Non Linear Dynamics of Ishikawa Iteration

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1 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober No Liear Dyamis of Ishiawa Iteratio Rajeshri Raa Asst. Professor Applied Siee ad Humaities Departmet G. B. Pat Egg. College Pauri Garhwal Yashwat S Chauha Asst. Professor Computer Siee & Egg. Departmet G. B. Pat Egg. College Pauri Garhwal Ashish Negi Asst. Professor Computer Siee & Egg. Departmet G. B. Pat Egg. College Pauri Garhwal ABSTRACT We itrodue i this paper the dyamis for Ishiawa iteratio proedure. The geometry of Relative Superior Madelbrot sets are explored for Ishiawa iterates. Keywords Complex dyamis Relative Superior Madelbrot Set Ishiawa Iteratio... INTRODUCTION Complex graphis of oliear dyamial systems have bee a subjet of itese researh owadays. These graphis are geerally obtaied by olorig the esape speed of the seed poits withi the ertai regios of the omplex plae that give rise to the ubouded orbits. The omplexity of the mathematial objets suh as Julia sets ad Madelbrot sets i spite of their deeitful simpliity of equatios that geerate them is truly overwhelmig. Perhaps the Madelbrot set is the most popular objet i the fratal theory. It is believed to be the most beautiful objet ot oly i the real but also i the omplex plae. This objet was give by Beoit B. Madelbrot i 979 ad has bee the subjet of itese researh right from its advet. Madelbrot set ad its various extesios ad variats have bee extesively studied usig Piard s iteratios. Reetly M. Rai ad V. Kumar[] itrodued the superior Madelbrot sets usig Ma iteratio proedure. We itrodue i this paper a ew lass of Madelbrot sets amed as Relative Superior Madelbrot sets usig Ishiawa iteratios. Our study shows that Relative Superior Madelbrot sets are exlusively elite ad effetively differet from other Madelbrot sets existig i the preset literature.. PRELIMINARIES Let{ z : 4...} deoted by { z } be a sequee of omplex umbers. The we say Lim z if for give M > there exists N > suh that for all > N we must have z M. Thus all the values of z lies outside a irle of radius M for suffiietly large values of. Let Q az az az... a z az ; a. The oeffiiets are be a polyomial of degree where allowed to be omplex umbers. I other words it follows that Q z z. Defiitio.: Let X be a oempty set ad : ay poit x iterates of a poit x that is; f X X. For X the Piard s orbit is defied as the set of O( f x ) { x ; x f ( x )...}. I futioal dyamis we have existee of two differet types of poits. Poits that leave the iterval after a fiite umber are i stable set of ifiity. Poits that ever leave the iterval after ay umber of iteratios have bouded orbits. So a orbit is bouded if there exists a positive real umber suh that the modulus of every poit i the orbit is less tha this umber. The olletio of poits that are bouded i.e. there exists M suh that Q M for all is alled as a prisoer set while the olletio of poits that are i the stable set of ifiity is alled the esape set. Hee the boudary of the prisoer set is simultaeously the boudary of esape set ad that is Madelbrot set for Q. Defiitio.: The Madelbrot set M for the quadrati Q z z is defied as the olletio of all C for whih the orbit of the poit is bouded that is M { C :{ Q ()};... isbouded } A equivalet formulatio is M { C :{ Q () does ot ted to as } We hoose the iitial poit as is the oly ritial poit ofq. 4

2 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober. ISHIKAWA ITERATION FOR RELATIVE SUPERIOR MANDELBROT SETS Let X be a subset of real or omplex umbers ad f : X X. For x X we ostrut the sequees{ } x ad { } y s f ( x ) ( s ) x y s f ( x ) ( s ) x... y s f ( x ) ( s) x where s ad y i X i the followig maer: s ad x s f ( y ) ( s ) x x s f ( y ) ( s ) x... x s f ( y ) ( s ) x where umber[]. s ad Defiitio.: The sequees is overget to o zero umber s is overget to o zero x ad y ostruted above is alled Ishiawa sequees of iteratios or relative superior sequees of iterates. We deote it RSO( x s s t ). by Notie that RSO( x s s t ) with s = is RSO x i.e. Ma s orbit ad if we plae RSO( x s s t ) redues too ( x t ). ( s t ) s s the We remar that Ishiawa orbit s is Relative Superior RSO( x s s t) with / orbit. Now we defie Madelbrot sets for futio with respet to Ishiawa iterates. We all them as Relative Superior Madelbrot sets Defiitio.: Relative Superior Madelbrot set RSM for the futio of the form Q z where = 4 is defied as the olletio of C for whih the orbit of is bouded i.e. RSM { C : Q () :...} is bouded. We ow defie esape riterios for these sets.. Relative Superior Esape Criterios for Quadratis: The followig theorem gives us a esape Criterios for futio proedure. Q z i respet to Ishiawa iteratio Theorem.: Let s assume that z / s ; z / s where s s ad is a omplex umber. Defie z ( s) z sq ( z ) z ( s) z sq ( z ) Where Q z a be a quadrati ubi or biquadrati polyomial i terms of s ad = 4. the z as. Proof: Let s tae Q ( s ) z s Q ( z ) s z ( s ) z s s z ( s ) z s where Q z z z ( s z ( s ) ) s z ( z z ( s z s ) s z z ( s z )... () z ( s) z sq ( z ) Now sie So ) z ( s) z sq ( z ) o substitutig () ( s) z s z ( s z ) z sz s z. s z s z ( z sz ) ( s z. s z s z ) z sz s z. s z s z z s z. s z z ( ss z ) sie s z so ss z there exists suh that ss z z z Cosequetly z z Thus the Ishiawa orbit of z uder the quadrati futio teds to ifiity. This ompletes the proof. Corollary.: Suppose that / s ; / s.the the Relative Superior orbit of Ishiawa RSO( Q s s ) esapes to ifiity. I the proof of the theorem we used the fats that z ad z / sas well as z / s. Hee the followig orollary is the refiemet of the esape riterio disussed i the above theorem. Corollary.(Esape Criterio): Suppose that z z ad z max{ / s / s } the z as. 44

3 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober Corollary.: Suppose that z max{ / s / s } z z ad for some.the z as. This orollary gives us a algorithm for omputig the Relative Superior Madelbrot sets of Q for ay. Give ay poit z we have omputed the Relative superior orbit of z. If for some z lies outside the irle of radius max{ / s / s } we guaratee that the orbit esapes. Hee z is ot i the Relative Superior Madelbrot sets. O the other had if z ever exeeds this boud the by defiitio of the Relative Superior Madelbrot sets deoted by RSM. We a mae extesive use of this algorithm i the ext setio.. Relative Superior Esape Criterio for Cubi Polyomials: We prove the followig theorem for the futio Qab () z z az b with respet to the Ishiawa iteratio. Theorem.: Suppose ( / ) / z b a s / z b ( a / s ) exists where s ; s ad a ad b are i omplex plae.defie z ( s) z sq ( z ) ab z ( s) z sq ( z ) = a b where Qab is the futio of s the z. Q ( s ) z s Q ( z ) Proof: Let s tae a b ( s ) z s ( z az b) s z s az z s z bs s z s az z s z bs z ( s z s a s ) s z z b z ( s z as s ) s z z { s z as s s } z { s z a } s z { z a / s } s z z a s { / } s z { z ( a / s )} z ( s) z sq ( z ) Now sie ab a b as ( s) z s. z.{ s ( z a ) } z sz s z. s ( z a ) s z z s z { s z. s ( z a ) s z } z s z s. s z ( z a ) s z z { s. s z a } z s. s (/ s. s z a ) z s. s (/ s. s z a ). { Sie ad z s s z ( a / s. s )} / z ( a / s) ad so ( / ) / z a s exists / z ( a / ss ) follows.therefore / z ( a / ss ) / ss suh that ss { z ( a / ss )}. Hee there exists suh that z z. Repeatig this argumet time we get z z. Therefore the Relative superior orbit of z uder the ubi polyomial Q ( z ) teds to ifiity. This ompletes the proof. ab Corollary.4: Suppose that / b ( a / s) ad / b ( a / s ) exists. The the Relative superior RSO( Q s s ) esapes to ifiity. orbit ab Corollary.5(Esape Criterio): Suppose / / z max{ b ( a / s) ( a / s ) } the z as. Corollary.5 gives a esape riterio for ubi polyomials. Corollary.6: Assume that / / max{ ( / ) ( / ) } z b a s a s for some.the z z ad z as. From Corollary.6 we fid a algorithm for omputig the Relative Superior Madelbrot sets of Qab ( z ) for ay a ad b.. A Geeral Esape Criterio: We will obtai a geeral esape riterio for polyomials of the form G z. Theorem.: For geeral futio G z = 4 where s s ad is the omplex plae. Defie z ( s) z sg ( z ) z ( s) z sg ( z ) 45

4 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober Thus the geeral esape riterio is max{ ( / s) ( / s ) }. Proof: We shall prove this theorem by idutio: For = we get G z. So the esape riterio is whih is obvious i.e. z max{ } G z z. So the esape riterio is For = we get z max{ / s / s } (Theorem.) For = we get G z z. So the result follows from Theorem. with a = ad b = suh that the esape riterio is / / max{ ( / ) ( / ) } z s s. Hee the theorem is true for = 4 Now suppose that theorem is true for ay. Let as / G z ad z ( / s ) as well / z s exists. The ( / ) G ( s ) z s G ( z ) where G z z s z s ( z ) s z s z z s z ( s z s s z ( z z {( s z s } s z z ( s z s s ) z ( s z ) z ( s) z sg ( z ) Now Sie.So ( s) z s z ( s z ) z sz s z. s z s z ( z s z ) ( ss z s z ) ( ss z z ) ss z z z ( ss z ) ( / ) ; ( / ) ) / / / z s z s ad z ( / ss ) / ( / ) z ss therefore ( ss z ) Hee for some we have ( ss z ). z z Thus z z Therefore the Ishiawa orbit of z uder the iteratio of z teds to ifiity. Hee / / max{ ( / ) ( / ) } z s s is the esape riterio. This proves the theorem. Corollary.7: Suppose that / ( / s) ad / ( / s ) exists. The the Relative Superior orbit RSO( G s s ) esapes to ifiity. Corollary.8: that / / max{ ( / ) ( / ) } z s s for some. The z z Assume ad z as. This orollary gives a algorithm for omputig the Relative Superior Madelbrot sets for the futios of the form G z = 4 4. GENERATION OF RELATIVE SUPERIOR MANDELBROT SETS: We geerate Relative Superior Madelbrot sets. We preset here some Relative Superior Madelbrot sets for quadrati ubi ad biquadrati futio. 4. Relative Superior Madelbrot Sets for Quadrati futio: Figure : Relative Superior Madelbrot Set for s=s'= Figure : Relative Superior Madelbrot Set for s= s'=. Figure : Relative Superior Madelbrot Set for s=. s'= 46

5 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober Figure 4: Relative Superior Madelbrot Set for s=.s'=.4 Figure : Relative Superior Madelbrot Set for s=. s'= Figure 5: Relative Superior Madelbrot Set for s=.4 s'=. Figure 4: Relative Superior Madelbrot Set s=. s'=.4 4. Relative Superior Madelbrot Sets for Cubi futio: Figure : Relative Superior Madelbrot Set for s=s'= Figure 5: Relative Superior Madelbrot Set for s=.4 s'=. Figure : Relative Superior Madelbrot Set for s= s'=.5 4. Relative Superior Madelbrot Sets for Bi-quadrati futio: Figure : Relative Superior Madelbrot Set for s=s'= 47

6 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober Figure : Relative Superior Madelbrot Set for s= s'=. 4.4 Geeralizatio of Relative Superior Madelbrot Set Figure : Relative Superior Madelbrot Set for s=s'==9 Figure : Relative Superior Madelbrot Set for s=.5 s'= Figure : Relative Superior Madelbrot Set for s=. s'=.4 =9 Figure 4: Relative Superior Madelbrot Set for s=. s'=.4 Figure : Relative Superior Madelbrot Set for s=.4 s'=. =9 Figure 5: Relative Superior Madelbrot Set for s=. s'=.4 Figure 4: Relative Superior Madelbrot Set for s=. s'=.4 =5 48

7 Iteratioal Joural of Computer Appliatios ( ) Volume 7 No. Otober Figure 5: Relative Superior Madelbrot Set for s=.4 s'=. =5 Figure 6: Relative Superior Madelbrot Set for s=. s'=.5 =5 5. REFERENCES [] R.Abraham ad C.Shaw Dyamis: The Geometry of Behaviour Part Oe: Periodi Behavior Part Two: Chaoti Behavior Aerial Press Sata Cruz Calif (98). [] Ala F.Beardo Iteratio of Ratioal futios Spriger Verlag N.Yor I.(99). [] V.Beride Iterative approximatio of fixed poits Editura Efermeide Baia Mare (). [4] SBeddigs ad K.Briggs Itreatio of quateriia maps It J. Bifur Chaos. Appl. Si. ad Egg.5 (995) [5] B.Braer ad J.Hubbard The Iteratio of Cubi Polyomials. Part IAta. Math. 66(998) 4-6. [6] G.V.R.Babu ad K.N.V.Prasad Ma iteratio overges faster tha Ishiawa iteratio for the lass of Zamfiresu operators Fixed Poit Theory ad its Appliatios Vol. 6 Art. ID (6). [7] P.W.Carlso Pseudo D rederig methods for fratals i the omplex plae Computer ad Graphis (5) (996) [8] R.M.Crowover Itrodutio to Fratals ad Chaos Madelbrot sets Joes ad Barlett Publishers (995). [9] R.L.Daveey A Itrodutio to Chaoti Dyamial Systems Spriger Verlag N.Yor.I.994. [] K.Heiz Beer ad Mihael Dorfler Dyamial Systems ad Fratals Cambridge Uiv. Press 989. [] R.A.Hohgre A First Course i Disrete Dyamial Systems Spriger Verlag 994. [] S.Ishiawa Fixed poits by a ew iteratio method Pro. Amer. Math. So.44 (974) [] M.A.Krasosel Two remars o method of suessive approximatios Uspehi. Math. Nau. (995) -7. [4] W.R.Ma Mea Value methods i iteratio Pro. Amer. Math. So.4 (95) [5] B.B.Madelbrot The Fratal aspets of iteratio of z z for omplex ad z A. N. Y. Aad. Si. 57(98) [6] J.R.Muers Topology: A First ourse Pratie Hall of Idia Publ. Ltd. N.Delhi 988. [7] M.O.Osilie Stability results for Ishiawa fixed poit iteratio proedure Idia Joural of Pure ad Appl. Math. 6(995) [8] M.O.Osilie Iterative ostrutio of fixed poits of multivalued operators of the aretive type Sohow J.Math. (996) [9] H.Peitge H.Jurges ad D.Saupe Fratals for lassroom Part Two Complex Systems AND Madelbrot Sets Spriger-Verlag N.Yor I. 99. [] H.Peitge H.Jurges ad D.Saupe Chaos ad Fratals Spriger-Verlag N.Yor I. 99. [] M.Rai ad V.Kumar Superior Madelbrot Set J Korea So. Math. Edu. Series DResearh i Maths. Edu. o.48(4) [] B.E.Rhoades Some fixed poit iteratios proedures It.J.Math.Si. 4(99) -6. [] B.E.Rhoades Fixed poits iteratios for ertai oliear mappigs J.Math. Aal. Appl. 8(994) 8-. [4] B.E.Rhoades ad S.M.Solutz O the equivalee of Ma ad Ishiawa iteratio methods It J. Math. Si. o.7 () [5] D.Roho A Geeralized Madelbrot Set for Bi- Complex Numbers World Sietifi Publishig Compay Fratals 8(4) ()