Stability and Fractal Patterns of Complex Logistic Map

Transcription

1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 4, No 3 Sofia 04 Pit ISSN: 3-970; Olie ISSN: DOI: 0.478/cait Stabilit ad Factal Pattes of Comple Logistic Map Bhagwati Pasad, Kuldip Katia Depatmet of Mathematics, Japee Istitute of Ifomatio Techolog, A-0, Secto-6, Noida, UP-0307 INDIA s: b_pasad0@ahoo.com, kuldipkatia.jiit@gmail.com Abstact: The itet of this pape is to stud the factal pattes of oe dimesioal comple logistic map b fidig the optimum values of the cotol paamete usig Ishikawa iteative scheme. The logistic map is show to have bouded ad stable behaviou fo lage values of the cotol paamete. This is well depicted via time seies aalsis ad iteestig factal pattes as well ae peseted. Kewods: Comple logistic map, Ishikawa iteatio, factals, Julia set.. Itoductio The ame logistic gowth model is essetiall due to Vehulst [0] which he used fo the studies o populatio damics. He itoduced logistic equatio fo demogaphic modellig b etedig the Malthus equatio of the cotiuous gowth of a populatio with a view to obtai a stable statioa fiite state (see [4]). It took moe tha huded eas to ecogize his foudig cotibutios towads the populatio damics ad o-liea scieces. This wok eceived wide attetio due to the geat implicatios of the simple lookig equatio i Chaos theo. I 963, Edwad Loez itoduced a equivalet vesio of the logistic model fo his famous weathe foecast model [3]. R. M a [5, 6], i 976, ecogized the impotace of the logistic model ad obseved that the cotiuous time model ma ot be suitable to eflect the ealities i most of the cases ad costucted a discete 4

2 vesio of this model. Theeafte, Feigebaum [8] ad the wok of othes appoved this model as the paadigm fo the peiod doublig oute to chaos [3]. The impotace of this model is due to its peculia behaviou fo chagig values of the paamete. It ehibits the fied poits, bifucatios ad chaos fo the successive values of the gowth paamete. This povides the basis of the mode chaos theo ad epesets the simplest cases of chaotic sstem [3]. The eteme sesitivit to the iitial coditios has made it to be ideal fo vaious applicatios. Ma vaiats ad geealizatios of this model have bee used to stud vaious phsical poblems. D e t t m e [6] poited out that the most obvious easo fo kowig about chaos is to ogaize ad possibl avoid it because the egulait ad stabilit disappeas oce the sstem becomes chaotic. Geeall feedback lieaisatio, vaiable stuctue cotolle, fuzz method ad eual etwoks etc. ae amog the vaious techiques used fo cotollig the chaos i the liteatue. Due to the advacemet of mode computatioal tools ad polifeatio of digital computes, ew vistas have bee opeed fo the stud ad aalsis of these hpe sesitive maps ad the liteatue is flouished with the papes sigifig thei impotace i chaos, factals, cptogaph, optimizatio, discete damics, populatio damics etc. (see fo istace, [5, 6,, 7, 8, 9,,, 6, 8] ad seveal efeeces theeof). K i t et al. [3] eploed the gaphical potetial of this map ad geeated factal figues compaable to the well kow Madelbot factals. The amed these attactig factal figues as Vehulst factals. Recetl R a i ad A g a w a l [7] studied the compaative behaviou of the comple logistic maps with Picad obit, Nolad obit ad Ma obit ad foud iteestig esults. Ou aim is to stud the stabilit of the logistic map fo Ishikawa iteates ad visualize the factal pattes of such map fo vaig values of the paametes. We use the Matlab tools fo all ou computatioal ad gaphical equiemet.. Pelimiaies Let (Y, d) be a metic space ad f be a tasfomatio o Y. The f ma be called a damical sstem i the sese of B a s l e [] ad it is deoted b {Y, f}. The obit of a poit i Y is defied as a sequece of iteates of f i the fom of {f (): 0,,, }. Diffeet iteative schemes have bee used i the liteatue to obtai the obits of such a damical sstem. The fuctio iteatio also kow as Picad iteatio is populal used i the liteatue ad its obit (Picad Obit PO) is epeseted as () PO(f, 0 ) : { : f( - ),,, }. This iteatio equies oe umbe as iput to etu a ew umbe as output ad populal called as oe step feedback machie. A two step feedback scheme equiig two umbes as iput to etu a ew umbe as output is used b R a i ad A g a w a l [6] fo the stud of the chaotic behavious of the logistic map. The -th,,, 3,, iteate of this is give as f( - ) ( ) -, whee 0 < ad the sequece { } is covegig awa fom 0. The obit geeated 5

3 usig this scheme is called Ma Obit (MO) o supeio obit (see [4]) ad it ma be epeseted i the followig mae: () MO(f, 0, ):{ : f( - ) ( ) -,,, 3, }. Now we defie a thee step feedback scheme essetiall due to I s h i k a w a [0]. Defiitio.. Let Y be a o-empt set ad f: Y Y. Fo a poit 0 i Y, costuct a sequece { } i the followig mae: (3) - f( ) ( ), f( ) ( ), fo,, 3,, whee 0 < ad 0 ad { } is coveget awa fom 0. The sequece { } costucted above will be called the Ishikawa iteatio of a poit 0 ad it is deoted b IO(f, 0,, ). We shall stud the Ishikawa Obit (IO) fo ad. It is emaked that (3) becomes () whe we put 0 i it ad () with is the Picad iteatio (). This scheme is widel studied b P a s a d ad K a t i a [3, 5] ad iteestig factal pattes ae geeated i [4] usig it. Defiitio. []. Let Y be a o-empt set ad f: Y Y. A poit p Y is called a peiodic poit of f of peiod, N (the set of atual umbes), iff f (p) p ad f k (p) p fo all k,,...,, whee f k (p) is the k-th iteate of poit p ude f. A peiodic poit of f of peiod is simpl a fied poit of f. Defiitio.3 [9]. Let f be a fuctio ad p be a peiodic poit of f with pime peiod k. The is fowad asmptotic to p if the sequece, f k (), f k (), f 3k (),... coveges to p. I othe wods, lim f k ( ) p. The stable set of p, deoted b 6 W s (p), cosists of all poits which ae fowad asmptotic to p. If the sequece, f k (), f k (), f 3k (),... gows without boud, the is fowad asmptotic to. The stable set of, deoted b W s ( ), cosists of all poits which ae fowad asmptotic to. The followig defiitio is motivated b Rai ad Agawal [6]. Defiitio.4 [6]. Let S R (the set of eal umbes), f: S S ad p is a peiodic poit of f with pime peiod k. Fo a poit 0 S ad p [0, ], costuct a sequece { :,,... } such that 0 ( ) 0 f( 0 ), k ( ) 0 f( 0 ), k ( ) k f( k ),... k ( ) k f( k ), ( )k ( ) (-)k f( (-)k ),...

4 k ( ) ( )k f( ( )k ). The 0 is called Ishikawa fowad asmptotic to p ad sequece { k } coveges to p. Defiitio.5 []. Let C be the comple plae, C ) ) ) C { } ad f : C C deote a polomial of degee geate tha. Let F f deote the set of poits i C ) whose obits do ot covege to the poit at ifiit. That is, ) F { z C { f ( z) } is bouded}. f : 0 This set is called the filled Julia set associated with the polomial f. The bouda of F f is called the Julia set of the polomial f ad is deoted b J f. 3. Discussios ad esults Vehulst postulated that the gowth ate at a time should be popotioal to the factio of the eviomet that is ot et used up b the populatio at that time ad thus fomulated the followig model: (4) p p ap ( p ), hee p measues the elative populatio cout at time ad a, the gowth ate at time measues the icease of the populatio i oe time step elative to the size of the populatio at that time []. Vehlust s model was futhe epessed b R. Ma i the followig mae: (5) ( ), whee (a eal umbe betwee 0 ad ) epesets populatio desit at time,, 3, ad (a o-egative eal umbe) is used fo the combied ate fo epoductio ad stavatio [9]. The quadatic tasfomatios of the tpe z z c, whee z ad c both ae fom comple plae C, ae widel studied b [, 7,, ] ad ma othes i the liteatue. The iteest is to kow the behaviou of the stuctue of the obit of the iteates of z whe z ad c va. Fo 0,,, the iteatio scheme of such map is (6) z z c. P e i t g e et al. [] have established the equivalece of the maps give above. Followig them, oe ca easil see that equatios (4) ad (6) ae idetical fo c ( a ) / 4, z ( a) / ap, wheeas (4) ad (5) ae idetical fo a p /( a), a. O simplifig, we fid that (5) ad (6) become idetical fo c ( ) / 4, z /. This fuctioal equivalece is useful to geeate the factal pattes fo the quadatic map give b (6) afte obtaiig c i tems of. 7

5 8 Fist we stud the behaviou of the map give b (5) whe ad ae comple umbes. Let i i,. Now we compute the values of ad at diffeet iteatio levels usig the iteative scheme (3) i the followig mae ( ) ( ) { } ( ) ( ) ( ) { } ( ),, whee ( ) ( ) { } ( ) ( ) ( ) { } ( ),, ad fid the optimum value of fo vaious choices of paametes,. We coside i i as the iitial choice fo ou epeimetal stud of the comple logistic map (5). We stud it i two cases fo the values of the paamete. Case I. Whe is puel eal, we compute the obits of the map fo fied ad ad go o vaig util the iteate of the map emais bouded. These theshold values of ae computed ude 000 iteatios ad show i the Table. Table. The optimum values of (whe 0) ude 000 iteatios I this case it is obseved that fo a fied ad vaig (fom towads zeo), the optimum value of the cotol paamete iceases supisigl to a maimum of Futhe, o fiig ad vaig, the optimum value of shows a icemet to some istat afte which it stats deceasig (see Table ). The coespodig factal pattes fo some adom values of (show bold) ae daw, although the same could be daw fo all the tabulated values of. The time seies aalsis showig the behavious of the map is also show i Fig. fo some specific choices of the paamete (show udelied).

6 (a), 0,.798 (b), 0.9,.9097 (c) 0.7, 0., 3.99 (d) 0.3, 0.9, (e) 0., 0., (f) 0., 0.9, 6.09 Fig.. Time seies at diffeet values of, (whe 0) Case II. I this case, we obtai the optimal values of a puel imagia fo the same choices of the paametes ad. We obseve that fo a fied ad vaig (fom towads zeo), the optimum value of the magitude of the cotol paamete iceases to a maimum of fo the same choice of ad. Futhe, o fiig ad vaig, the optimum value of shows a icemet to some istat afte which it stats deceasig (Table ). Table. The optimum values of (whe 0) ude 000 iteatios

7 The coespodig factal pattes fo some adom values of (show bold) ae daw, although the same could be daw fo all the tabulated values of. The time seies aalsis showig the behavious of the map is also show as Fig. fo some specific choices of the magitude of the paamete (show udelied). (a), 0,.038i (b) 0.7, 0.3,.8535i (i) fo whole age of (ii) zoomed fo (c) 0.3, 0.3, i (i) fo whole age of (ii) zoomed fo (d) 0., 0., 0.774i Fig.. Time seies at diffeet values of, (whe 0) 0

8 We also stud the behaviou of the map fo geeal comple b takig some selected values of ad. The optimum value of is plotted fo 7 diectios with a agula icemet of 5 degees (Fig. 3). (a), 0 (b), 0.9 (c) 0.5, 0. (d) 0.7, 0. (e) 0., 0 (f) 0., 0.9 Fig. 3. Plots of the optimum values of (with icemet 5 degees) ude 000 iteatios Now we stud the factal aalsis of the map defied i (5) ad geeate the factal pattes fo (6) b obtaiig the value of the paamete c fom the tabulated value of (Tables -) usig c ( )/ 4. Devae [7] defied the escape citeia fo Picad iteate of the comple quadatic map ad obseved that the obit escapes whe z c >. So, whe z c > fo some, the z as. Attactive factal pattes ae obtaied b them o the basis of this escape algoithm. We eted it fo the Ishikawa iteates of the comple quadatic maps ad foud that the obit escapes whe z > ma{ c,/, / } whee 0 <, 0. Theefoe, if we costuct a sequece {z } usig Ishikawa iteatio with z > ma c,/,/ fo some, the z as. We follow the colou { }

9 schemes of P i e t g e ad S a u p e [] alog with the above defied escape citeio fo ou stud of the comple logistic map. The colouig scheme of the gaphics peseted i the figues depeds upo the ate of escape to ifiit. A poit z 0 is coloued black if the obit of z 0 ot escaped withi the fist 00 iteates, ed is used to deote poits which escape to ifiit fastest. Shades of oage, ellow ad gee ae used to colou poits which escaped less quickl ad shades of blue ad violet epeset the poits which escaped, but ol afte a sigificat umbe of iteatios. This colouig scheme is well depicted i the gaphical pattes give i Figs 4 ad 5. (a), 0,.798 (b) 0.9, 0., (c) 0.7, 0.7, (d) 0.7, 0.5, (e) 0.3, 0.7, (f) 0.3, 0.3, 7.08 (g) 0., 0.9, 6.09 (h) 0., 0., Fig. 4. Julia sets fo eal

10 (a), 0,.038i (b), 0.9,.050i (c) 0.9, 0.,.358i (d) 0.7, 0.5,.7636i (e) 0.3, 0.7,.887i (f) 0., 0.9, i 4. Coclusio (g) 0., 0., 0.755i (h) 0., 0, i Fig. 5. Julia sets fo puel comple We obseve that fo a fied ad vaig (fom towads zeo), the optimum value of the magitude of the cotol paamete (i puel eal case) iceases supisigl to a maimum of wheeas it iceases to a maimum of i case of puel imagia fo the same choice of ad. Futhe, o fiig ad vaig the optimum value of shows a icemet to some istat afte which it stats deceasig (see, Tables ad ) fo both the cases. The time seies aalsis of the comple logistic map cofims the bouded behaviou of the logistic map eve fo the highe values of fo specific choices of the paametes ad. 3

11 Ackowledgemets: The authos would like to thak the leaed efeees fo thei valuable commets ad suggestios fo the impovemet of this mauscipt. Refeeces. B a s l e, M. F. Factals Evewhee. Secod Ed. Revised with the Assistace of ad a Foewod b Hawle Risig, III. Bosto MA, Academic Pess Pofessioal, B a s l e, M. F. Supefactals. Cambidge, Cambidge Uivesit Pess, A. Bude, S. Havli, Eds. Factals i Sciece. Spige-Velag, C a m a c h o, E. F., C. B o d o s. Model Pedictive Cotol. Beli, Spige, C o w o v e, R. M. Itoductio to Factals ad Chaos. Joes & Balett Publishes, D e t t m e, R. Chaos ad Egieeig. IEE Review, Septembe 993, D e v a e, R. L. A Fist Couse i Chaotic Damical Sstems: Theo ad Epeimet. Addiso-Wesle, F e i g e b a u m, M. Quatitative Uivesalit fo a Class of No-Liea Tasfomatios. J. Statistical Phsics, Vol. 9, 978, H o l m g e, R. A. A Fist Couse i Discete Damical Sstems. Spige-Velag, I s h i k a w a, S. Fied Poits b a New Iteatio Method. Poc. Ame. Math. Soc., Vol. 44, 974, No, J u l i e, C. S. Chaos ad Time-Seies Aalsis. Ofod Uivesit Pess, K e l l e, K. Ivaiat Factos, Julia Equivaleces, ad the (Abstact) Madelbot Set. Beli Heidelbeg New Yok, Spige-Velag, Kit, J., D. Costales, A. Vadebauwhe d e. Piee-Facois Vehulst s Fial Tiumph. I: M. Ausloos, M. Diick Eds. The Logistic Map ad the Route to Chaos: Fom the Begiigs to Mode Applicatios. Spige-Velag, M a, W. R. Mea Value Methods i Iteatio. Poc. Ame. Math. Soc., Vol. 4, 953, No 3, M a, R. M. Simple Mathematical Models with Ve Complicated Damics. Natue, Vol , No 459, M a, R. M., G. F. O s t e. Bifucatios ad Damic Compleit i Simple Biological Models. The Ameica Natualist, Vol. 0, 976, No 974, Mooe, A., J. G. Keatig, D. M. Heffe a. A Detailed Stud of the Geeatio of Opticall Detectable Watemaks Usig the Logistic Map. Chaos, Solitos ad Factals, Vol. 30, 006, No 5, M o a, P. A. P. Some Remaks o Aimal Populatio Damics. Biometics, Vol. 6, 950, No 3, P a e e k, N. K., V. P a t i d a, K. K. S u d. Image Ecptio Usig Chaotic Logistic Map. Image ad Visio Computig, Vol. 4, 006, No 9, Pastij, H. Chaotic Gowth with the Logistic Model of P.-F. Vehulst. I: M. Ausloos, M. Diick, Eds. The Logistic Map ad the Route To Chaos: Fom the Begiigs to Mode Applicatios. Spige-Velag, P e i t g e, H., H. J u g e s, D. S a u p e. Chaos ad Factals: New Foties of Sciece. Spige-Velag, H. Peitge, D. Saupe, Eds. The Sciece of Factal Images. Spige-Velag, P a s a d, B., K. K a t i a. A Compaative Stud of Logistic Map Though Fuctio Iteatio. I: Poc. It. Co. Emegig Teds i Egieeig ad Techolog. ISBN: , Kuuksheta, Idia, 00, P a s a d, B., K. K a t i a. Factals via Ishikawa Iteatio. CCIS, Spige, Beli, Heidelbeg, Vol. 40, 0, No, P a s a d, B., K. K a t i a. A Stabilit Aalsis of Logistic Model. Iteatioal Joual of Noliea Sciece, Vol. 7, 04, No, R a i, M., R. A g a w a l. A New Epeimetal Appoach to Stud the Stabilit of Logistic Map. Chaos, Solitos ad Factals, Vol. 4, 009, No 4, R a i, M., R. A g a w a l. Geeatio of Factals fom Comple Logistic Map. Chaos, Solitos ad Factals, Vol. 4, 009, No, S a l a i e h, H., M. S h a h o k h i. Idiect Adaptive Cotol of Discete Chaotic Sstems. Chaos, Solitos ad Factals, Vol. 34, 007, No 4,