1 : : : : 991 : : : 3.3.31 3.6.15

3 4.1 4. 4.3 1 11 3.1 11 3. 11 3.3 1 3.4 13 15 4.1 15 4. 17 8 5.1 8 5. 8 5.3 3 5.4 3 5.5 31 3 3 3

3 (1) () (3) (4) Henon Logistic (5) Chaotic Synchronization of Discrete System and Its Application in Secure Communication Abstract In this paper, some problems of chaotic synchronization and its application in communication are studied on the bases of the theory of nonlinear dynamics. In recent years, making use of the random-like characteristic of the chaotic signal to realize secure communication becomes a hot point in the chaotic application study. In the process of completing this thesis, for further understanding the concept of chaos and its application and development, particularly the application in communication, we make some research works below: Firstly, the conceptions and definitions of chaos are summarized; Secondly, We research into the basic theory of the chaotic synchronization on the bases of the stability theory of dynamical system, and introduce some methods of synchronous at the same time; Thirdly, We summarize the applications in the communication, make comparisons to the modern communication and chaotic communication; Fourthly, We study the direct inverse system, and make research to the Henon system and the Logistic system. We design a synchronization method to retrieve the chaotic carrier to recover the information signal. And its secret ability is increased by making use of the modular operation. The results of our study shows that the chaotic synchronization method has the good immunity to interference and can be realized easily. It also shows that the chaotic secure communication system is fine in security. Finally, Results of the paper. Key words chaos discrete dynamical system synchronization secure communication

4 Internet ogistic 5 A.L.Hodgkin A.F.Huxley Aihara Hodgkin Huxley 6 8. Lorenz 1864 1983 1 Lorenz NASA ISEE 3/ICE ( ) Christini Poon ( )

5.1 1963 Lorenz << >> chaos ( ). 1975 Li Yorke 3 Li-Yorke.1 f : I I R, I R (1) f I Per( f ) () ), S I \ Per( f X 1 X S( X 1 X ) lim t t sup f ( X 1) f ( X ) > t t t inf f X ) f ( ) = lim t ( 1 X f t ( ) = f ( f ( f ( ))) t (3) X 1 S f P I lim t t sup f ( X 1) f ( P) > t f S.1 f :[,1] [,1],

6 x, f ( x) = x, x < 1/ ; 1/ x < 1. x = 6 / 7 f ( x ) = / 7, f ( x ) = 4 / 7, 3 f ( x ) = 6 / 7, x = 6 / 7 f 3 Li-Yorke 3 I I S I I S F F Per( f ) x [,1] k x. a1a a k = = a k /, a k {,1} k = 1 f f ( x) = k = 1 a. aa =. aa 3 3 k 1 a k + 1 / a a k k, a 1 =, a = 1 a = 1 a m k k x 1 =. a1a aka1a ak a1a ak, 1 k k x x < 1 /, i = 1,, m k i f [ 1],1 x =. a a a x =. a a a 111. f k ( x) =, + 1 f k ( x) =, x =. a1a a a1 a a a1a a a1 a a, 1 k 1 k x k k k Li-Yorke 1.. S () (3) S () 3. (3) S ( ] k

7 () 4. Li-Yorke S S 1964 Sarkovskii 3 5 7 9 3 5 7 3 5 7 9 3 n 5 n 7 n 9 n m 3 16 8 4 1 f p p q q Sarkovskii 5 7 n Li-Yorke Sarkovskii 3.1 1976 P.Kloeden Li-Yorke 1989 Devaney. X f X X (1) f () f X (3) f f Devaney (1) Devaney () ( Devaney (3) Devaney (1) () ( ). Devaney () lorenz =1 =8/3 =8. Lorenz =1 =1/3 =5 () =1 =8/3 =8 Devaney

8.3 f. (1) (3) f Devaney (1) Smale () kolmogorov (3) (4) Lyapunov (5) (6) (7) (8) Shil nikov.4 f( ) S (1) S f () F S (3) x S { } f k ( x) k z + S (4) f (1) ( ) Koch Sierpinski () S (3) S (4) Li-yorke S f ( ) (1) () (3) (1) () (3)

9 (4) (5) (6) (7) (1) 7. x' = ryz y' = rxz z' = z r = x + y x( t) = x cosu y y( t) = y cosu + x z( t) = z expt sin u sin u u = r z (expt 1), r = x + y () Lorenz.5 f : s s x s x n n U y U n > d( f ( x) f ( y)) > f.3 x ' = x x t) = x expt ( t = ln f : R R,

1 f ( x) = x x R δ =1 ε > y x y < ε x y n ε n δ n ε =.4.4 x' = x y' = x + y x1, x x < ε n 1 x ε.4 ε + nα (.5) Lyapunov Lyapunov ( ) Lorenz f ( ) Chua

11 Tom.3 ( ) Lorenz Dufing. x = F( x n+ 1 n ) x xn x n+ 1 F Logistic Henon

1 3.1 ( ) ( ) 1963 Lorenz 199 Pecora Carroll 3. n X R ; t b >, X x ' = F( t, X ) (3.1) + + n n t R, R F :[ R R + ] R S S = {( t, X ) t t, X X b} (3.) ( t, X 1),( t, X ) S F(t X) Lipschitz K> (3.1) X t; t X ) F( t, X K X X (3.3) (, 1 ) F( t, X ) 1 X ( t = X 3.1 F(t X) X t; t, X ) ( (1) b >, 1 t t ) X t; t X [ t X ( t) ] S ) (, 1, t t X 1 X 1 X b1 () ε > δ ( ε, F, X ) < δ b1 X 1 X δ (3.4) X ( t; t, X 1) X ( t; t X ) ε t t (3.5)

13 3. (1) F(t x) X t; t, X ) ( () δ = δ ( F, X ), < δ b1 X 1 X δ (3.6) X t; t, X ) X ( t; t, X ) t (3.7) ( 1 (3.1) X t; t X ) (, 3.3 R n D( t ), ), X D ( t X t; t, X ) D( t ) ( (3.1) D( t ) R n (3.1) X t; t, X ) ( 3.4 (.1) t V(X t) X D( t ) (1)V(X t) D( t ) X t () V '( X, t) D t ) X t ( (3.1) V(X t) Lyapunov D( t ) X V (X ). (3.1). V '( X, t) V '( X, t) = ( ) 3.3 3.5 x ' = F( t, X ) (3.8) y ' = G( t, Y ) (3.9) n X, Y R t X ( t; t, X ) Y ( t; t, Y ) n R

14 D ( t ), X Y D( ), t X t; t, X ) Y ( t; t, Y ) t (3.1) ( F( t, X ) G( t, Y ) D( t ) D( t ) n R ) ( t, X F G( t, Y ) F( t, X ) G( t, Y ) F = G F( t, X ) G( t, Y ) F G F( t, X ) G( t, Y ) Lyapunov 3.4 Lyapunov ( ) 3.4 D-B ( ) Pecora Carroll Lyapunov Kocarev Parlitz D-B 1995 D-B

15 Pecroa-carroll Lyapunov Lyapunov ( ) Lyapunov

16 ( ) M M Gold M Gold Logistic δ 1/N 4.1 Feldmann 4.1 y ( t) = E( u( t), X ) (4.1) ' 1 u ( t) = E ( y( t), Z) (4.) E( ) E ( ) X Z 1 X Z. u u ' y ( ) u (4.1)

17 (4.) 4.1.1 E ' 1 1 u = E ( y, Z) = E ( E( u, X ), Z) = u (4.3) (4.3) X = Z Logistic x( n + 1) = ax( (1 x( ) (4.4) y( n + 1) = u( x( (1 x( ) (4.5) ' x( n + 1) u ( t) = (4.6) x( (1 x( ) u( t) 4 ) u(t 4. Frey y ( n + ) = F[ ax( n + 1) + bx( + u( ] (4.7) u ( t) = F[ x( n + ) ax( n + 1) bx( ] (4.8) a, b R F( x) = ( x + 1) mod 1 u( t) [ 1,1 ) u ( t) = u( t) (4.5) (4.6) y ( n + 1) = [ ax( (1 x( + u( ]mod1 (4.9) u ( = y( n + 1) ay( (1 y( ) (4.1) ( ) 4.1. X ( t) Z( t) t

18 x(t) Z X Z X 4. Oppenheim Pecora Carroll Lorenz Logistic Henon 4..1 Logistic x( n + 1) = µ x( (1 x( ) (4.11) x( 1 µ 4 Lyapunov 3.58 µ 4 µ * = 4 { } s( x '( n + 1) = 4x( (1 x( ) + s( (4.1) x' ( n + 1) > 1 x( n +1) x' ( n + 1) 1 (4.1) (4.13) x ( n + 1) = x'( n + 1) mod1 (4.13) x ( n + 1) = [4x( (1 x( ) + s( ]mod1 (4.14)

19 (4.14) Logistic Logistic x( n + 1) = 4x( (1 x( ) (4.14) x ( m) = x( ( m > s ( m) = s( x( m + 1) x( n + 1) (4.14) (4.14) x' ( n + 1) x '() = x() < x() < 1 { x'( } y( = x'( mod1 = x( z( n + 1) = 4y( (1 y( ) = 4x( (1 x( ) (4.15) z( n +1) x' ( x' ( n + 1) s ^ ( = x'( n + 1) z( n + 1) = s( (4.16) s^( 4. Logistic (4.15)

(4.14) 4.. Henon x( n + 1) = 1 ax ( + y( (4.17) y( n + 1) = bx( 4 a =.3,1.8 b 1. Henon a =.3, b = 1.4 1.5 x 1.5,.4 y. 4 (4.17) Henon x ( n + 1) = 1 1.4x ( +.3x( n 1) (4.18) x( ), x(1) [ 1.5,1.5 ] (4.18) x'( n + 1) = 1 1.4x ( +.3x( n 1) + s( x'( n + 1) mod1.5 x'( n + 1) x( n + 1) = [( x'( n + 1))mod1.5] x'( n + 1) < x'( mod1.5 y( = [( x'( ) mod1.5] z( n + 1) = 1 1.4y ( +.3y( n 1) x'( x'( < (4.19) (4.) s ^ ( = x'( n + 1) z( n + 1) (4.1) Henon 4.3 Henon

1 4..3 Logistic Henon Logistic u=4; Henon a=.3 b=1.4 1. Hz 4.4 Logistic 4.5 4.6 4.4 Logistic Matlab 4.5 Logistic

4.6 Logistic 4.7 Logistic

3 4.8 4.9 4.1

4 4.11 Henon Matlab 4.1 Henon Matlab 4.13 4.14 4.11 Henon 4.1 Henon Matlab 4.13 Henon

5 4.14 Henon 4.15 Henon 4.16 Henon

6 4.17 Henon 4.18 4.19 4..4 x ( n + 1) = f ( x(, µ ) x(, x( n + 1) I (4.) I [, a] I [ a, a] b µ c (4.)

7 x'( n + 1) = f ( x(, u) + u( t) x'( n + 1) mod a x( n + 1) = [( x'( n + 1)) mod a] x'( mod a y( = [( x'( ) mod a] s^ ( = x'( n + 1) f ( y(, µ ) x'( n + 1) x'( n + 1) < xn ' x ' < n Henon x( n + 1) = 1 ax y( n + 1) = bx( ( + y( (4.3) (4.4) (4.5) x ( n + 1) = 1 ax ( + bx( n 1) (4.6) ( ) Takens Takens Frey y( y( n +1) u(t) a b n 1 + b c i x i (4.5) (4.6) = i y( y( n +1) ci (

8 ) ( ) (

9 5.1 1 13 BW (BW) (BW) (BW) ( ) (1) DES IDEA DES () (3) RAS DSA Fiat-Shamir Diffie-Hellman 1.. 3. 4. 5. 9 Pecora Carroll Oppenheim Robust Oppenheim

3. 5..1 m m p m p=15( ) 1111 11 11 1111 11 11 1111 11 11 p m m ( ) m 5.1 m X m Y E E m Y X 5.. m m m 4 (1) m () (3) (4) M p 5..3

31 5.3 ( ) 1.. (1) () (3) (4) CDMA CDMA (m ) Gold N=55 m 16 m m δ. 5.4 khz ( MHz GHz) MHz

3 5.5.?

33 3! [1],. 1993. [],. 1994. [3] Ian Stewart Does God Play Dice? The Mathematics of Chaos. Basil Blackwell 199. [4],.. [5]. 1. [6], MATLAB. 1. [7], MATLAB. 1. [8], Simukink.. [9],. 33(1999)55. [1],. 3()5. [11],. 4(3)44. [1],. 17()37. [13],. 19(1998)47. [14],. ( ) ()6. [15],. 9()6. [16],. (1)1. [17] S.Boccaletti.etal, The Synchronization of chaotic Systems. Physics Reports 366() 1-11.