary the filled superior Julia set [9]. (Superior Mandelbrot sets) A Superior Mandelbrot set SM for a funtion the form Q (z) = z n +, n = 1, 2,..., is defined as the olletion C for whih the superior orbit the point 0 is bounded, SM={ C: { Q (0): k=0, 1,.. } is bounded in SO} [8]. k To ompute SJ and SM sets, Rani and Kumar [10] gave new and onsiderably improved esape riterions and alled them superior esape riterions. For a omplex map the form Q = z n +, the esape riterion is max {, (2/s) 1/n-1 } and for k the ubi polynomial Q a,b (z) = z 3 + az + b, the esape riterion is max{ b, ( a + 2/s) 1/2 }. Similarly for the quadrati polynomial Q a,b (z) = z 2 +, the esape riteria is max {, 2/s}. Superior Triorn and Multiorn: A superior multiorn, for the omplex-valued polynomial A= z d +, for d 2, is the olletion all ε C, for whih superior orbit SO(A, z 0, s ), with z 0 = 0 is bounded, i.e. A = { ε C: { A (0), d = 0, 1, 2, } is bounded with respet to SO}. k A superior multiorn with d=2 is known as superior triorn [7]. Suppose x 0 is a fixed point for F. Then x 0 is an attrating fixed point if F (x 0 ) <1. The point x 0 is a repelling fixed point if F (x 0 ) >1. Finally if F (x 0 ) =1, the fixed point is neutral [14]. Note that the initial value z 0 should be infinity, sine infinity is the ritial point z for transendental polynomial. However instead starting with z 0 =, it is simpler to start with z 1 =, whih yields the same result. A ritial point z f(z)+ is a point where f (z)=0. The point z in Mandelbrot set for transendental funtion has an orbit that satisfies imag(z) > 50, then the orbit z esapes[14]. 3. FIXED POINT CALCULATION USING MATLAB AND RESULT DISCUSSION The proess fratal generation uses iterative method on Mandelbrot set i.e. f(z)=z n ->, where z and both are omplex quantities and n>1. We are onsidering fratal generation proess for transendental Mandelbar set using esape time method. The fratal images are onstruted as a twodimensional array pixels. Eah pixel is represented by a pair (x, y) o-ordinates. If the fratal image is onstruted with various values z keeping the value onstant is known as z plane images on the other hand if fratal image is onstruted for various values keeping the value z onstant is known as plane images. Here we are analyzing the work done by various researhers in the fratal world. In the Dynamis Mandelbrot Set with Transendental Funtion Shafali, Gunjan & Ashish studied the fratal images transendental polynomial z-> sin(z n )+ in z plane and plane using Mann iteration. The main attration this paper was the existene unique images like earthen lamp (diya), Lord Vishnu ion et by the iteration quadrati, ubi and bi-quadrati funtion. Here we have seen that the visual harateristis images are hanged as the value onstant s varies with polynomial. The following tables show the number iterations needed to onverge the image to fixed point TABLE 1: Orbit -> sin(z 2 )+ at s=0.3 & (C= i) Here we skipped 10 iteration and the value onverges to a fixed point after 17 iterations Fig 1: Orbit at s=0.3 for (C= i) 25

3 International Journal Computer Appliations ( ) TABLE 2: Orbit -> sin(z 2 )+ at s=0.5 & (C= i) The value onverges to a fixed point after 6 iterations TABLE 3: Orbit -> sin(z 2 )+ at s=1 & (C= i) The value onverges to a fixed point after 6 iterations Fig 2: Orbit at s=0.5 for (C= i) Fig 3: Orbit at s=1 for (C= i) In Fixed Point Results Transendental Superior Antifratals Shafali and Ashish analyzed transendental polynomial with onjugay using two step feedbak proesses. The obtained images are having the features multiorn with transendental funtion. They have used Mann iteration to produe fratal images funtion z->sin ( z d +) and transformation the funtion for quadrati, ubi and biquadrati polynomial. All the fratals possess two ommon properties for apiee polynomial. One is all the images generated using funtion sin( z d +) is having number arms (d+1) for power d and an added is for even power d, images are symmetrial about real axis whereas for odd power d, images are symmetrial about both real and imaginary axis. The tabular representation fixed point alulation is given below TABLE 4: Orbit -> sin ( z 2 +) at s=0.1 & (C= i) Here we skipped 23 iterations and the value onverges to a fixed point after 30 iterations Fig 4: Orbit at s=0.1 for (C= i) 26

4 International Journal Computer Appliations ( ) TABLE 5: Orbit -> sin ( z 2 +) at s=0.5 & (C= i) The value onverges to a fixed point after 8 iterations Fig 5: Orbit at s=0.5 for (C= i) TABLE 6: Orbit -> sin ( z 2 +) at s=1 & (C= i) The value onverges to a fixed point after 2 iterations Fig 6: Orbit at s=1 for (C= i) In Midgets Transendental Superior Mandelbar Set Shafali and Ashish explored the presene midgets in Mandelbar set for osine funtion using Mann iteration i.e. z->os( z d +). The geometrial shape resembles to both multiorn as well as transendental polynomial funtion. Another interesting feature is the existene midgets in terms triorn, multiorn and Mandelbrot sets with different number bulbs on external rays key image. It is observed that all the midgets are either basi Mandelbrot set or triorn irrespetive the degree polynomial for onstant 0<s<1. However the midgets are dependant on the degree polynomial at s= TABLE 7: Orbit -> os( z 2 +) at s=0.1 & (C= i) Here we skipped 10 iterations and the value onverges to a fixed point after 18 iterations Fig 7: Orbit at s=0.1 for (C= i) 27

5 International Journal Computer Appliations ( ) TABLE 8: Orbit -> os( z 2 +) at s=0.5 & (C= i) The value onverges to a fixed point after 8 iterations TABLE 9: Orbit -> os( z 2 +) at s=1 & (C= i) The value onverges to a fixed point after 6 iterations 4. GENERATION OF SUPERIOR JULIA SETS Transendental funtion with superior Julia or AntiJulia set follows the law having 2d wings, where d is the power z. Most the images for all polynomials possesses symmetry about both x and y axis. The fratal for z->os ( z d +) formed with large entral body having rotational and refletion Fig 8: Orbit at s=0.5 for (C= i) Fig 9: Orbit at s=1 for (C= i) symmetry along with axes symmetry. Here enormously stunning images transendental superior Julia and AntiJulia set have been generated for different power polynomial. The omplement the superior antijulia set is displayed in blue olour. It appears that antijulia set onsists unountably many urves or hairs extending to infinity in the plane. Fig 10: = sin(z 2 )+, s=0.5 and = , i Fig 11: = sin ( z 2 +), s=0.5 and = , i Fig 12: = os ( z 2 +), s=0.5 and = i 28

6 International Journal Computer Appliations ( ) 5. CONCLUSION In this paper, we have reviewed the researh work done by Ashish, Shafali and Gunjan in the field fratal. The study have shown the striking properties and esape riteria for transendental funtion and generated the orresponding fratals using Mann iteration in whih most the images are having symmetry along x axis and y axis. The analysis represents the beauty midgets in the form Mandelbrot set and multiorn appeared in superior antimandelbrot set for the funtion os ( z d +). In their study we observed fixed points 6. REFERENCES [1] Ashish Negi & Mamta Rani, Midgets Superior Mandelbrot set, Chaos, Solitons and Fratals 36 (2008) [2] C. Pikover, Computers, Pattern, Chaos, and Beauty, St. Martin s Press, NewYork, [3] Eike, Lau and Dierk Shleiher, Symmetries fratals revisited, The Mathematial Intelligener, Vol 18, No. 1, 45-51, DOI: /BF [4] G. Julia, Sur 1 iteration des funtions rationnelles, J Math Pure Appli. 8 (1918), [5] Kin-Keung Poon, Fatuo-Julia Theory on Transendental Semigroups, Bull Austrad. Math So. Vol. 58 (1998) [ ]. [6] J. Milnor, Dynamis in one omplex variable; Introdutory letures, Vieweg (1999). [7] M Rani,, Superior Antifratals,volume 1, IEEE, /10/2010. [8] M Rani and V Kumar. Superior Mandelbrot sets, J. Korea So. Math. Edu. Ser. D; Res. Math. Edu. (8)(4)(2004), (44) [9] M Rani, V. Kumar. Superior Julia set. J Korea So Math Edu Ser D Res Math Edu 2004;8(4): (43) [10] M. Rani. Iterative proedures in fratals and haos, PhD Thesis, Department Computer Siene, Gurukula Kangri Vishwavidyalaya, Hardwar, India, [11] N. Shizuo and Dierk Shleiher, On multiorns and uniorns: I. Antiholomorphi dynamis, Hyperboli omponents and real ubi polynomials, Internat. J. Bifuration. Chaos Applied Siene Engineering, (13)(10)(2003), funtion for different values onstant s. An observable fat is that the number iterations required to ahieve onvergene is dependant to the value onstant s. As the value s raise to 1, the image onvergene is faster to the fixed point and the fratals onverged to asymptotially stable fixed points. Fratal iterates provides envision for fixed points to a given funtion, allowing us to determine properties with out omputing it. In future researher an apply Mann iteration to study the speed onvergene new Fratal Models using different funtion polynomial. [12] Peitgen H, Jürgens H, Saupe D. Chaos and fratals: new frontiers siene. New York: Springer-Verlag; [13] Philip AGD. Wrapped midgets in the Mandelbrot set. Computer Graphis 1994;18(2): [14] R. L. Devaney, A First Course in Chaoti Dynamial Systems: Theory and Experiment, Addison-Wesley, MR Zbl [15] R. L. Devaney, Complex Exponential Dynamis, Boston University, August 6, {5} [16] Romera M, Pastor G, Montoya F. On the usp and the tip a midget in the Mandelbrot set antenna. Phys Lett A 1996; 221(3 4): MR (97d:58073). [17] Shafali Agarwal, Gunjan Srivastava, Ashish Negi, Dynamis Mandelbrot Set with Transendental Funtion. International Journal Advaned Computer Siene & Appliations, Vol. 3 No ( ). [18] Shafali Agarwal, Ashish Negi, Fixed Point Results Transendental Superior Antifratals. Proeedings International Conferene Advanes in omputing, ommuniations and informatis. (In Press) [19] Shafali Agarwal, Ashish Negi, Midgets Transendental Superior Mandelbar Set. International Journal Computer Siene Issues, Impat Fator (In Press) [20] W. D. Crowe, R. Hasson, P. J. Rippon, and P. E. D. Strain-Clark, On the struture the Mandelbar set. Nonlinearity (2)(4)(1989), MR Zbl [21] WR. Mann, Mean value methods in iteration. Pro Am Math So 1953;4: