110 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Several Chaotic Approaches of One Dimensional Doubling Map Md. Zakir Hosena*, Md. Mohiuddin Zillub, Payer Ahmedc aLecturer, Department of Mathematics, Jagannath University, Dhaka-1100 , Bangladesh bLecturer, Department of Mathematics, European University of Bangladesh, Dhaka- 1216, Bangladesh cProfessor, Department of Mathematics, Jagannath University, Dhaka-1100 , Bangladesh aEmail: zakirhosen2008@gmail.com bEmail: zillujnu123@gmail.com cEmail: dr.payerahmed@yahoo.com Abstract In this paper, we study basic dynamical behavior of one-dimensional Doubling map. Especially emphasis is given on the chaotic behaviors of the said map. Several approaches of chaotic behaviors by some pioneers it is found that the Doubling map is chaotic in different senses. We mainly focused on Orbit Analysis, Sensitivity to Initial Conditions, Sensitivity to Numerical Inaccuracies, Trajectories and Staircase Diagram of the Doubling map. The graphical representations show that this map is chaotic in different senses. The behavior of the said map is found irregular, that is, chaotic. Keywords: Approaches; Orbit; Sensitivity; Staircase Diagram; Trajectories; Transitivity. 1. Introduction Dynamical Systems is a branch of mathematics that attempts to understand processes in motion. There are many branches of Dynamical Systems but Chaos is one of them and it explains how very small changes in the initial configuration of a system model may lead great discrepancies over time. Called ‘Butterfly Effect’, this phenomena accounts our accurate prediction for a long period of time. A dictionary definition of Chaos is a ‘disordered state of collection; a confused mixture.’ With the Advancement of chaos research, more and more chaos phenomena have been discovered in the mathematics, engineering and other fields. Many researchers have focused on the demonstration of chaos phenomena using simulation. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 111 The graphical analysis of the Doubling map significantly has contributed to our understanding of the essence of chaos. The use of the term “Chaos” was first introduced into dynamical systems by Li and Yorke [7] for a map on a compact interval. Another explicit definition of chaos belongs to Devaney [11]. Then Robinson gave a refined definition [1]. The remarkable feature of the Doubling map is in the simplicity of its form and the complexity of its dynamics. It is the simplest model that shows chaos. The Doubling map ( ) 2 if 0 < 0.5 2 1 if 0.5 <1. x x D x x x ≤ =  − ≤ is very interesting and representative model of dynamical systems. The dynamical behavior of the Doubling map is very complicated. We have shown that Doubling map is chaotic in the sense of Devaney, Li-Yorke, Lyapunov and Wiggin. In this paper, we discussed chaotic dynamical behavior of one dimensional Doubling map. The Doubling map is topologically conjugate to the Logistic map Shift map and also semi-conjugacy to Tent map. We also have shown that the dynamical behavior of Doubling map is chaotic considering the Initial Seeds, Orbit Analysis, Sensitivity to Initial Conditions, Sensitivity to Numerical Inaccuracies, Trajectories and Staircase Diagram. These are the main results of this article. 2. Methodology There are some of senses to analyze the chaos like Henri Poincare′ , R. L. Devaney, Li and Yorke, Lyapunov Exponent, Wiggins etc. In our research we tried to present one dimensional Doubling map in the sense of R. L. Devaney, Li and Yorke, Lyapunov Exponent, Wiggins etc. We use mathematical software’s like Mathematica, MATLAB to analyse the numerical results to be found in the future so that we can describe the graphical representation of our mathematical research. 3. Mathematical Preliminaries We need some basic definitions to be used elsewhere of this article. We mention these definitions as follows. 3.1 Basic Definitions 3.1.1 Orbit Given 0x R∈ ( 0x is called the seed or initial value of the orbit), we define the orbit of 0x under f to be the sequence{ }2 0 1 0 2 0 0, ( ), ( ), , ( ),n nx x f x x f x x f x= = =L L . For example, Let ( )f x x= and 0 16,x = then the orbit of 0x under f to be the sequence {16,4,2,1.414, }.L American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 112 We see that the point on this orbit tend to 1. 3.1.2 Periodic Orbit or Cycle The point 0x is called periodic if 0 0( )nf x x= for some 0>n , where n is called the prime period of the orbit. 3.1.3 Periodic Orbit with Prime Period n If the point 0x is periodic with prime period ,n then the periodic orbit with prime period n to be the sequence { 0x , ,),( 0 Lxf ),( 0 1 xf n− 0x , ,),( 0 Lxf L),( 0 1 xf n− }. For example, let 3)( xxf −= and 10 =x , then 1)1( −=f , 1))1(()1(2 == fff . Thus 1 is periodic point with prime period 2. Similarly, -1 is periodic point with prime period 2 and these orbits are: {1, 1, 1, 1, }− − L and },1,1,1,1{ L−− . 3.1.4 Transitivity A dynamical system is transitive if for any pair of points x and y and any 0>ε there is a third point z within ε of x whose orbit comes within ε of y. In other words, a transitive dynamical system has the property that, given any two points, we can find an orbit that comes arbitrarily close to both. 3.1.5 Sensitivity to Initial Conditions A dynamical system f depends sensitively on initial conditions if there is a 0β > such that for any x and any 0>ε there is a y within ε of x and a k such that the distance between )(xf k and )(yf k is at least .β In this definition it is important to understand the order of the quantifiers. The definition says that, no matter which x we begin with and no matter how small a region we choose about x. we can always find a y in this region whose orbit eventually separates form that of x by at least .β The distance β is independent of x. as a consequence, for each x, there are points arbitrarily nearby whose orbits are eventually “far” that of x. The idea of sensitive dependence on initial conditions is very important topics in the study of dynamical system. If a particular system possesses sensitive dependence, then for all practical purpose, the dynamics of this system defy numerical computation. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 113 Mathematically, A continuous map :f X X→ has sensitive dependence on initial conditions if there exists 0δ > such that, for any x X∈ and any neighborhood ( ) of N x x , there exist ( )y N x∈ , 0n ≥ such that ( ) ( )( ), ,n nd f x f y δ> ( )where ,X d is a compact metric space. 3.1.6 Topologically Transitivity Consider the metric space 𝑋𝑋 and the continuous map :f X X→ .We say that f is topological transitive if for every pair of non-empty open sets U and V in X there exists a positive integer k such that ( )kf U V f∩ ≠ . 3.1.7 Topological Conjugacy Let :f A A→ and :g B B→ be two continuous mappings. Then and f g are said to be topologically conjugate if there exists a homeomorphism :h A B→ such that h f g h=o o . The homeomorphism h is called a topological conjugacy between and f g . 3.1.9 Compact Space: Let 𝑋𝑋 and 𝑌𝑌 be metric spaces. We say that the metric space 𝑋𝑋 is compact if every open cover of 𝑋𝑋 has a finite sub cover, i.e. if {𝐼𝐼𝑖𝑖}𝑖𝑖∈𝐼𝐼 is a collection of open sets of 𝑋𝑋 such that 𝑋𝑋 ⊂ i I Ii ∈ U then we have that 1 n i i X I = ⊂U .Also compact spaces on the real line can be thought as closed and bounded intervals. 3.1.10 Lyapunov Exponent Let :f →¡ ¡ be a continuous and differentiable map. Then x∀ ∈ ¡ we define the (local) Lyapunov exponent of x say ( )xλ as ( ) ( ) 1 ' 0 1lim log n i in i x f x X n λ − →∞ = = ∀ ∈∑ ¡ . 3.1.11 Scrambled Set Consider an interval I and the continuous map :f I I→ .Then an uncountable subset 𝑆𝑆 of I containing no periodic points of f is said to be scrambled if: 1. Any Periodic point 𝑝𝑝 of f and any point x I∈ satisfies ( ) ( )lim sup 0n n n f x f p →∞ − > . 2. ( ) ( ) ( ) ( )x,y , lim sup 0 and lim inf 0.n n n n n n X f x f y f x f y →∞ →∞ ∀ ∈ − > − = 3.2 Several Definitions of Chaos American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 114 In the following we give some important mathematical definitions of chaos by some pioneers. 3.2.1 Devaney’s Definition [9] (R. L. Devaney 1989) Let X be a metric space. A continuous function :f X X→ is said to be chaotic on X if f has the following three properties: (C-1) Periodic points are dense in the space X (C-2) f is topologically transitive (C-3) f has sensitive dependence on initial conditions Mathematically, (C-1) ( )kP f = { : ( ) ( )}kx X f x x k∈ = ∃ ∈N is dense in .X (C-2) For , :U V non empty open sets∀ − of X, k∃ ∈N such that .)( f≠∩VUf k (C-3) 0>∃δ (Sensitive constant) which satisfies: ),(),,( εε xNyxNandXx ∈∃∀∈∀ and 0≤∃ k such that [ ]( ( ), ( )) 4 .k kd f x f y δ> In other words, a continuous map :f X X→ on a compact metric space X is called chaotic in the sense of Devaney – or just D-chaotic if there exists a compact invariant subset Y (called a D-chaotic set ) of X with the following properties: ( ) | i f Y is transitive, ( ) ( )| ii P f Y Y= , ( ) | iii f Y has sensitive dependence on initial conditions. 3.2.2 Li and Yorke Definition [7] A continuous map :f X X→ on a compact metric space ( ), X d is called chaotic in the sense of Li and Yorke – or just L/Y-chaotic – if there exists an uncountable subset S (called a scrambled set ) of X with the following properties: ( ) ( ) ( )( ) lim , 0 for all , , ,n n i sup d f x f n y x y S x y →∞ > ∈ ≠ ( ) ( ) ( )( ) lim inf , 0 for all , , ,n n n ii d f x f y x y S x y →∞ ∈= ≠ ( ) ( ) ( )( ) > 0 lim sup , for all , , periodic. n n n iii d f x f y x S p X p →∞ ∈ ∈ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 115 In other words, let 𝐼𝐼 be an interval and let :f I I→ be a continuous map with a periodic point of period three. Then f is said to be chaotic in the sense of Li-Yorke or L-Y chaotic if f has an uncountable scrambled set. 3.2.3 Lyapunov Definition [1] Consider the continuous and differentiable map :f →¡ ¡ .Then f is said to be chaotic according to Lyapunov or L-chaotic if: 1. f is topologically transitive. 2. f has a positive Lyapunov exponent. 3.2.4 Wiggin’s Definition [13] Let :f X X→ be a continuous map and X be a metric space .Then the map f is said to be chaotic according to Wiggins or W-chaotic if: 1. f is topologically transitive. 2. f exhibits sensitive dependence on initial conditions. 4. Theorems and Propositions 4.1 Theorem (Period Three implies Chaos). Suppose :F →¡ ¡ is continuous. If F has a periodic point of prime period 3, then F has periodic points of all other periods. Proof. First we need to established two lemma that follows from the continuity of F . Fixed Point Lemma: Suppose I and J are closed intervals such that I J⊂ .If ( )J F I⊂ then F has fixed point in I Pre-image Lemma: Suppose I and J are closed intervals such that ( )J F I⊂ . Then there exists a closed subinterval I I′ ⊂ such that ( )J F I ′= . Let F has a 3-cycle given by a b c a→ → → . We will assume a b c< < , the other cases are handled similarly. Let [ ]0 ,I a b= and [ ]1 ,I b c= . Then we have ( )1 0I F I⊂ and ( )0 1 1I I F I∪ ⊂ .First we note that since ( )1 1I F I⊂ , F has a fixed point in 1I , by the fixed point lemma. Now we will find a period 2-cycle. First we have that ( )1 0I F I⊂ , so by the pre-image lemma, there exists a subset 0 0A I⊂ such that ( )1 0I F A= . On the other hand ( )0 1I F I⊂ , so in fact ( )2 0 0I F A⊂ . Then by the fixed point lemma, there is a fixed point for 2F in 0I . So we have a point of period 2 for F. In fact, since the iteration of this American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 116 point leaves 0I it cannot be a fixed point for F , so it has prime period 2. We will find a periodic cycle of period n for all n>3 by involving the Pre-image Lemma n times. Since ( )1 1I F I⊂ , there is a closed subinterval 1 1A I⊂ such that ( )1 1I F A= . Now again we have ( )1 1A F A⊂ , so there is a closed interval 2 1A A⊂ such that ( )1 2A F A= . Thus ( )2 1 2I F A= . Continue this process for n-2 steps to produce the following nested collection of closed subintervals: 2 3 2 1 1...n nA A A A I− −⊂ ⊂ ⊂ ⊂ ⊂ , such that ( )1i iA F A= + and ( )2 1 2 .n nI F A− −= Now let’s bring in 0I . We have ( )2 1 0nA I F I− ⊂ ⊂ , So there is a closed subintervals 1 0nA I− ⊂ such that ( )2 1n nA F A− −= . Finally we also have ( )1 0 1nA I F I− ⊂ ⊂ , so there is a closed subinterval 1nA I⊂ such that ( )1n nA F A− = . What we have accomplished is the following: 1 1 1...F F F F n nA A A I−→ → → → Where 1nA I⊂ and ( )1 n nI F A= . But by the fixed point lemma, this means that there is a fixed point for nF in nA .Lets call this fixed point 0x . Then 0x is a periodic point of period n for F . In fact, since the first iterate of 0x lies in 1I , while the next 1n − iterates lie in 1I we know that 0x has prime period. 4.2 Proposition The doubling map ( ) 12 for 0 2 12 1 for 1 2 x x D x x x  ≤ <=   − ≤ <  is chaotic on [ )0,1 in the sense of Devaney. Proof. First, we check that D has a dense set of periodic points. We claim that every rational number px q = with q odd is a periodic point for D .This follows by observing that D is a bijection from the set 1 2 1, ,..., q q q q  −     to itself: clearly D maps this set into itself, and it is also surjective because 2k kD q q   =     and ( )1 / 22 1 k qk D q q  + −− =      But the rational numbers with odd denominator are dense in [ )0,1 so D has a dense set of periodic points. Second, we claim that D has a dense orbit, so (in particular) it is transitive. To do this, let { 1 2 3 0. 01 00011011000001...111... length length length α = 14 2 43 1 44 2 4 43 be the base-2 decimal constructed by listing all sequences of length 1, then all sequences of length 2, then all sequences of length 3, and so forth. Note that, in base ( )2, D x is obtained simply by deleting the first digit of the base-2 decimal expansion of α (i.e., it acts essentially as the shift map).So in particular, for any sequence of digits, there is a shift of α that begins with that sequence of digits. Now let 1 2 30. ...x d d d= and 0ε > . We will show there is some shift of α within ε of x. Choose n with 2 n ε− < . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 117 Then there is a positive integer k such that ( )kD α begins as 1 2 10. ... n nd d d d + , so that ( )kD α and x can only differ past the n + 2nd decimal place. Then ( ) 2 2 2 2 k n i i n D xα ε ∞ − = + − ≤ < <∑ as required. Finally, we show that D has sensitive dependence. We will show that the value 1 3 β = will satisfy the requirements of the definition. First, observe that if ,a b are both in 10, 2     or 1 ,1 2     , then ( ) ( ) 2D b D a b a− = − . Also, if 10, 2 a  ∈   and 1 ,1 2 b  ∈   then one of b a− and ( ) ( )D b D a− is at least 1 3 , since if 1 3 b a− < then ( ) ( ) ( )1 2D b D a b a− = − − is larger than 1 3 .Therefore, if x, y are any two distinct points, the value of ( ) ( )n nD y D x− will double at each stage until the points x and y land in opposite halves of [ )0,1 ,at which point either ( ) ( )n nD y D x− will exceed 1 3 or ( ) ( )1 1n nD y D x+ +− will. Thus, for any two distinct points x and y, their orbits will eventually be a distance of at least apart after iterating some number of times [3] . 4.3 Proposition The Doubling map ( ) 12 for 0 2 12 1 for 1 2 x x D x x x  ≤ <=   − ≤ <  is chaotic on [ )0,1 in the sense of Lyapunov exponent. Proof. The give Doubling map is ( ) 2 ; 0 0.5 2 1 ; 0.5 x 1 x x D x x ≤ < =  − ≤ < In this case, ( ) 2 1D x′ = > , so the Doubling map is not attracted to a sink and this is not asymptotically periodic. It is easy to compute the Lyapunov Exponent as ( ) ( )1 1 1 1 1lim ln lim ln 2 ln 2 0 n n n n i i h x D x n n→∞ →∞ = = ′= = = >∑ ∑ Since each orbit of D(x) is asymptotically periodic and Lyapunov Exponent ( )1 0h x > , then theDoublingmap is Chaotic on [0,1) [1]. 4.4 Proposition American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 118 The Doubling map ( ) 2 ; 0 0.5 2 1 ; 0.5 x 1 x x D x x ≤ < =  − ≤ < is chaotic with prime period three or Li and York Sense. We know the doubling map is ( ) 2 ; 0 0.5 2 1 ; 0.5 x 1 x x D x x ≤ < =  − ≤ < Let the initial seeds 0 1 1 1 1 1 2 11, , , , , , ,......... 3 5 6 9 7 13 24 x = 0 1When ; 3 x = ( ) 1 2 1 2 1 2 , , , , , ,......... 3 3 3 3 3 3 i       , 0 1When ; 5 x = ( ) 1 2 3 1 3 2 , , , , , ,......... 5 5 5 5 5 5 ii       0 1When ; 6 x = ( ) 1 1 2 1 2 1 , , , , , ,......... 6 3 3 3 3 3 iii       , 0 1When ; 9 x = ( ) 1 2 4 8 7 5 1 2 , , , , , , , ,......... 9 9 9 9 9 9 9 9 iv       0 1When ; 7 x = ( ) 1 2 4 1 2 4 1 2 , , , , , , , ,......... 7 7 7 7 7 7 7 7 v       , 0 1When ; 11 x = ( ) 1 2 4 8 5 10 9 7 3 6 1 , , , , , , , , , , ,......... 11 11 11 11 11 11 11 11 11 11 11 vi       We see that (i), (ii), (iii), (iv), (vi) have the period 2, 4, 2, 6, 9 respectively but (v) has prime period 3. So the Doubling Map is chaotic in the sense of Li-Yorke [7]. 4.5 Proposition Doubling Map is chaotic in the sense of Wiggin. Proof. Consider the Doubling Map ( ) [ ) [ ): 0,1 0,1D x → given by ( ) 2 ; 0 0.5 2 mod1 2 1 ; 0,5 1 x x D x x x x ≤ < = =  − ≤ ≤ First I will prove that ( )D x is transitive using symbolic dynamics .We let ∑ be the metric space of all infinite sequences containing 0 's and 1's equipped with the metric ( ) ( )0 1 2 1, ....... 2 i iis s s s s sρ τ τ= − ∀ = and ( )0 1 2......τ τ τ τ= ∈Σ and we define :σ Σ → Σ given by ( ) ( )0 1 2 1 2 3........ ........s s s s s sσ = . Then there exist a point ( )0100011011000001....x = created by blocks of 0 ' and 1's s which has a dense orbit. So σ is transitive and then ( )B x is transitive [13]. Now I will prove that ( )B x has a dense set of periodic points. We have that ( ) ( ) { } ( ) 1 1Fix 0 1 2 1.B Per B Per f= = ⇒ = = − The second iterated map 2B is given by ( ) ( )2 2 1 24 mod1 and 0, , 3 3 B x x Per B  = =     ( ) 2 2 3 2 1.Per B⇒ = = − Generalizing this result the n-th iterated American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 119 map is given by ( ) 2 mod1.n nB x x= So ( ) 1 2 2 20, , ,...... 2 1 2 1 2 1 n n n n nPer B  − =   − − −   ( )and 2 1n nPer B = − .Now ( ) [ ) ( )lim so 0,1 and 0,nn Per B x N xεε →∞ = ∞ ∀ ∈ ∀ > will contains a periodic point. Hence the periodic points of B are dense. Since all the conditions for the three chaotic definitions are satisfied, so the map ( )D x is W- chaotic [7].Note: This map can be stated as chaotic map in the sense of J. Banks etc. al. because of Transitivity + Density implies Sensitivity [6]. 4.6 Proposition Let D is defined by ( ) 2 0 0.5 2 1 0.5 1 x if x D x x if x ≤ < =  − ≤ < ( )D x is conjugate to Logistic Map. Let h be the homeomorphism defined by ( ) 2 2h x Sin xπ= and we put ( ) ( )( )1x h D h xϕ −= o o . Then we have, ( ) ( )( )( ) ( ) 1 1 1 1 , 0 1 . .0 1 2 2 1 1 1 2 , 0 . .0 2 2 2 2 x h D h x arcSin xh D Sin x i e x arcSin xh Sin x i e x ϕ π π π π − − − =   = ≤ < ≤ <         = × ≤ < ≤ <        11 1 1 , 0 . .0 2 2 2 arcSin xh Sin x i e x π π −   = ≤ < ≤ <        ( ){ } ( ) ( ){ } ( ) ( ){ } ( ) 21 1 2 21 2 21 1 2 . 2 2 1 2 1 4 1 Sin Sin x Sin Sin x Sin Sin x Cos Sin x x Sin Sin x x x π π   −= ×      −= − −= −= − = − And in the other case we have, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 120 ( ) ( )( )( ) ( ) ( ) 1 1 1 1 1 , 0 1 . .0 1 2 2 1 2 1 , 0.5 1 . .0 1 2 2 1 1 1 , 0.5 1 . .0 2 2 x h D h x arcSin xh D Sin x i e x arcSin xh Sin x i e x arcSin xh Sin x i e x ϕ π π π π π π − − − − =   = ≤ < ≤ <       = × − ≤ < ≤ <       = − ≤ < ≤ <        ( ){ } 2 1 2 1 1 2 1 2 2 Sin Sin x Sin Sin x π π π − −   = −      = − ( ){ } ( ) ( ){ } ( ) ( ){ } ( ) 2 1 2 1 1 2 2 1 2 2 . 2 1 4 1 Sin Sin x Sin Sin x Cos Sin x x Sin Sin x x x − − − − = = = − = − Hence we have shown that D is chaotic map [9]. 4.7 Proposition Let D is defined by ( ) 2 0 0.5 2 1 0.5 1 x if x D x x if x ≤ < =  − ≤ < ( )D x is Semi-Conjugacy with Tent map. We will show that the doubling map is semi-conjugate to the tent map T via T itself! That is, we will show that [ ] [ ] [ ] [ ] 0,1 0,1 0,1 0,1 D T T T → ↓ ↓ → Thus T will be chaotic. This is because orbits under iteration of D map to dynamically equivalent orbits under T. In fact we now prove by induction that ( )1 1n nT D T− =o For all 0n > . Suppose Equation-1 is true for : .n k= Then 1 1k k k kT D T T T D T T− −= ⇒ =o o o o And since T D T T=o o , we have 1k kT D T +=o American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 121 which completes the inductive proof. We remark that (1) gives an explicit formula for ( )nT x since we already know that ( )1 12 mod1n nD x x− −= . We now show that D is semi conjugate to T via T , or in other words, that T D T T=o o . There four cases to consider for T To . ( ) ( ) ( ) ( )1 10 0 2 2 2 44 2x T x T T x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = = =o ( ) ( ) ( ) ( )1 1 1 1 2 2 2 2 2 44 2 2x T x T T x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = = − = −o ( ) ( ) ( ) ( )31 1 1 2 2 2 2 2 2 4 22 4 2x T x T T x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = − = − − = −o ( ) ( ) ( ) ( )3 11 0 2 2 2 2 2 4 44 2x T x T T x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = − = − = −o Similarly, there are four cases for T Do : ( ) ( ) ( ) ( )1 10 0 2 2 2 44 2x D x T D x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = = =o ( ) ( ) ( ) ( )1 1 1 1 2 2 2 2 2 44 2 2x D x T D x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = = − = −o ( ) ( ) ( ) ( )31 1 0 2 1 2 2 1 4 22 4 2x D x T D x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = − = − = −o ( ) ( ) ( ) ( )3 11 1 2 1 2 2 2 1 4 44 2x D x T D x T x x x≤ ≤ ⇒ ≤ ≤ ⇒ = − = − − = −o We have to be a little bit careful at x=1/2 since D is not continuous there, and also at x=1 since we have not yet defined D(1). But the reader may check that ( ) ( )1 1 0,2 2T D T T= =o o and that ( ) ( )1 1 0T D T T= =o o provided we defined ( )1D to be either 0 or 1. It is also straight forward to check that both T To and T Do are continuous on [ ]0,1 . So what we have shown is that ( ) ( ) 14 if 0 4 1 12 4 if 4 2 314 2 if 2 4 34 4 if 14 x x x x T D x T T x x x x x  ≤ ≤   − ≤ ≤= =  − ≤ ≤   − ≤ ≤ o o And so D is conjugate to T via T for the graph of .T T T D=o o [9] 4.8 Proposition American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 122 Doubling Map 12 ; 0 2(x) 12 1 ; 1 2 x x D x x  ≤ <=   − ≤ <  is Conjugate to Shift Map. Proof. The doubling Map is defined by 12 ; 0 2(x) 12 1 ; 1 2 x x D x x  ≤ <=   − ≤ <  Let’s represent each [ )0,1x∈ by its binary expansion: 31 2 4 2 1 2 3 2 3 40. ... ... 2 2 2 2 bb b bx b b b= = + + + + where each { }0,1 .ib ∈ For 1 2nx = , represent x with a binary expansion ending in 0’s, rather than 1’s. Then if 1 0b = , we know [0, 1/2)x∈ Similarly, 1 1b = , we know [ )1 .1/ 2,x∈ Suppose [0, 1/2)x∈ is given. 32 4 2 2 3 4 2 3 4 00. 0 ... ... 2 2 2 2 bb bx b b b= = + + + + .Then ( ) 32 4 2 2 3 4 2 3 4 02 0. 0 ... ... 2 2 2 2 bbbD x b bx b = = = + + + + Consider 20. 01010110.....Then 1/ 4 1/16 1/ 64 1/128 ...x x= = + + + + and ( ) 2 2 / 4 2 /16 2 / 64 2 /128 1/ 2 1/ 8 1/ 32 1/ 64 ...D x x= = + + + = + + + + 1/64... 20. 1010110....= Now suppose [ )1/ 2,1x∈ is given. Then 32 4 2 2 3 4 2 3 4 10. 1 ...= ... 2 2 2 2 bb bx b b b= + + + . Then ( ) 2 2 3 4 2 2 3 42 1 1. ...-1=0. ...D x x b b b b b b= − = Hence on [ )0,1 , ( )D x is equivalent to ( )xσ , the shift map on two symbols. We have already seen that the shift map ( )xσ is chaotic on the entire space of sequences of two symbols; hence D(x) is chaotic on the entire interval [0, 1) [chaos notes]. 4.9 Proposition Let ( )fN x be Newton iteration function associated to .1)( 2 += xxf Then )(xN f is conjugate to ).(xD Proof. Let    <≤− <≤ = 12/112 2/102 )( xx xx xD be chaotic on [0,1), and let       −= + −= x x x xxxN f 1 2 1 2 1)( 2 be defined on ¡ Define the conjugacy function :[0,1)h → ¡ by ).cot()( xxh π= Then we have ( ) ( )( ) ( ) 2 2cos ( ) sin ( )cot cot 2 2sin( ) cos( ) x xh D x D x x x x π ππ π π π − = ⋅ = =o ( )1 cot( ) tan( ) 2 x xπ π= − )(xhN f o= . Therefore, ( )fN x is conjugate to ( ).D x That is, we have the following commutative diagram: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 123 [0,1) [0,1) f D N h h → ↓ ↓ →¡ ¡ Thus the proof is complete [9]. 4.10 Proposition The doubling function D is chaotic on the unit circle. Proof. We will again make use of a semi-conjugacy. Define the function ]2,2[: 1 −→CB be given by ).cos(2)( θθ =B Since )cos(θ is the x-coordinate of the point θ on 1C , the map B is given geometrically by projecting points vertically from the circle o the x-axis, and then stretching by a factor of 2 ( Fig. 10.5)… Note that B is two-to-one except at the points 𝜋𝜋 and 0 on 1C .Consider the diagram 1 1 ?[ 2,2] [ 2,2] DC C B B → ↓ ↓ − → − As before we ask which function completes the diagram. We have, ( )( ) 2cosB D θ θ=o So we must find the function that takes, ( )θcos2 ↦ ( )2cos 2θ However, we may write, ( ) ( )2 2 22cos 2 2(2cos ( ) 1) (2cos( )) 2 Q xθ θ θ −= − = − = Then the required diagram is ( )2 1 1 [ 2, 2] [ 2,2] D Q x C C B B − → ↓ ↓ − → − So the required function is our friend the quadratic function 2)( 2 −= xxQc . Thus D and 2−Q are semi conjugate. It is not difficult to mimic the arguments given above to the complete the proof [10]. 5. Results and Discussion: Dynamical Behavior of Doubling Map 5.1 Chaotic Behavior of Doubling Map Considering Initial Seeds American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 124 The Doubling Map is given by 12 ; 0 2(x) 12 1 ; 1 2 x x D x x  ≤ <=   − ≤ <  Considering the initial seeds 0 0 0 0 0 0 0 0 1 1 1 1 1 30.3, 0.7, , , , , , . 8 16 7 14 11 22 x x x x x x x x= = = = = = = = For each of the following seeds discuss the behavior of the resulting orbits under D(x). (i) 0 0.3x = ( ) ( ) ( ) ( ) ( )Since 0.3 0.6, 0.6 0.2, 0.2 0.4, 0.4 0.8 and 0.8 0.6D D D D D= = = = = The orbit of 0.3 is eventually periodic with period 1 and period 4. We write 1 40.3 per D∈ (ii) 0 0.7x = . ( ) 4Since 0.7 0.4,and since 0.4 perD D= ∈ from exercise before it follows that 1 40.7 per D∈ . (iii) 0 1 8 x = ; 1 2 4 1mod1 0 8 8 8 =a a a .But 0 is fixed by D .Therefore 3 1 1 per fix . 8 D D∈ ⊆ (iv) 0 1 16 x = ; 1 2 4 8 0. 16 16 16 16 a a a a Therefore, 4 1 1 per 16 D∈ . (v) 0 1 7 x = ; 1 2 2 4 4 1Since , , and , 7 7 7 7 7 7 D D D     = = =            we have that 3 1 per 7 D∈ . (vi) 0 1 14 x = ; 1 2 1Since . 14 14 7 D   = =    But it was shown in (e) that 3 1 per 7 D∈ .Therefore, 1 3 1 per 14 D∈ . (vii) 0 1 11 x = ; 1 2 4 8 5 10 9 7 3 6 1 11 11 11 11 11 11 11 11 11 11 11 a a a a a a a a a a , we see that 10 1 per 11 D∈ . (viii) 0 3 22 x = 1 10 10 3 6 3 3;since ,we have that per 22 22 11 22 D per D D  = = ∈ ∈    But we can see that for initial seed 0 1 7 x = , this is a periodic point with prime period-3 and therefore the Doubling Map is Chaotic by the Period Three Theorem [7]. 5.2 Orbit Analysis of Doubling Map by Newton’s Iteration The behavior of orbit of Newton iteration function associated to Doubling function American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 125 The doubling function 2 0 1/ 2 ( ) 2 1 1/ 2 1 x x D x x x ≤ < =  − ≤ < is chaotic on ].1,0[ Obviously, D has two roots 0 and 2/1 which are on the real line. Figure 1: The graph of doubling function Figure 2: Chaos in doubling function (1) The Newton iteration function associated to )(xD is ( ) .0 2 2 =−= xxxN Note that N has fixed point, at the root 0 of .D We compute ,2)0( =′D and we see that .0)0( =′N Thus the fixed point 0 of N is attracting fixed point. Obviously, orbit of any real point under N converges to fixed point .0 (2) The Newton iteration function associated to )(xD is ( ) . 2 1 2 12 = − −= xxxN N has fixed point, at the root 2/1 of .D We compute ,2)2/1( =′D and we see that .0)2/1( =′N Thus the fixed point 2/1 of N is attracting fixed point. Obviously, orbit of any real point under N converges to fixed point 1 2 [8]. 5.3 Sensitivity to Numerical Inaccuracies of Doubling Map The Doubling Map ( ) 2 ; 0 0.5 2 1 ; 0.5 x 1 x x D x x ≤ < =  − ≤ < is very sensitive to numerical inaccuracies. To see this, we calculate 100 values from the map, the first by using normal decimal numbers and then by using high-precision numbers. In the latter case, we start with numbers that have a precision of 65 digits: vals1 = NestList�Piecewise��{2#,0 ≤ # < 1 2⁄ }, {2# − 1, 1 2⁄ ≤ # < 1}��&,0.003,100�; vals2 = NestList[Piecewise[{{2#,0 ≤ # < 1 2⁄ }, {2# − 1, 1 2⁄ ≤ # < 1}}]&,0.0003`65,100]; Values corresponding to vals2 are thick. From approximately iteration 50 on, the values differ greatly. In calculating vals2, we started with numbers having 65 digits of precision. During the calculation, many digits were lost so that the last value 0.6128only has a precision of approximately 38.20719793550678.Thus we know that all the digits of vals2 are correct. This means that the values in vals1are incorrect from approximately 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 126 iteration 50 on. This demonstrates the sensitivity to numerical inaccuracies of the doubling map. Thus, if we calculate long sequences from the doubling map, it is important to use a high enough precision during the calculation. From the plot of vals2, we see that the series behaves quite chaotically. It is known that chaotic models are very sensitive to numerical inaccuracies. Figure 3: The initial x=1/7 and 0.1/7 `65, n=100 Figure 4: The initial x=1/3 and 0.1/5`65, n=100 Figure 5: The initial x=0.003 and 0.0003`65, n=100 We see that Figure 3 and Figure 4 are regular which means that Numerical Accuracy. But the Figure 5 shows that the behavior is irregular which means Numerical Inaccuracies. In other words, it is called chaotic behavior of Doubling Map [5]. 5.4 Sensitivity Analysis to Initial Value of Doubling Map Chaotic models are also very sensitive to the initial value. To show this, compute, 50 iterations using starting points 0.02 + 10−𝑖𝑖, i = 1, …, 25. Then plot the 20th value of each of the 25 series. Also plot the 50th value of each of the 25 series: 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 0.6 20 40 60 80 100 0.2 0.4 0.6 0.8 20 40 60 80 100 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 127 Figure 6: The initial x=0.02, n=50 Figure 7: The initial x=0.02`50, n=50 Figure 8: The initial x=0.02, n=50 Figure 9: The initial x=0.02`50, n=50 From the first plot Figure 6 & 7, we see that even if the starting point differs from 0.02 by 710− or more (see the first seven points in the plot), the value of 𝐷𝐷20 significantly differs from the value that results when starting from 0.02. From the second plot Figure 8 & 9, we see that if the starting point differs from 0.02 by 1610− or more, the value of 𝐷𝐷50 differs significantly from the value that result when starting from 0.02 [5]. 5.5 Trajectories of Doubling Map We first calculate a solution set by starting from various points and iterating the equation ntimes. The starting points are chosen between x01and x02in steps of dx0. We get the following trajectories: Figure 10: Trajectory for x01 =0.0001, x02=0.11, step size dx0=0.01, n=20. 5 10 15 20 25 0.2 0.4 0.6 0.8 1.0 20th value 5 10 15 20 25 0.2 0.4 0.6 0.8 1.0 20th value 5 10 15 20 25 0.2 0.4 0.6 0.8 1.0 50th value 5 10 15 20 25 0.2 0.4 0.6 0.8 1.0 50th value 5 10 15 20 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 128 Figure 11: Trajectory for x01 =0.0001, x02=0.11, step size dx0=0.01, n=30. Figure 12: Trajectory for x01 =0.0001, x02=0.11, step size dx0=0.001, n=10. We observe that if we take the starting values x01 and x02 are same even step size also same (10 and 11 figure) then the trajectories turns to chaotic with the increasing the number of iteration. If the step size is very small, the trajectories also chaotic in the Figure 12, [5]. 5.6 Orbit Analysis of Doubling Map Graphically The Doubling Map is defined by 1 12 for 0 2( ) 12 -1 for 1 2 n n n n n n x x x D x x x +  ≤ <= =   ≤ <  Taking initial seeds from [ )0,1 such that ( ) ( ) ( ) ( ) ( )0 0 0 0 1 1 1 1 11 , , , , 3 5 7 11 13 i x ii x iii x iv x v= = = = = ( ) ( ) ( )13 23 37, , 29 59 69 vi vii viii . ( ) ( ) 0 0 1 1 2 2 2 For , then , , ,..., ,... ; 3 3 3 3 3 1 1 2 4 2 4 2 4 For , then , , , , ,..., , ... ; 5 5 5 5 5 5 5 5 i x ii x  =      =     ( ) 0 1 1 2 4 6 2 4 6 2 4 6 For , then , , , , , , ,..., , , ,.... ; 7 7 7 7 7 7 7 7 7 7 7 iii x  =     5 10 15 20 25 30 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 129 ( ) ( ) ( ) 0 0 0 1 1 2 4 8 6 10 2 2 2 4 8 6 10 For , then , , , , , , ,..., , , , , , ,... . 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 9 5 10 7 1 2 4 8 3 6 12 11 For , then , , , , , , , , , , , , ,... . 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 2 For , then , 29 29 iv x v x vi x  =      =     = ( ) ( ) 0 0 6 23 17 5 10 20 11 22 15 1 2, , , , , , , , , , ,... . 29 29 29 29 29 29 29 29 29 29 29 23 23 46 33 7 14 28 56 53 47 35 11 22 44 For , then , , , , , , , , , , , , ,... . 59 59 59 59 59 59 59 59 59 59 59 59 59 59 37 37 5 10 20 40For , then , , , , 69 69 69 69 69 69 vii x viii x        =     = 11 22 44 19 38 7 14 28, , , , , , , , ,... . 69 69 69 69 59 69 69 69       Figure 13: 0x =1/3, n=1000 Figure 14: 0x =1/5, n=1000 Figure 15: 0x =1/7, n=1000 Figure16: 0x =1/11, n=1000 Figure17: 0x =11/13, n=1000 Figure18: 0x =13/29, n=1000 Figure19: 0x =23/59, n=1000 Figure 20: 0x =37/69, n=1000 The sequence behave as follows: (i) there is period one behavior, (ii) there is period two behavior, (iii) there is a 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 130 period three sequence, (iv) there is a period five sequence, (v) there is period twelve and in case (vi), (vii), (viii) are contained some list of iteration, but other method need to be used to established the long term behaviour of the sequences [2]. 5.7 Staircase Diagram Again if we iterate the Doubling Map taking nearby initial seeds from [ )0,1 graphically, we can get the following graph: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 2 0 1 0 2 0 1 0 2 0 1 0 2 0 [1/ GoldenRatio,20], [1/ GoldenRatio,25], [1/ GoldenRatio,50], [1/ GoldenRatio,51], [1/ GoldenRatio,200], [1/ GoldenRatio,250], [1/ GoldenRatio], [1/ GoldenRatio,3 a x N a x N b x N b x N c x N c x N e x N e x N = = = = = = = = ( ) 020], [1/ GoldenRatio,500].f x N= Figure 21: For ( )1 0 [1/ GoldenRatio,20]a x N= Figure 22: For ( )2 0 [1/ GoldenRatio,25]a x N= Difference: 0.× 10−21 Figure 23: For ( )1 0 [1/ GoldenRatio,200]c x N= Figure 24: For ( )2 0 [1/ GoldenRatio,250]c x N= Difference: 0.× 10−201 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 131 Figure 25: For ( )1 0 [1/ GoldenRatio]e x N= Figure 26: For ( )2 0 [1/ GoldenRatio,302]d x N= 16Difference :1.110223024625156 10−× Each of the diagrams (Figure13-26) can be reproduced by using Mathematica. Figure show that the doubling map displays sensitivity to initial conditions and can be described as being chaotic. The iterative path plotted in Figure 23, 24, 26 appears to wander randomly. It clearly shows the sensitivity to initial conditions. It is clear from the diagrams is that the three basic properties of chaos and they are mixing, periodicity, sensitivity to initial conditions, and they are all exhibited for certain values. Indeed, a now famous result due to Li and [3] states that if a system displays period-three behaviour, then the system can display periodic behaviour of any period. Li and Yorke then go on to prove that the system can display chaotic phenomena [2]. 6. Conclusion Chaotic behavior of one dimensional Doubling map introduces an interesting and exciting part of Dynamical Systems. In this paper, we have introduced one dimensional Doubling map which is shown as chaotic map in several senses by some pioneers. We are trying to find the dynamical behaviors of others one dimensional maps with applications and establish some mathematical formulas concerning chaotic dynamical systems. Acknowledgement It is starting with thanking the Almighty Allah. Not only this paper but also my whole life is blessed by the grace of Him. I acknowledge with a humble respect to my supervisor Professor Dr. Payer Ahmed, Department of Mathematics, Faculty of Science, Jagannath University, Dhaka and Professor Dr. Md. Rezaul Karim, Chairman, Department of Mathematics, Jagannath University, Dhaka for delivering right directions to complete this research work properly. I am also gratitude to my teacher A.B.S. Manik Munsi, Lecturer, Department of Mathematics, Jagannath University, Dhaka for his constructive guides during my research works. 7. Constraints/Limitations of the study There are some limitations we see in our article. First of all we have to face software problem while preparing 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 132 Orbit analysis and analyzing staircase diagram of the Doubling map. First we tried to see the said analysis using Mathematica 5.0 version but it did not working showing graph. After that we took an attempt to use Mathematica 11.0 version. Then we have seen that it worked very smoothly and got our expected result. Therefore Software version problem can be treated as our major limitation of our paper. The 2nd limitation is that as the Doubling map does not come any parameter so we did not have desired chaotic sound like others chaotic maps such as Logistic map and Tent map. Of course high speed super computer is very much needed to run the Doubling map related program to run quickly. 8. Recommendations We would like to notice here that Mathematica 11.0 version and some other upgraded versions are recommended to use while performing graphical Analysis, Orbit analysis, Staircase diagram etc. To run the program more smoothly and to find actual chaotic sound a powerful high speed super computer are recommended for such a research so that the ambitious researchers can get their expected results. References [1] C. Robinson, Dynamical Systems, Stability, Symbolic Dynamical and Chaos, CRC Press, Boca Eaton, 1995 [2] Dynamical Systems and Ergodic Theory-The Doubling Map, Corinna Ulcigrai, University of Bristol. [3] Evan Dummit, 2015, v. 1.00, Dynamics, Chaos, and Fractals (part 3): Chaotic Dynamics. [4] Evgeny Demidov, Chaotic 1 D maps. [5] Heikki Ruskeepaa, Mathematica Navigator, 3rd Edition. [6] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney’s Definition of chaos, American Mathematic Monthly 99 (1992), 332-334 [7] Li, T.-y., and Yorke, J., “Period Three Implies Chaos.” American Mathematical Monthly 82 (1975), 985-992. [8] P. Ahmed, Chaotic homeomorphisms in one-dimensional dynamics, Proc., Iwate Univ., Japan, May 5- 6, 2001. [9] R. L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addision- Wesley, New York,1992. [10] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second Edition, Addision-Wesley, RedwoodCity,1989. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 43, No 1, pp 110-133 133 [11] R. L. Devaney and L. Keen, Chaos and Fractals: The Mathematics Behind the Computer Graphics, American Mathematical Society, Providence, 1989. [12] S. Kolyada and L. Snoha, Some aspects of topological transitivity-A survey, 1997. [13] S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics Series, Volume 2, Spring-Verlag, Berlin, 1992. [14] Wolf, A. “Quantifying Chaos with Lyapunov exponents,” in Chaos, edited by A. V. Holden, Princeton University Press, 1986. 3.2 Several Definitions of Chaos 4.3 Proposition The Doubling map is chaotic on in the sense of Lyapunov exponent. 4.5 Proposition Doubling Map is chaotic in the sense of Wiggin. 4.6 Proposition Let is defined by is conjugate to Logistic Map. 5.1 Chaotic Behavior of Doubling Map Considering Initial Seeds But we can see that for initial seed, this is a periodic point with prime period-3 and therefore the Doubling Map is Chaotic by the Period Three Theorem [7]. 5.2 Orbit Analysis of Doubling Map by Newton’s Iteration The behavior of orbit of Newton iteration function associated to Doubling function The Doubling Mapis very sensitive to numerical inaccuracies. To see this, we calculate 100 values from the map, the first by using normal decimal numbers and then by using high-precision numbers. In the latter case, we start with numbers that have a p... Figure 4: The initial x=1/3 and 0.1/5`65, n=100 Figure 5: The initial x=0.003 and 0.0003`65, n=100 We see that Figure 3 and Figure 4 are regular which means that Numerical Accuracy. But the Figure 5 shows that the behavior is irregular which means Numerical Inaccuracies. In other words, it is called chaotic behavior of Doubling Map [5]. 5.4 Sensitivity Analysis to Initial Value of Doubling Map Figure 6: The initial x=0.02, n=50 Figure 7: The initial x=0.02`50, n=50 Figure 8: The initial x=0.02, n=50 Figure 9: The initial x=0.02`50, n=50 5.5 Trajectories of Doubling Map 5.6 Orbit Analysis of Doubling Map Graphically The Doubling Map is defined by