A Study for the (μ,s) n Relation for Tent Map


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1 Applied Mathematical Scieces, Vol. 8, 04, o. 60, HIKARI Ltd, A Study for the (μ,s) Relatio for Tet Map Saba Noori Majeed Departmet of Mathematics College of Educatio for pure sciece/ IbAlHaitham Uiversity of Baghdad Baghdad, Iraq Copyright 04 Saba Noori Majeed. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract This work study the (μ,s) relatio for the tet map μ x 0 x < T(x) μ = μ( x) x where T μ (x):[0,] [0,] ad μ [0,], the (μ,s) relatio obtaied by dt μ (x) elimiatig x from the equatio S =, ad obtaied a equatio directly dx relatig μ ad S for period poits of T μ, also we foud the stability regio of T μ accordig to the parameter μ usig S, ad this had bee drive for periods,, 3 ad i geeral for ay positive iteger, fially we show that the (μ,s) relatio are varia for T μ uder the trasformatio μ μ for Z +. Keywords: Tet map, slop, periodic pots,stablety Itroductio Discrete dyamical systems refer to the study of dyamical systems where the space X is typically a compact cotiuum ad the time is Z or N, the word discrete i the phrase discrete dyamical systems emphasizes the fact the time is ot represeted by cotiuum like R or R + as it happes i the case of differetial
2 300 Saba Noori Majeed equatios, The law of evolutio of a poit x i the space X is give by f (x), a selfmap o X which is either a homomorphism or just cotiuous. A poit x is a periodic poit if f m (x) = x for some iteger m. If f l (x) is a periodic poit for some l (x it self eed ot to be periodic poit the x is a evetually periodic poit), []. I mathematics, the tetmap with parameter μ is the real valued fuctio f m defied by [] μ x 0 x < T(x) μ = () μ( x) x or T μ (x) = μ( x ) the ame beig due to the fact, that the graph of T μ has the shape of tet, see fig., for the value of the parameter μ withe 0 ad, T μ maps the uit iterval [0,] ito itself, thus defiig a discretetime dyamical system o it, [3]. I particular, iteratig a poit x 0 i [0,] gives rise to a sequece: x + = T μ (x ) where μ is a positive real costat. The tetmap ad the Logisticmap which defied by f β (x) = βx( x) () are topologically cojugate, [] thus the behavious of the two maps are i this sece idetical uder iteratio see fig., Li ad Shei i [4] proved that the (μ,s) relatio of the Logisticmap with form f β (x) = βx( x) are ivariat uder the trasformatio of β β, i this work we show that the tetmap varaied ad trasform μ μ, so the (μ,s) relatio determie the relatio betwee μ ad S as varaied, i additio we foud that the rage of μ cotrollig the stability of the periodic poits for T μ ad this result came out from testig the relatio betwee μ ad S. What is the (μ,s) relatio We write f (x) = f (f (x)) the secod iterate of x for f ad f (x) = f (f (x)) the th iterate of x for f. We say a x is period poits of f if f (x) x = 0, ad if i additio, x f i (x), where i =,,,, [5]. The slop fuctio of f (x) is defied by d f (x) S = (3) dx For ay parametric fuctio (i.e. depeds o a parameter like μ) equatio (3) will relate S with μ uder a equatio, we call this equatio the (μ,s) relatio. Also the slop S i goig to give us the stability of f depedig o the defeetios of stability i.e. S < the the periodic poit of f (x) is stable, [5].
3 A study for the (μ,s) relatio for tet map 30 Takig the requiremet to the (μ,s) relatio we ca determie the rage of μ for the existece of stable period poits. 3 The (μ,s) relatio of T,T,T μ μ μ ad T μ For μ x 0 x T(x) μ = μ( x) < x dt μ(x) S= = m μ, we see that S = μ ad S = μ this meas μ trasformed to dx μ or μ μ o the other had T μ (x) has stable fixed poit if S < which yields to μ < or < μ < we have that <μ< thus whe <μ< the Tetmap is stable. Now for 4 μ x 0 x < μ 4 μ( μ x) μ x < 4 T μ (x) = μ( μ + μ x) x < 4 μ 4μ ( x) x< 4μ (4) See []. S = 4μ that is 4μ 4μ ad the stable regio lies i < 4μ < we get μ < so we have i μ<, agai for T 3 (x) we derive the geeral 4 4 formula of T 3 (x) i order to relate S ad μ as followig, [5]:
4 30 Saba Noori Majeed 3 8 μ x 0 x < μ 8 μ( 4 μ x) μ x < 8 4 μ( μ + 4μ x) x < 4 μ 4 μ ( μ x) x < μ 8 3 μ T μ (x) = 4 μ ( μ+ μ x) x < + μ 3 μ(4μ 4μx ) + x < μ 4 3 μ( 4μ + 4μ x) x < 4 8 μ 3 ( x) x (5) We have S = 3 this meas 3 trasform ito 3 ad the stability regio lies i < 3 3 < leads to μ < the μ<. 8 8 I geeral i this work we derived the iteratio of the Tetmap Z + ad we got the followig geeral formula: μ T (x) = μ( μ + μ x) x < T μ for μ x 0 x < μ μ( μ x) x < μ M M (6) μ μ ( x) x μ
5 A study for the (μ,s) relatio for tet map 303 We have from (3) S = μ that is μ μ ad the stable regio is < μ < so we have that μ < the μ< we may therefore summarize this situatio as follows: Propositio (The (μ,s) Relatio of T μ ) For ay positive iteger we have: () μ trasform ito μ. () For μ< the T μ is stable. (3) If is odd the the regio of stability is real. m i (4) If is eve the the regio of S stability icludes complex values. (5) From (4) the trasformatio which we refer to i () happeed oly whe is odd. i.e. μ μ whe is odd umber. Graphical Results After a simple compariso, we fid that there is a clear similarity betwee the bifurcatio diagram of both of T μ (x) ad T μ (x) set out i the two figures 3 ad 4. Note fig.3 represet the T μ (x) bifurcatio while fig.4 represet T μ (x). Refereces. Jack, K. ad Ko c ak, H., (99), Dyamics ad Bifurcatios, Spriger Verlage, U.S.A., p.0.. Gulick, D., (007), "Ecouters with Chaos ad Fractals", Mc. GrawHill, Ic., U.S.A., p Luch, S., (004), Dyamical System with Applicatio Usig Matlab, Berkhauser, U.K., p Li, C.L. ad Shei, M.L., (007), Logistic map f (x) = βx( x) is topologically cojugate to map f (x) = ( β)x( x), Tamkag Joural of Sciece ad Egieerig, Vol.0, No., p Devay, R.L., (99), A First Course i Chaotic Dyamical Systems, Theory ad Experimet, Perseus Books Publishig, L.L.C., U.S.A., p. 0,39,46.
6 304 Saba Noori Majeed Figure () Tet Map T m (x) Figure () Cojugacy betwee Tet Map ad Logistic Map
7 A study for the (μ,s) relatio for tet map 305 Figure (3) T m (x) Bifurcatio Diagram Received: April 4, 04 Figure (4) T  m (x) Bifurcatio Diagram
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