4 47 Lorenz [9-11]. M.. [16]... 1 [16] ẋ 1 = a( ) ẋ 2 = b + c (1) ẋ 3 = h + 1 a, b, c, h. a = 2, b = 14, c = 1.6, h = Lyapunov L 1 = , L

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1 JOUNAL OF APPLIED SCIENCES Electronics and Information Engineering Vol. 28 No. 4 Jul. 21 DOI: /j.issn ,2 1, Lyapunov. Lyapunov.... Lyapunov TP271 TN (21) Circuit Design as Implementation of Four Dimensional Hyper-chaos and Its Projective Synchronization LIU Ming-hua 1,2, FENG Jiu-chao 1,3 1. School of Electronic and Information Engineering, South China University of Technology, Guangzhou 51641, China 2. College of Mathematics and Physics, Jinggangshan University, Ji an 3439, Jiangxi Province, China 3. Guangzhou Auto College, South China University of Technology, Guangzhou 51641, China Abstract: We obtain a four-dimensional system by adding a nonlinear state feedback controller to a threedimensional chaotic system. The dynamical behaviors of this 4D system are investigated, including the Lyapunov exponent spectrum and bifurcation diagram. The results show that it is a 4D hyperchaotic system with two positive Lyapunov exponents. Projective synchronization for the 4D hyperchaotic system is realized based on the stability criterion of linear systems. An electronic circuit is designed and implemented with projective synchronization of the 4D hyperchaotic system. The circuit consists of four parts: anti-adders, integrators, inverters and multipliers. Experimental observations are in agreement with numerical simulations. Keywords: hyperchaos, Lyapunov exponent,, circuit experiment össler [1] Lyapunov [2-4].. össler [1] Chua [5]. [6-15]. Chen Chen [7] L L [8] Lorenz (No ) (No.U8351) liuminghua2@sina.com fengjc@scut.edu.cn

= d + d. (1) (2), 1. a = 2, b = 14, c = 1.6, h = 2.

4 2 2 4 3 2 1 1 2 3 7 6 5 4 3 2 1 7 6 5 4 3 2 1 2 1

2 4 47 Lorenz [9-11]. M.. [16]... 1 [16] ẋ 1 = a( ) ẋ 2 = b + c (1) ẋ 3 = h + 1 a, b, c, h. a = 2, b = 14, c = 1.6, h = Lyapunov L 1 = , L 2 =, L 3 = ) 4 2) 3) Lyapunov Lyapunov. 3 (1) ẋ 1 = a( ) + ẋ 2 = b + c (2) ẋ 3 = h + 1 ẋ 4 = d + d. (1) (2), 1. a = 2, b = 14, c = 1.6, h = 2.8, d = 1.2 [17] Q Lyapunov L 1 = 1.42, L 2 =.273 1, L 3 =, L 4 = Lyapunov 4 Lyapunov (2) x2 1 d = 1.2 (2) Figure 1 Phase portraits of hyperchaotic attractors of system (2) with d =

48 28 2 (2) 2.1 (2) (,,, ) (,,, ) (2). 2.2 (2) V = ẋ 1 + ẋ 2 + ẋ 3 + ẋ 4 = a + c h + d a + c h + d < (2) e (a c+h d)t. t. 2.3 ẋ 1 = ẋ 2 = ẋ 3 = ẋ 4 =, (2) 3 O(,,, ), ( adh(b + c) P +, b + ad adh(b + c), ad(b + c) + b) adh(b + c) ), a(c, ( adh(b + c) P, b + ad adh(b + c), ad(b + c) a(c + b) adh(b + c) ),.

3 (2) 2.1 (2) (,,, ) (,,, ) (2). 2.2 (2) V = ẋ 1 + ẋ 2 + ẋ 3 + ẋ 4 = a + c h + d a + c h + d < (2) e (a c+h d)t. t. 2.3 ẋ 1 = ẋ 2 = ẋ 3 = ẋ 4 =, (2) 3 O(,,, ), ( adh(b + c) P +, b + ad adh(b + c), ad(b + c) + b) adh(b + c) ), a(c, ( adh(b + c) P, b + ad adh(b + c), ad(b + c) a(c + b) adh(b + c) ),. O (2) Jacobian a a 1 b c J O = (3) h d det(j λi) =, f(λ) = (λ d)(λ + h)(λ 2 + (a c)λ ac ab) = (4) λ 1 = d, λ 2 = h, f(λ) = λ 2 + (a c)λ ac ab = (5) a c > ac ab > (6) a = 2, b = 14, c = 1.6, h = 2.8. O (2) (2) 4 Lyapunov L i (i = 1, 2, 3, 4) L 1 L 2 L 3 L 4. a, b, c, h ( a = 2, b = 14, c = 1.6, h = 2.8) d [ 3, 1.3) d Lyapunov 2 (a) (b). L 1,2,3 L d L 1 L 2 d L 3 L 4 (a) Lyapunov (a) Lyapunov exponents spectrum d (b) (b) Bifurcation diagram 2 (2) Lyapunov Figure 2 Lyapunov exponents spectrum and bifurcation diagram of system (2) (2) Lyapunov. (2) d 1) d [ 3 2.4] L 1 =, L 2,3 <, 2) d

4 4 49 ( ] L 1 >, L 2 =, L 3,4 <, Lyapunov 3) d ( ] L 1,2 >, L 3 =, L 4 < Lyapunov. 3 (2) (2) (7) ẋ 1 a ẋ 2 b k = 1 + ẋ 3 h x 3 ẋ 4 k 2 a + (c + k 1 ) x 1 (7) (d + k 2 ) + ẏ 1 a ẏ 2 b k = 1 ẏ 3 h ẏ 4 k 2 a + 1 (c + k 1 ) α x 1 (d + k 2 ) + y 1 y 2 y 3 y 4 + (8) t e 1, e 2, e 3, e 4 lim t e i = (i = 1, 2, 3, 4),. a = 2, b = 14, c = 1.6, h = 2.8, d = 1.2, k 1 = 2, k 2 = 4 4 (1., 1., 1., 1.) (.2,.2,,.2).1. α = 2 α = 2, 3.., y 1, y , y 5 2 1, y 3 (a) (a) Phase portrait of same-phase 5 x y 2, 2 1 5, y k 1 k 2 α(a ). e i (t) = x i (t) αy i (t)(i = 1, 2, 3, 4), ė 1 a e 1 ė 2 b k = 1 e 2 (9) ė 3 h e 3 k 2 e 4 ė 4 e 1 (t) = m 1 e at, e 2 (t) = bm 1 a e at + m 2 e k1t, e 3 (t) = m 3 e ht, e 4 (t) = m 4 e k2t, m 1, m 2, m 3, m 4. a, h k 1, k 2 e (b) (b) Phase portrait of anti-phase e 1 e 2 e 3 e t (c) (c) Error curve of same-phase

5 41 28 e e 1 e 2 e 3 e t (d) (d) Error curve of anti-phase 3 α = 2 α = 2 Figure 3 Simulation results of with α = 2 and α = 2 4 [18-19] (7) (8) 4 α 2.. TL82CP AD633JN ±15 V. ( ). 4. [2] α = k 5k 1k f 1k 5k C - 1k 1k y 1 5k 1k 2k f 1k 5k C -y 1 1k 1k y 1-7.1k 5k 7.9k.4k.1x f 1k C 5k 1k 1k -y 1 y 2 7.1k 5k 15.9k.8k.1x f 1k C 5k -y 2 1k 1k y k.4k -.1x f 1k 5k C 1k - 1k y k.8k f 1k 5k C -y 3 1k 1k y 3-25k 19.2k 1k.4k x f 5k C 1k 1k y 4-25k 38.5k.8k -.1x f 1k 5k C -y 4 1k 1k y 4 (a) (a) Circuit diagram of drive system 4 Figure 4 Experimental observations of hyperchaotic attractors (b) (b) Circuit diagram of response system

hyperchaotic attractors (a) x4y4 ᗕব䞣ৠℹᯊᑣ

synchronization (b) x4y4 ᗕব䞣ৠℹⳌ (b) Phase

6 刘明华等四维超混沌系统及其投影同步的电路实现第4期 (a) x1=.5 V/Ḑ, x2=.5 V/Ḑ (b) x1=.5 V/Ḑ, x3=.5 V/Ḑ (c) x1=.5 V/Ḑ, x4=2 V/Ḑ (a) x1=.5 V/div, x2=.5 V/div (b) x1=.5 V/div, x3=.5 V/div (c) x1=.5 V/div, x4=2 V/div (d) x2=.5 V/Ḑ, x3=.5 V/Ḑ (e) x2=.5 V/Ḑ, x4=2 V/Ḑ (f) x3=.5 V/Ḑ, x4=2 V/Ḑ (d) x2=.5 V/div, x3=.5 V/div (e) x2=.5 V/div, x4=2 V/div (f) x3=.5 V/div, x4=2 V/div 图 5 电路实验中观察到的超混沌吸引子 Figure 5 Experimental observations of hyperchaotic attractors (a) x4y4 ᗕব䞣ৠℹᯊᑣ (a) Timing sequence diagram of x4y4 synchronization (b) x4y4 ᗕব䞣ৠℹⳌ (b) Phase portrait of x4y4 synchronization 图 6 α = 2时同相投影同步实验结果 Figure 6 Experimental observations of the same-phase with α = 2 (a) x4y4 ᗕব䞣ৠℹᯊᑣ (a) Timing sequence diagram of x4y4 synchronization (b) x4y4 ᗕব䞣ৠℹⳌ (b) Phase portrait of x4y4 synchronization 图 7 α = 2 时反相投影同步实验结果 Figure 7 Experimental observations of the anti-phase with α = 2 411

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1 2 EI ( 2005) 3 EI ( 2005) 4 2006 3 1 2 3 EI ( 2005) http://www.ei.org.cn/twice/coverage.jsp ISSN 0567-7718 Acta Mechanica Sinica/Lixue Xuebao 1006-7191 Acta Metallurgica Sinica (English Letters) 0253-4827