1

. , ,

:

.

. xn+1 =xn2 +a

Chua.

, Hopf,

( )

..

.

1.

. . (York 1975) 1, , , , . (state-space), Hamilton Jacobi, ( ) (Poincare, Andronov, Hopf) 2. 1960 Edward Lorentz, 3, 70 Feigenbaum Los Alamos (). ( ) 4. , . .( Chua, Colpitts, Van der Pol)

5 , ,

.

: ; ; ; ; . Poincare-Bendixon,

x = (x) , t () {f(x)=0}, ) , ) {x(t)=x (t+T)}6,7.

,

. ? (nonperiodic)

(nonexplosive) , , ,

(nonpredictable)! Sheinerman 8.

2

. , , . . : : (1) H 9, (2) H (intermittency)

10, (3) (- ) 11.

2.

12 , , ()

() (continuos time) (discrete time)

x = f (x) [ x =f(x,y) y =g(x,y)] x (k+1)= f (x(k)) xk+1 =f (xk)

x(t0)=x0 x(k0)=x0

, x=[x1,x2, xn] , f g t (x0,y0), k : 0,1,...n.

(fixed point) x

f( x) = x (1)

x (stable), f ( x ) 1 (2)

T x (unstable), f ( x ) 1 (3)

.

x(k)=f k (x) (4)

f k (x)

f k

(x) =f[f[f[..f[x]..]]] k (5)

f , k, x, f[x], f[f[x]]......

x

f k( x ) = x (6)

k (7) x.

f 2k

(x) = f k

[f k (x)] =f

k (x)= x (7)

, x k, 2k, 3k,

4k, .

x (prime period), k

(7) x.

4. . f(x) x, f 2 (x) x , f 3 (x) x, f 4 (x) = x.

Sarkovskii13, f 3, f

k k.

3

p, p. A

f (p) 1 p, {p, f(p), f2(p),

f3(p),...f

k-1(p)}. A . f (p) 1, {p, f(p),

f2(p), f

3(p),...f

k-1(p)}

( ), 14.

3. f[x]=x2+a

3.1 ()

f(x)= x2+a .

: 1) o (fixed

points) f(x)=x , 2) (. f(x) ).

x2+=x (1+ 1-4)/2 (1- 1-4)/2,

1/4. 2x f(x) .

1+ 1-4 1, ,

1- 1-4 1, -3/4

1/4. , 1-4 2. x (x ) , =1/4 ( .1)

, =1/4 , 1/4.

(bifurcation), f(x) =1/4

y=x (saddle node bifurcation).

, (fixed points) ,

(stable) : -3/4 1/4

3.2.

, -3/4,

, . = -3/4 .

3/4 1, 2.

1.5 1 0.5 0 0.5

2

0

2

2.0

2.0

y1

i

z1

i

y2

i

z2

i

0.51.5 a

i

4

, : ? Mathcad xn+1 =xn

2 +a

n (. 2), x0 = -0.5 = -0.8.

(n 100) xn+1. K Nest [f,-0.50, n] Mathematica x0 n=110 -0.276502, n=111 0.723547, n=112 0.27648, n=113 0.723559. , . n=1000 0.276393, n=1001 0.723607, n=1002 0276393, n=1003 0.723607.

x0 = -0.5, = -0.8 n , xn+1

.

f 2k (x) = f k [f k (x)] =f k (x)= x. f[f[x]]=x f[f[-0.276393]]=f[-0.723607]=-0.276393,

x=-0.276393 k=2.

2k, 3k, 4k, . 0.723607. ,

= -3/4

(period-doubling pitchfork bifurcation). 2, f[f[x]]=x.

(x^2+)^2+ = x

(1+ 1-4)/2, (1- 1-4)/2, (-1+ -3-4)/2 (-1- -3-4)/2.

f[x]=x. 1 2 2, ,

-3/4.

1, 2 , f[f[x]] . 1, f[1]=2 f[2]=1,

f[f[x]] (1)=f [f[1]]*f [1]=f [2]*f [p1]=22*21=(-1- -3-4)* (-1+ -3-4)=4+4

4+4 1 -5/4 -3/4 2, .

0 50 100 150

0.6

0.4

xn

n

0.6 0.4

0.6

0.4

xn 1

xn

5

3.3. 4

1 2 5/4 4, 2 2 .. xn+1 =xn

2 +a

6

( n=1000:1017 x0 =0.50)

[0.447814, -1.19046, 0.0262008, -1.39031,

0.541972, -1.09727, -0.187006, -1.35603,

0.447814, -1.19046, 0.0262008, -1.39031,

0.541972, -1.09727, -0.187006, -1.35603,

0.447814, -1.19046]

( n=1000:1020 x0 =0.50)

[0.444978, -1.19599, 0.036404, -1.39267,

0.545543, -1.09638, -0.191945, -1.35716,

0.447876, -1.19341, 0.0302208, -1.39309,

0.546691, -1.09513, -0.194692, -0.194692,

-1.3561, 0.444994, -1.19598,0.036368,-1.39268]

=-1.40 ( n=1000:1017 x0 =0.50) xn+1 32 f 32

(x)= x

[0.46702, -1.18189, -0.00313082, -1.39999, 0.559973, -1.08643, -0.219668, -1.35175,

0.427217, -1.21749, 0.0822715, -1.39323, 0.541094, -1.10722, -0.174069, -1.3697,

0.476078, -1.17335, -0.0232501, -1.39946, 0.558487, -1.08809, -0.216055, -1.35332,

0.431476, -1.21383, 0.073379, -1.39462, 0.544952, -1.10303, -0.183332, -1.36639,

0.46702, -1.18189]

2, 4, 8, 16, 32, 64 . ( = 1.4)

. = -1.5 = -2

. :

=-1.75

0.5, -1.5, 0.5,- 1.5, 0.5

2

=-1.76

1.3356, 0.0238308, -1.75943,

1.3356, 0.0238308, -1.75943,13356

3

=-1.77

0.0828164, -1.76314, 1.33867, 0.0220113,

-1.76951, 1.36118, 0.0828164, -1.76314, 1.33867

6

=-1.63

1.01663,-059646,-1.27424,-0.00632493,

-1.62996,1.02677,-0.575744,-1.29852,

0.0561505,-1.62685,1.01663,-0.59646,..

10

=-1.48

0.706669,-0.980619,-0.518386,-1.21128,

-0.0218103,-1.47986,0.709914,-0.976022,-0.527382,

-1.20187,-0.0355118,-1.47874, 0.706669,-0.980619..

12

3.4. x0

>-2 , , x0

[-2, 2] f (x) n

[-2, 2].

7

f(x) . f . , , f 15

a = -1.95 x0= -0.50 xn+1 = xn

2 +a ( ),

xn+1 .

(pattern), ( ) . . x0 , .. [-0.50 -0,50001] f[-0,5]-f[-0.50001] (..20) .

, xn+1 ( ) f (x) ( )

x. (.4) n 1000 1019 .

( n, ) , - ( Lorenz). . , / n, ! ( Cantor), .

4. Hopf

4.1.

x1 = x1 + x2 - x13

x2 = - x1 (8)

x: f[-0,5]-f[-0,50001]

20 n=[1000,1019]

-4

-2

0

2

4

1 4 7

10

13

16

19

8

x =[ x1 x2 ] T

, (fixed points) f(x)=0.

x = 0. Df =[Df1 Df2]

T,

Df1=[ f1/ x1 f1/ x2]

Df2=[ f2/ x1 f2/ x2]

Df(0), 1/2*(1+i 3), 1/2*(1-i 3),

, . f( x )=0, .

V(x)=x12 + x2

2, V(x) 0, x 0.

V(x) Lyapunov, 0 . ?

dV(x)/dt = ( V/ x1)( x1/dt)+ ( V/ x2)( x2/dt) = 2x1

2 (1-x1

2 ) (9)

dV(x)/dt 0. x1 1. , x1

, 0. 0 .

. , x(t) , x1. 0 . . 16.

4.2.

? (8),

ax0 ., x=0.

? , ,

1/2*(+ 2-4), 1/2*(- 2-4). , ,

x=0 . , 0 . Lyapunov 2x1

2(a-x1

2). a

, Lyapunov, 0, . , V(x) Lyapunov, . , Poincare-Bendixson, . . . (0.5, 0.5) , 0. , ., - + , 0, , =0 1,

. 0 ,

, Hopf. Hopf . , . , Hopf =2, . (torus), .17

4.3. . .

9

. Mathcad ( Runge-Kutta), [0.0001,0.0031] x n=1000 0 40.

[x1 , x2] , [zn,1, zn,2],

[zn,1, zn,2]. 0 ( x0),

(5):

5. CUA:

.

Hopf . Hopf , ( Van der Pol, Colpitts

18,19

.) Chua 20, 21, 22, 23, 24. , , . 25.

m 0.1 x0.5

0.5n ..0 500

D( ),t x

.m x0

x1

x0

3

x0 Z rkfixed( ),,,,x 0 20 500 D

0 200 400

0

1

Z,n 2

n

0 200 400

0

1

Z,n 1

n

0.5 0 0.50.5

0

0.5

Z,n 1

Z,n 2

10

, , 26. 27:

x=c1*(y-x-g(x)) (10) y=c2*(x-y+z) (11) z= -c1*y (12)

g(x)=m1*x+0.5*(m0-m1)*( x+1 - x-1 )

H g(x) - . c1, c2, c3, m0, m1 . . Chua Mathcad.( [x,y,z] [x0, x1, x2]) . [1, 1, 1] [0, 0, 1]. [x0, x1, x2] ( [z1, z2 ,z3]) .

[1,1,1]

. 0.0000002 c3, . , (.6, .7). c3=41.90 o , c3=42.90 , c3=43.90 , c3=44.90 , c3=45.90 , c3 =46.90 , c3 =47.90 , c3 =47.90 . c3 =44.90 c3 =47.90 . . H .

c1 15.6 c2 1.0 c3 64.4683668m0

8

7m1

5

7

g( )x .m1 x0

.m0 m1

2x0

1 x0

1x

1

1

1

n ..0 25000

D( ),t x

.c1 x1

.c1 x0

.c1 g( )x

.x0

x1

x2

c2

.c3 x1

Z rkfixed( ),,,,x 0 40 25000D D1

Z,n 1

D2

Z,n 2

D3

Z,n 3

11

0 5000 1 104

1.5 10410

0

10

Z,n 3

n

2 0 2

0Z ,n 1

Z,n 2

5 0 52

0

2

Z,n 2

Z,n 3

2 0 210

0

Z,n 3

Z,n 1

3 2 1 0 1 2 34

3

2

1

0

1

2

3

Z,n 3

Z,n 1

12

0,0000002 c3 ( c3=64.4683666 (. 7).

[0, 0, 1]

c3 , (1) (c3=50), (2) (c3=33.8), (3) (c3=33), (4) (c3=25.59)`.

3 2 1 0 1 2 34

3

2

1

0

1

2

3

Z,n 3

Z,n 1

13

1 0.5 0 0.5 1 1.5 2 2.5 34

3

2

1

0

1

2

Z,n 3

Z,n 1

1 0.5 0 0.5 1 1.5 2 2.5 34

3

2

1

0

1

2

Z,n 3

Z,n 1

14

1 0.5 0 0.5 1 1.5 2 2.5 34

3

2

1

0

1

2

Z,n 3

Z,n 1

2 1 0 1 2 34

3

2

1

0

1

2

3

Z,n 3

Z,n 1

15

6.

. -, . , . . . . , , .

1 J Gleick, Chaos, (Viking, New York, 1987), . 65.

2 J.L. Moiola & G. Chen, Hopf bifurcation analysis: A frequency domain approach. Singapore, Ney Jersey: World

Scinetific Series of Nonlinear Science (Ed. L.O. Chua), 1996. 3 . . Lorentz, J. Almos, Science, 20 , (1963), 130.

4 A. , , (.

. ), . , : . .. , 1989, .61. 5 H. K. Khalil, Nonlinear Systems, 2

nd ed. New Jersey, Prentice Hall, 1996.

6 . R Scheinerman, Invitation to Dynamical Systems, New Jersey: Prentice Hall, 1966, .153.

7 H. K. Khalil, Nonlinear Systems, 2

nd ed. New Jersey, Prentice Hall, 1996.

8 . R Scheinerman, Invitation to Dynamical Systems, New Jersey: Prentice Hall, 1966, .153.

9 M. Feigenbaum, Journal of Statistical Physics, 19, 25 (1978), J. Stat. Phys 21, 669(1979).

10 Y. Pomeau & P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun.

Math. Phys. 74, 189 (1980). 11

D. Ruelle & F. Takens, On the nature of turbulence, Commun. Math. Phys, 20, 167 (1971). 12

, .. -. 2001 13

. R Scheinerman, Invitation to Dynamical Systems, New Jersey: Prentice Hall, 1966, . 188-199. 14

. . . 174. 15

O....202-216 16

.. . 158-160. 17

A. , .., .117. 18

M. P. Kennedy, Chaos in the Colpitts oscillator IEEE Trans. Circuits Syst. I. 1994, 41, (11), . 771-774. 19

.P. Kennedy, On the relationship between the chaotic Colpitts oscillatos and Chuas oscillator, IEEE Trans.

Circuits Sust. I, 1995, 42, (6), . 376-379. 20

L. O. Chua, Chuas circuit: an overview ten years later, J. Circuits Syst. Comput. 1994, 4, (2), 117-159. 21

L.O. Chua, M. Itoh, L. Kocarev & K. Eckert, Chaos synchronization in Chua circuit, R.N. Madam (E.):

Chuas circuit: A paradigm for Chaos, World Scientific Publ. Co, Singapore, 1993, . 309-324. 22

L.O Chua, Global unfolding of Chuas circuit , IEIECE Trans. Fundamentals Electronics, Communications and

Computer Sciences, vol. E76-A, no. 5, 704-734, May 1993. 23

M.P.Kennedy, Robust Op Amp Implementation of Chuas Circuit, Frequenz, vol. 46, n0. 3-4, pp. 66-80, 1992. 24

M.T. Abuelma Atti, Chaos in an autonomus active-R circuit, IEEE Trans. Circuits Syst. I: Fundam. Theory appl.,

1995, 42, (1), . 1-5. 25

, 7-8/94, . 109-110 26

X. Rodet, Sound and Music from Chuas Circuit, Journal of Circuits, Systems and Computers, vol. 3, n0. 2, 49-61,

1992. 27

..Alligood, T.D. Sauer & J.A. Yorke, Chaos: An introduction to dynamic systems, New York, Spinger, 1977,

.375