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]).
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** = 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
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−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. 2 2 b −−4az sin()π − − Definition 4 [23] (Discrete Entropy): Let Aa= {},, a be aa2 1 L a finite set endowed with a linear ordering ≤ , FA: → A be a 2 b 4ab− sin()π z − −+ b2 16 ρ = bijection, and π ∈ Sn be a permutation on {}0,1, ,n − 1 with Z ()z aa2 2 ≤≤nL. The following is defined: 42−b bb−− 4 ≤−sin()π z − ≤ ππ01n− 2aa22 aa Qπ () n=∈{ a AF: ()() a << F a} and 0, otherwise Qnπ () qnπ () = 2 cos()ππzz sec()() ,1sin−≤ π z ≤ 1 ∑ Qnτ () = τ∈Sn 0, otherwise Then, the discrete entropy of F with the order n ≥ 2 is defined 11 as 1, −≤≤z = ()n 1 22 hF() = − qlog q (10) δ − ∑ ππ 0, otherwise n 1 π∈Sn 1 ab The best way to encapsulate the information contained in Therefore, the random variable Zx=arcsin −− is ()2 ()n ()n π 24 max hδδ() F, , h() F , nmax = max{ nh :δ() F≠ 0} into a single uniformly distributed in []−0.5,0.5 . number is by taking its arithmetic mean. We call
nmax 1 n = () IV. THE CHAOTIC CHARACTERISTICS OF THE QPSM hFδ () ∑ h() F (11) nmax −1 n=2 In this section, we analyse the statistical properties and the discrete entropy of F . entropy of the QPSM from the perspectives of the statistical histogram, information entropy, and discrete entropy. First, B. Bifurcation Diagrams and Statistical Properties of a several definitions of entropy are introduced. Quadratic Polynomial Map Based on Eq. (12), suppose that ab=−=1.8, 2.8 . Then, c = 0.8 , and we can obtain a specific quadratic polynomial
ISSN: 1998-4464 102 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 map: 18000 2 (a) f() x= ax ++ bx c 16000
(13) 14000 2 1 17 =−1.8xxx + 2.8 + 0.8, ∈− , 39 12000 This map evolves chaotic motions through period-doubling 10000 bifurcation while the system parameters vary. Figs 1(a)–(c) Count 8000 show the bifurcation diagrams. 6000 4000
2000
0 0 0.5 1 1.5
2000 (b)
1500
1000 Count
500
0 -0.5 0 0.5
Fig. 2 histogram: (a) histogram of the sequence before uniformity; (b) histogram of the sequence after uniformity. D. Entropy of the QPSM In order to calculate the information entropy of an ergodic sequence with the length of N , we can divide the range of values into M equal intervals and count the number of samples
that fall into each interval. This is denoted as nii ( =1, 2, , M) . Therefore, the probability of each interval can be considered as M pii= nN/,∑ p i= 1. According to the maximum information i=1 entropy principle, the maximum information entropy is
log2 M . Here, we let N = 600000 and calculate the information entropy of the original and homogenized sequences for the different M values. Table 1 lists the results. Table. 1 Information and maximum entropies before and after uniformity Before After Maximum ()NM, Fig. 1 bifurcation diagrams: (a) the bifurcation diagram of uniformity uniformity information parameter a in system (12); (b) the bifurcation diagram of (600000,100) 6.3508 6.6438 6.6439 parameter b in system (12); (c) the bifurcation diagram of (600000,200) 7.3342 7.6437 7.6439 (600000,300) 7.9124 8.2285 8.2288 parameter c in system (12) (600000,600) 8.6412 8.9652 8.9658 C. Statistical Histograms of the QPSM For all M values, the information entropy of the The histograms of the sequences generated by both the homogenized sequence is close to the maximum entropy original and homogenized maps are shown in Fig. 2 (a) and (b), log2 M , which suggests that the homogenization method is respectively. The distribution of the homogenized chaotic effective. sequence can clearly be observed to be approximately uniform. The concept of discrete entropy proposed by Kocarev in 2007 is used to measure the chaotic extent of a sequence by considering the permutations of successive nn()≥ 2 points. In this paper, we let n = 9 . Then, we can calculate the discrete
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entropies of the original and homogenized sequences with The DE before and after uniformity 1 respect to the different coefficients, and compare then with the (d) K entropy. 0.5 Let a ∈−[]1.8,0 . The homogenized system can be written as 0 xn()()()+=1 axx2 + 2.8 xn + 0.8 DE before uniformity 1 ab (14) -0.5 zn()+=1 arcsin −xn() − y=0 after uniformity π 24 -1 The K entropies (i.e. positive Lyapunov exponent) and -1.5 discrete entropies of the original and homogenized sequences -1.8 -1.6 -1.4 -1.2 -1 -0.8 are shown in Figs 3(a) and 3(d). Similarly, let
∈∈ bc[][]1.8,2.8 , 0,0.8 . Fig. 3 shows the simulation results. As The DE before and after uniformity 1 shown in Figs 3(a), (b), and (c), the discrete entropy of the original sequences approximated the K entropy with a 0.5 constant offset. To some extent, the discrete entropy can be used (e) to measure the chaotic extent of a system. As shown in Figs 0
3(d), (e), and (f), the discrete entropies of the original and DE -0.5 homogenized systems are identical. The simulation results show before uniformity y=0 after uniformity that the uniformity of the distribution of the homogenized -1 system was improved with the same chaotic property. The KE and DE of the original map 1 -1.5 1.8 2 2.2 2.4 2.6 2.8
(a) 0.5 The DE before and after uniformity 1 0
0.5 (f)
KE & DE -0.5
0
-1 KE y=0 -0.5 DE DE before uniformity -1.5 -1 -1.8 -1.6 -1.4 -1.2 -1 -0.8 y=0 after uniformity -1.5 The KE and DE of the original map 1 -2 0.3 0.4 0.5 0.6 0.7 0.8
0.5 (b) Fig. 3(a), (b), (c) K entropy and discrete entropy before 0 uniformity; (d), (e), (f) discrete entropy of the before and after uniformity
KE & DE -0.5
KE -1 V. CONCLUSION y=0 DE In this paper, we propose a method for determining the -1.5 1.8 2 2.2 2.4 2.6 2.8 coefficients of the QPSM and prove that the QPSM is topologically conjugate to the tent map. We then derive an
The KE and DE of the original map analytical expression of the PDF of the QPSM and present an 1 approach to homogenize the QPSM. The information entropy, 0.5 K entropy, and discrete entropy of the QPSM were analysed (c) and simulated. The theoretical results showed that the 0 information entropy was highest for the sequence generated by -0.5 the homogenized system, and the discrete entropy remained the
KE & DE same. The simulation results show that the discrete entropy is -1 KE similar to the K entropy with a constant offset. In other words, -1.5 y=0 DE homogenization improves the information entropy of the QPSM
-2 while maintaining the chaos properties unchanged. 0.3 0.4 0.5 0.6 0.7 0.8 The construction and optimization methods of chaotic system proposed in this paper can be used to generate noise-like
ISSN: 1998-4464 104 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 sequences, which can be applied to communication and radar aeras. Jun Tang received the B.S and M.S. degree in communication and information system form Lanzhou University of Technology and Xidian University, China, in 2000 and 2007 respectively. He is now pursuing Ph.D. REFERENCES degree in communication and information in Xiamen University. He has been in Xiamen University of Technology since 2007 and is currently an assistant [1] A. L. Goldberger, "Non-linear dynamics for clinicians: chaos theory, professor with the School of Photoelectricity and Communication Engineering. fractals, and complexity at the bedside," Lancet, vol. 347, pp. 1312-1314, His research interests include chaotic signal processing, radar signal processing 1996.K. Aihara, "Chaos and Its Applications," Procedia Iutam, vol. 5, pp. and multimedia signal processing. 199–203, 2012.
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