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Θεωρία του Χάους και ο ρόλος των αρχικών συνθηκών
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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
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