A non-trivial relation between some many-dimensional chaotic discrete dynamical systems and some one-dimensional chaotic discrete dynamical systems

Computer Physics Communications 179 (2008) 628–633

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Computer Physics Communications www.elsevier.com/locate/cpc

A non-trivial relation between some many-dimensional chaotic discrete dynamical systems and some one-dimensional chaotic discrete dynamical systems Reza Mazrooei-Sebdani, Mehdi Dehghan ∗ Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 28 February 2008 Received in revised form 16 April 2008 Accepted 28 May 2008 Available online 12 June 2008 PACS: 05.45.Jn 05.45.-a 05.45.Gg

Chaotic maps are very useful in practical applications. In this paper, we present a method for constructing the many-dimensional chaotic discrete dynamical systems using semiconjugacy property. The chaotic property in one dimension may be influenced the chaotic property in higher-dimensions. In fact, using the one-dimensional chaotic maps and semiconjugacy property, we construct some many-dimensional chaotic discrete dynamical systems. These systems may be used as random number generators in Monte Carlo simulations. Also, these systems may be used in practical applications such as chaotic cryptography and evolutionary algorithms. © 2008 Elsevier B.V. All rights reserved.

Keywords: Discrete dynamical systems Chaotic map Semiconjugacy

1. Introduction The only aspect that almost appears in all literatures of chaotic maps is sensitivity to initial conditions (in the sense of Devaney, in the sense of Li–Yorke or in physical literatures and other applied models). Also, in many applied literatures the main requirement for a system to be chaotic is the existence of at least one positive Lyapunov exponent. This property shows the sensitivity to initial conditions. In this paper, the purpose of a chaotic map is any map which has sensitivity to initial conditions [1–20]. Random number generators are used in many problems of physics and physical chemistry specially in Monte Carlo simulations. For example, the Monte Carlo simulations were carried out to obtain the fluorescence anisotropy decay equations due to the rotational diffusion of dipoles on a spherical surface in [21]. For another example, the Monte Carlo simulations are used for translational diffusion of fluorescent probes on a sphere in [22]. Currently there is an intensive search for new methods for the generation for random numbers due to which are performed using Monte Carlo methods. The problem is that when the generators are not very good, incorrect results are obtained. The applying chaos theory to randomness has produced important works recently. In this paper, we present a non-trivial method for constructing the chaotic dis-

*

Corresponding author. Tel.: +9821 64542503; fax: +9821 66497930. E-mail addresses: [email protected], [email protected] (R. Mazrooei-Sebdani), [email protected] (M. Dehghan). 0010-4655/$ – see front matter doi:10.1016/j.cpc.2008.05.010

©

2008 Elsevier B.V. All rights reserved.

crete dynamical systems. These systems may be used as random number generators in Monte Carlo simulations [21–30]. In recent years the chaotic maps have been used for designing some cryptography systems. The close relationship between chaos and cryptography makes chaotic encryption a natural candidate for secure communication and cryptography. A number of chaotic cryptosystems have been studied so far. A current situation in existing digitized chaotic cryptosystems is that multi-dimensional chaotic maps are usually used for image encryption as the elements of an image are managed in two-dimensional way. On the other hand, one-dimensional chaotic maps are widely applied in document encryption [31–40]. Also, some evolutionary algorithms are presented, here use the chaotic maps. Some new researches in optimization are based on chaotic maps [41–43]. Chaotic maps are very useful in practical applications. There are many chaotic maps in one dimension. The dynamics of onedimensional maps may be influenced the dynamics of higher dimensional maps. In fact, according to [1] using the semiconjugacy, we can transfer the dynamics of one-dimensional maps to manydimensional maps. In this paper, using semiconjugacy and onedimensional chaotic maps, we construct many-dimensional chaotic maps. Consider “Theorem 3.3.3” of [1]. According this theorem, if we consider a one-dimensional chaotic map and a suitable semiconjugacy, then we can construct a many-dimensional chaotic map (in fact, using semiconjugacy, we can construct many-dimensional maps which have sensitivity to initial conditions and thus chaotic

R. Mazrooei-Sebdani, M. Dehghan / Computer Physics Communications 179 (2008) 628–633

in the sense of this paper). This method is non-trivial for constructing many-dimensional chaotic maps (see examples of Section 3). This may be a general method for construction the chaotic maps. In Section 2, we will explain the method, and then in Section 3, we will present some examples. Numerical examples are presented in Section 4. The applications of these sequences in Monte Carlo Simulations are explained in Section 5.

Let F ∈ C ( X , X ), where X ⊂ R is nonempty and C ( X , X ) is the set of continuous functions mapping X to X . When there is a nonconstant function H ∈ C ( X , R) such that p

H ◦ F = φ ◦ H,

(2.1)

on X for some φ ∈ C ( H ( X ), R), we say F is a ( X , H , φ)-semiconjugate map of R p . The mapping H is named a linked map and φ is the factor map. Where there is no confusion, we may also use the term R-semiconjugate in referring to F . Also, the sequence F n (x0 ) for each x0 ∈ X is a R-semiconjugate trajectory. The following diagram presents the superpositions (2.1): F

X

H

H(X)

β

1

1

μ α x( g (xy, y) ) α (1 − xα y β ) α

Some dynamics of the factor map influence the dynamics of the original map. For example, stability and boundedness of factor map influence the stability and boundedness of original map. For more details, you can refer to [1]. The chaotic behavior of a factor map influences the chaotic behavior of original map (a chaotic map, i.e., any map which has sensitivity to initial conditions). Let F be a ( X , H , φ)-semiconjugate map, with H ∈ C 1 ( X , R) and X ⊂ R p a compact and convex set. According to Theorem 3.3.3 of [1], if φ is chaotic in H ( X ), then F is chaotic in X . Let F be ( X , H , φ F )-semiconjugate and let G be ( X , H , φG )-semiconjugate. Then according to Lemma 3.2.1 of [1], F ◦ G is ( X , H , φ F ◦ φG )-semiconjugate and for each positive integer n, the iterate F n is ( X , H , φ nF )-semiconjugate, i.e., H ◦ F n = φ nF ◦ H (for every function g, and n ∈ N g n = g ◦ g ◦ · · · ◦ g ).







n times

We can easily show that, if L (φ) denotes Lyapunov exponent of the map φ then L (φ n ) = nL (φ). So, if φ is sensitive dependence on initial conditions, then so is φ n , for every n. Thus, if F be a ( X , H , φ)-semiconjugate map, with H ∈ C 1 ( X , R) and X ⊂ R p a compact and convex set, and φ , is chaotic, then so is φ n , for every n, and so is F n , for every n. So, we can obtain many chaotic higher-dimensional mappings. This procedure for construction the chaotic maps is easy and useful. In fact, if we use this method and also the above property of semiconjugate maps, then we can construct several chaotic maps in higher-dimensions. The examples show that, this method is non-trivial (see the examples of next section).

1−

μ

1−

4

 b  1,

we can set

 g (x, y ) =

μ 4

2

+ (x − b )

β1 .

So, the two-dimensional map

⎛

⎜ F (x, y ) = ⎝

1

μ α x(

[4

1 +(x−b)2 ] β

⎞ ⎟ ⎠,

[ μ4 + (x − b)2 ] β

with

μ ∈ [3.9, 4]. Also let X , is defined as





X = (x, y ) ∈ [0, 1]2 : 0  xα + y β  1 .

g (x, y )β  4(xα + y β )(1 − xα − y β ),



F (x, y ) =

1

[4(xα + y β )(1 − xα − y β ) − g (x, y )β ] α g (x, y )

1 1 g (x, y ) = μ β y (1 − xα − y β ) β .

Thus the two-dimensional map



F (x, y ) =

1

1

1

1

μ α x(1 − xα − y β ) α μ β y (1 − xα − y β ) β

,

is chaotic. Example 3.3 (Using Logistic map). Let

with

μ ∈ [3.9, 4]. Now the following map of R3

⎛ F (x, y , z) ⎞ 1 F (x, y , z) = ⎝ F 2 (x, y , z) ⎠ ,

1/β

(x, y )∈ X



,

satisfies H ◦ F = φμ ◦ H , on the compact and convex set X and so is chaotic. We can set

on [0, 1], is a well-known example of one-dimensional chaotic maps for μ ∈ (3.9, 4). Let H (x, y ) = xα y β , and X = [0, 1]2 (α , β > 0 and for every u, u α , u β , u 1/α and u 1/β are valid). Then for every g ∈ C ( X , R), with sup g (x, y )  1,

(x, y ) ∈ X ,

the two-dimensional map, F : X → X , with definition,

φ(t ) = μt (1 − t ),

and

(3.1)

We can show easily that X is a compact and convex set. (α , β > 0 and for every u, u α , u β , u 1/α and u 1/β are valid.) Then for every positive continuous function g ∈ C ( X , R), with

φμ (t ) = μt (1 − t ),

< inf g (x, y ),

α , β  1, H (x, y ) = xα + y β

φ(t ) = μt (1 − t ),

and φμ be the logistic map

(x, y )∈ X

1

) α (1 − xα y β ) α 1

Example 3.1 (Using Logistic map). The Logistic mapping

4

β

y μ

H (x, y , z) = x2 + y 2 + z2 ,

μ

,

satisfies H ◦ F = φμ ◦ H , on the compact and convex set X and so is chaotic. If

3. Examples





g (x, y )

Example 3.2 (Using Logistic map). Let and φμ be the Logistic map

H ◦ F (X)

φ



F (x, y ) =

is chaotic on X .

F (X)

H

the two-dimensional map, F : X → X , with definition,



2. Semiconjugacy and chaotic maps

629

F 3 (x, y , z) where

630

R. Mazrooei-Sebdani, M. Dehghan / Computer Physics Communications 179 (2008) 628–633







When r > 1, Chebyshev polynomial map T r : [−1, 1] → [−1, 1] of degree r is a chaotic map with its invariant density

F 1 (x, y , z) = z 4(1 − x2 − y 2 − z2 ) cos ψ(x, y , z) , F 2 (x, y , z) = and F 3 (x, y , z) =









4(x2 + y 2 )(1 − x2 − y 2 − z2 ) cos ψ(x, y , z) ,



f ∗ (x) = √



4(x2 + y 2 + z2 )(1 − x2 − y 2 − z2 ) sin ψ(x, y , z)

and also ψ ∈ C (R, R), is ( X , H , φμ )-semiconjugate map with X being the closed unit ball, i.e.,



3

2

2



2

X = (x, y , z) ∈ R : x + y + z  1 .

2x,

0x

2(1 − x),

1 2

     T r (xα y β )   g (x, y )β   1,

β

F 1 (x, y )

(x, y ) ∈ X ,

1

where C > −2, is a constant number. Thus the two-dimensional map,



1

F 1 (x, y ) =

⎩

β y )α g (x, y )

,

1

(2x2α y 2β + C ) β

1

Example 3.6 (Using Chebyshev polynomials). Let xα + y β and X , is defined as





Then X is a compact and convex set. (α , β > 0 and for every u, u α , u β , u 1/α and u 1/β are valid). Then for every g ∈ C ( X , R), with

 xα y β  1,

   T r (xα + y β ) − g (x, y )β   1,

g (x, y ) = sin β (π xα y β ).

   g (x, y )  1,

Thus the two-dimensional map

the two-dimensional map,

F 1 (x, y )





,

F 2 (x, y )

(x, y ) ∈ X ,

⎧ 1 ⎪ ⎨ 2 α x(

β

y

0  xα y β  12 ;

)α ,

1

sin β (π xα y β ) 1

2 α (1 − xα y β ) α sin− α (π xα y β ), 1

1

1 2

 xα y β  1,

and

g (x, y )

 ,

satisfies (2.1), on the compact and convex set X and so is chaotic. We can set 1 

g (x, y ) =

1

 T r (xα + y β ) − sin ψ(x, y ) β , 1

2β

⎛

Example 3.5 (Using Chebyshev polynomials). The Chebyshev polynomial of degree r is defined as

2

T 2 (x) = 2x − 1,

T r +1 (x) = 2xT r (x) − T r −1 (x),

....

F (x, y ) = ⎝

⎞

1

[ 12 T r (xα + y β ) + sin ψ(x, y )] α 1 1

2β

1

[ T r (xα + y β ) − sin ψ(x, y )] β

⎠,

is chaotic.

−1  x  1.

The recurrent formulas are T 1 (x) = x,

1

where ψ ∈ C ( X , R). Thus the two-dimensional map

1

F 2 (x, y ) = sin β (π xα y β ).

T r (x) = cos(r ∗ arccos x),

(x, y ) ∈ X ,

[ T r (xα + y β ) − g (x, y )β ] α

F (x, y ) =

is chaotic, where

T 0 (x) = 1,

(3.2)

and also

1

⎪ ⎩

α , β  1, H (x, y ) =

1 2

β

1

2 α (1 − xα y β ) α g (x, y )− α ,

We can set

F 1 (x, y ) =

,

X = (x, y ) ∈ [−1, 1]2 : |xα + y β |  1 .

F 2 (x, y ) = g (x, y ).

F (x, y ) =



0  xα y β  12 ;

and



1

(2x2α y 2β + C )− α (2x2α y 2β − 1) α

is chaotic.

,

satisfies (2.1), on the compact and convex set X and so is chaotic, where

⎧ 1 ⎨ 2 α x(

,

g (x, y ) = (2x2α y 2β + C ) β ,

F (x, y ) =



F 2 (x, y )



g (x, y )

the two-dimensional map,



1

satisfies (2.1), on the compact and convex set X and so is chaotic. Specially, if r = 2, we can set

on [0, 1], is a well-known example of chaotic maps in one-dimension. For this map the Lyapunov exponent is ln 2 > 0. Let H (x, y ) = xα y β and X = [0, 1]2 (α , β > 0 and for every u, u α , u β , u 1/α and u 1/β are valid). Then for every positive continuous function g ∈ C ( X , R), with

F (x, y ) =

(x, y ) ∈ X ,

g (x, y )− α T rα (xα y β )

F (x, y ) =

1 ; 2

 x  1,

T (xα y β )  g (x, y )β  1,

,

for Lyapunov exponent λ = ln r > 0. Let H (x, y ) = xα y β and X = [−1, 1]2 (α , β > 0 and for every u, u α , u β , u 1/α and u 1/β are valid). Then for every g ∈ C ( X , R), with



Example 3.4 (Using tent map). The tent mapping T (x) =

1 − x2

the two-dimensional map,

So, F is a chaotic map on X .



1

...,

Remark. In the above examples, let φ is the factor map. According to Section 2, for each positive integer n, φ n is chaotic and the iterate F n is ( X , H , φ n )-semiconjugate. Therefore for every n, F n is chaotic.

R. Mazrooei-Sebdani, M. Dehghan / Computer Physics Communications 179 (2008) 628–633

631

Fig. 1.

Fig. 2.



4. Numerical results

yn+1 =

Let us now turn to numerical investigations for some examples in previous section. According to Example 3.1, the two-dimensional discrete dynamical system

1

xn+1 = μ α xn



 αβ

yn 1

[ μ4 + (xn − 1)2 ] β



β  α1

1 − xnα yn

;

μ 4

+ (xn − 1)2

β1 ,

is chaotic for μ ∈ (3.9, 4), with initial conditions in X = [0, 1]2 . In the both of Figs. 1 and 2, we consider the parameters as

α = β = 1,

μ = 3.95.

In Fig. 1, the initial conditions are x0 = 0.1, y 0 = 0.1. But in Fig. 2, the initial conditions are x0 = 0.1, y 0 = 0.5. You can compare the figures. Also for the first case the sequences {xn }, { yn } are shown

632

R. Mazrooei-Sebdani, M. Dehghan / Computer Physics Communications 179 (2008) 628–633

Fig. 3.

Fig. 4.

in Figs. 3-1, 3-2 and for the second case for the first case the sequences {xn }, { yn } are shown in Figs. 4-1, 4-2. You can compare them. 5. Monte Carlo simulations Random number generators are used in many problems of physics and physical chemistry specially in Monte Carlo simulations. For example, the Monte Carlo simulations were carried out to obtain the fluorescence anisotropy decay equations due to the rotational diffusion of dipoles on a spherical surface in [21]. For other example, the Monte Carlo simulations are used for translational diffusion of fluorescent probes on a sphere in [22]. The specialists in Monte Carlo method [28,29] face the problem of testing pseudorandom number generators. How to create a sequence of numbers indistinguishable from those produced by a truly random process. This question possesses practical importance because a very large number of calculations are performed using the Monte Carlo method with pseudorandom numbers. Almost all known generators (in some cases) give rise to incorrect results because they deviate from randomness [30]. There is a lack of mathematical basis for generation of randomness [28,29]. Most of the algorithms used in Monte Carlo calculations are quite predictable. There is also the quasi-Monte Carlo method [28]. The advantage of this method is that the generated points cover the phase space very uniformly and the numerical integration has very fast convergence rates. However this algorithm is unappropriate for many Monte Carlo calculations because the generated points are very regular and predictable. The applying chaos theory to randomness has produced important works recently [21–30]. In this paper, we present a non-trivial method for constructing the chaotic discrete dynamical systems. These systems may be used as random number generators in Monte Carlo simulations as random number generators. In fact, we can construct many random number generators that we can use them in Monte Carlo simulations.

6. Conclusion This paper, was proposed a method for constructing the manydimensional chaotic discrete dynamical systems using semiconjugacy. This method is easy and practical. The examples in Section 3 showed that, this method is non-trivial. For illustration the results, we presented a numerical example in Section 5. These systems may be used as random number generators in Monte Carlo simulations. Also, these systems may be used in practical applications such as chaotic cryptography and evolutionary algorithms. In fact, in this fields, we need good random number generators. It does not matter how chaotic a nonlinear map can be, it needs truly random. The method in this paper can be used for testing many random number generators. References [1] H. Sedaghat, Nonlinear Difference Equations, Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003. [2] M. Dehghan, C.M. Kent, R. Mazrooei-Sebdani, N.L. Ortiz, H. Sedaghat, Dynamics of rational difference equations containing quadratic terms, J. Difference Equations Appl., in press. [3] D.M. Chan, E.R. Chang, M. Dehghan, C.M. Kent, R. Mazrooei-Sebdani, H. Sedaghat, Asymptotic stability for difference equations with decreasing arguments, J. Difference Equations Appl. 12 (2006) 109–123. [4] M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2002. [5] S.N. Elaydi, Discrete Chaos, Chapman and Hall/CRC, Boca Raton, FL, 1999. [6] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, second ed., Addison–Wesley, Reading, MA, 1989. [7] T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975) 985–992. [8] Z. Xiong, Symmetry breaking, multiple bifurcations, quasi-periodicity and chaos in a class of delay difference equations, Chaos Solitons Fractals, in press. [9] M. Peng, Bifurcation and chaotic behavior in the Euler method for a Uçar prototype delay model, Chaos Solitons Fractals 22 (2004) 483–493. [10] A. Uçar, On the chaotic behaviour of a prototype delayed dynamical system, Chaos Solitons Fractals 16 (2003) 187–194.

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