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Construction and Chaos Properties Analysis of a Quadratic Polynomial Surjective Map

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Construction and Chaos Properties Analysis of a Quadratic Polynomial Surjective Map INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 Construction and Chaos Properties Analysis of a Quadratic Polynomial Surjective Map Jun Tang, Jianghong Shi, Zhenmiao Deng randomly distributed sequences [16]–[18]. Abstract—In this paper, a kind of quadratic polynomial surjective In this paper, we deduce the relationships between the map (QPSM) is constructed, and the topological conjugation of the coefficients of a quadratic polynomial surjective map (QPSM) QPSM and tent map is proven. With the probability density function and prove the topological conjugacy of the QPSM and tent map. (PDF) of the QPSM being deduced, an anti-trigonometric transform We give the PDF of the QPSM, which is then used to design a function is proposed to homogenize the QPSM. The information entropy, Kolmogorov entropy (KE), and discrete entropy (DE) of the transform function to homogenize the QPSM sequence. Finally, QPSM are calculated for both the original and homogenized maps we estimate entropies of the QPSM such as the information, with respect to different parameters. Simulation results show that the Kolmogorov, and discrete entropies for both the original and information entropy of the homogenized sequence is close to the homogenized map. theoretical limit and the discrete entropy remains unchanged, which This paper is organized as follows. Section II presents a suggest that the homogenization method is effective. Thus, the method for determining the QPSM coefficients, and the homogenized map not only inherits the diverse properties of the original QPSM but also possesses better uniformity. These features topological conjugation of the QPSM and tent map is proven. make it more suitable to secure communication and noise radar. Section III gives the derivation for the PDF of the QPSM and Keywords—chaotic map, entropy, topological conjugation, presents a homogenization method. Section IV presents an homogenization. analysis on the statistical and chaos characteristics of the QPSM and the results of some simulations. Finally, Section V I. INTRODUCTION concludes the paper. HAOS is seemingly random and irregular movement that Coccurs in a deterministic system without any random II. THE CONSTRUCTION OF THE QPSM factors [1]–[3]. As a new science, chaology is usually thought to have begun with the publication of “Period three implies chaos” A. The Relationships between the Coefficients of the QPSM by Li and Yorke in 1975 [4], which is the first use of “chaos” as Reference [19] proposed an equivalent proposition for the scientific terminology. As a complex dynamic phenomenon in a determination of period-3 points of a real coefficient nonlinear system, chaos exists throughout nature. Because of its polynomial by decomposing it into a complex field and complex behaviour, chaos has attracted great interest from presented a necessary and sufficient condition of determining scholars in various fields [5]–[8]. period-3 points of a quadratic polynomial. Because of ergodicity and mixing [9], chaos can be Lemma 2.1 [19]: The necessary and sufficient condition of characterized by methods of probability and statistics. That is, determination 3-periodic points of a quadratic polynomial map the probability density function (PDF) of a chaotic map can be f() x= ax2 ++ bx c,0 a ≠ is used to analysis the properties of chaos. So far, however, only a b2 −4 ac −≥ 27 b (1) few simple chaotic maps have known PDFs [10], [11]. If the quadratic polynomial is a surjective map, we can further Reference [12] demonstrated that the logistic map and conclude the following: Chebyshev map are topologically conjugate to the tent map and i. One of the fixed points x* of f is the boundary of the presented PDFs of the logistic and Chebyshev maps. Producing chaotic sequences with good uniformity and randomness is a map interval I ; very important subject [13–[15]. At present, there are many ii. Let ()xycc, be the vertex coordinate of f ; then, ways to transform chaotic sequences into uniformly and * fy()c = x should hold. Without loss of generality, if we suppose a < 0 , then we can obtain the following equation according to the above The research was supported in part by the National Natural Science conditions: Foundation of China under Grant 61471012, in part by the Open-End Fund of BITTT Key Laboratory of Space Object Measurement. Jun Tang is with the School of Information Science and engineering, Xiamen University, Xiamen, China (corresponding author to provide e-mail: [email protected]). Jianghong Shi is with the School of Information Science and engineering, Zhenmiao Deng is with the School of Information Science and engineering, Xiamen University, Xiamen, China (e-mail: [email protected]). Xiamen University, Xiamen, China (e-mail: [email protected]). ISSN: 1998-4464 100 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 ** = xf= ()x If we let f() hx() h()2 x, we get (2) 2 =− * 2 fx(()4ac b /4a) k = 1 a By solving the above equation, we can get one fixed point b *2 2 k = − x=( b −4 ac − 2 b +−+ 1 b 1) /2 a and cb=() −−2 b 8 /4 a. 2 2a In this case of b2 −4 ac −≡ 28 b , the condition of lemma 2.1 is In other words, when hx()()()=2cosπ x / a −∈ b / 2 a , x [] 0,1 , met. Therefore, the QPSM is chaotic, and its map interval is hgx()() = f() hx() . Therefore, fx() and gx() are I=()()4 − ba /2 , −+ 4 ba /2 . Thus, the QPSM can be topological conjugacy via hx() . expressed as 44−bb −− f() x= ax2 ++ bx c() a,, b x ∈ , (3) III. PDF AND HOMOGENIZATION OF THE QPSM 22aa 2 A. PDF of the QPSM where c() ab,=() b −− 2 b 8 /4 a. Lemma 3.1 [21]: If the maps fx() and gx() are B. Topological Conjugation of the QPSM topological conjugate via hx() and ρ ()x is the PDF of In order to prove that Eq. (3) is topologically conjugate to the g tent map, we first introduce the definition of a topological gx() , then the PDF of fx() is conjugation. −1 −1 dh() x Definition 2.2 [20] (Topological Conjugation): Two maps ρρfg()()x= () hx dx fI: → I and gJ: → J are topological conjugate if there =2 ++ exists a homeomorphism hI: → J such that Theorem 3.2: The PDF of f() x ax bx c() a, b is hf= gh (4) a 16 Lemma 2.3: The QPSM f() x= ax2 ++ bx c() a, b is π −4a22 x − 4 abx −+ b2 16 topologically conjugate to the tent map. Proof: The PDF of the tent map is Proof: The tent map is given by ρT ()()xx=1, ∈ 0,1 1 According to Lemma 3.1, we can obtain 2,xx 0≤≤ 2 − 1 ab gx() = (5) hx1 () = arccos x+ 1 π 24 2−2,xx ≤≤1 2 and −1 Let hx()()= k12cosπ x +∈ k , x [] 0,1 . Obviously, hx() is a −1 dh() x ρρfT()()x= () hx continuous and reversible function. dx According to definition 2.2, 1 ab 1 dxarccos+ kcos() 2π xk+ , 0 ≤≤ x π 24 12 2 =1 × hgx()() = dx 1 kcos() 2ππ− 2 xk + , ≤≤ x 1 122 a 16 = k12cos() 2π xk+ (6) = π 22 2 = hx()2 −4a x − 4 abx −+ b 16 22 2 On the other hand, Therefore, 16a /(π − 4 a x − 4 abx −+ b 16 ) is the PDF of 2 = ++ 2 f() h() x ah()()() x bh x c a, b f() x= ax ++ bx c() a, b . Obviously, the distribution of the 2 = ak12cos()π x+ k sequence produced by Eq. (3) is non-uniform. + b k12cos()()π x+ k + c ab, B. Homogenization of the QPSM =22ππ++ 2 Theorem 3.3: If the PDF of a random variable X is a k1cos () x k22 kk 12 cos() x (7) + bk12cos()()π x+ bk + c a, b 22cos() 2π x + 1 =a k1 ++k22 kk 12 cos()π x + 2 bk12cos()()π x+ bk + c a, b ISSN: 1998-4464 101 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 −a 16 A. Definitions of Entropy , π 22 2 = −4a x − 4 abx −+ b 16 Definition 4.1 (Information Entropy): Let Ss{}12,s , , sn 44−bb −− be an information source and pi be the probability of si ρ X ()xx= ≤≤ 22aa showing up in S . The information entropy of S is then defined 0, otherwise as n HS() = −∑ piilog p then the random variable i=1 1 ab When pn=1/ for all in∈{}1, , , HS() achieves the ZX=arcsin −− (8) i π 24 maximum log2 n . In other words, the maximum entropy follows a uniform distribution in the interval []−0.5,0.5 . probability distribution is the uniform distribution. Proof: The PDF of the random variable Z is Definition 4.2 [22] (Kolmogorov Entropy). Consider the F z= PZ ≤ z Z ()() trajectory Xt()()()= xt1 ,, xd t of a dynamical system on a 1 ab strange attractor and suppose that the d -dimensional phase =Parcsin − Xz −≤ π 24 space is partitioned into boxes of size ld . The state of the ab system is now measured at intervals of time τ . Let P be the =PX − −≤sin()π z (9) ii0 n 24 joint probability that Xt()= 0 is in box i0 , Xt()=τ be the 2 b =PX ≤−sin()π z − joint probability that it is in box i ,, , and Xt()+ nτ be the aa2 1 2 b joint probability that it is in box in . The K -entropy is defined −−sin()π z = aa2 ρ ()x dx ∫−∞ X as 1 If the derivation operation is applied to both sides of Eq. (9), K= −limlim lim PPlog ∑ ii01NN−− ii 01 τ →00lN → →∞ τ then we can get the PDF of Z : N ii01, N − 32 cos()π z For one-dimensional maps, K is simply the positive Lyapunov , π 2 exponent.
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