Post on 05-Oct-2015
description
1
1
-
1.1
1.1.1
[5, 50, 123].
n
ddt
tx
F x c= ( , , ) (1.1)
x n t,
x(t) {(x1(t), x2(t), ...,xn(t)} (1.2)
xRn. Rn .
xi (i=1,...,n) ,
, , , ,
.
. c (c1,...,ck) ci (i=1,....,k)
.
.
2
0.
,
. ,
,
F: Rn Rn (1.1).
.
, (1.1)
.
( ) (
) .
-,
.
1.1.2 F: Rn Rn ,
[5, 50, 123].
ddtx
F x c= ( , ) (1.3)
.
(
t ).
. (
) ,
( ) .
F
(1.1).
3
.
, ,
x1(t), x2(t), ...,xn(t), t ,
.
, .
(1.1) .
,
xk+1=f(xk) (1.4)
1.1.3
1.1.3.1
F xi (i=1,...,n).
ddt
Ax
x= (1.5)
, (nXn) .
n i , n
yi (1.5)
x y( )t C eii
nt
ii=
=
1
(1.6)
n Ci .
H xs x.
s =0
.
() ()
4
xs
, [5, 50, 123].
1, 2 .
:
.
1.
1. , (unstable node)
2. , (stable node)
3. (suddle point)
2.
1. (unstable (dicritical node or star ))
2. (stable (dicritical node or star))
. (1,2=ib)
1. (unstable focus)
2. (stable focus)
3. (vortex point)
1.1
, 1.2
.
(trace) .
.
5
1.1 , .
6
1.2 ( ) .
7
1.1.3.2
F xi (i=1,...,n) [5,
50, 123].
ddtx
x)= F( , x(to)=xo (1.7)
.
, ,
.
.
,
.
, t ,
,
, ,
t ,
.
xs
. xs
x, :
x = xs + x x
( )
.
1.1.3.3
xk+1=f(xk), [50, 123]. x
xk+1 = xk = x ,
x = f( x ) (1.10)
,
,
.
1.1.4 qi
(i=1,...N). (3)
,
3. (6)
, (3) (3) ,
6 .
Hamiltonian H
= + (1.11)
ddt
iq =Ht
, ddt
ip = Ht
(1.12)
Hamiltonian
[123].
.
9
,
,
.
, .
.
.
Hamiltonian
[123].
.
, t
.
,
. ,
(regular) (strange)
.
. ,
,
.
(1.7), ,
: ()< 0, (1.9).
,
: () = 0.
1. 1.5
,
.
10
d xdt
x xy
d ydt
y y x
2
2
2
22 2
2=
= (1.13)
.
x=(m+n)/2 y=(n-m)/2
d ndt
n n
d mdt
m m
2
22
2
22
=
= + (1.14)
,
(1.13) [123].
,
.
d xdt
x xy
d ydt
y y x
2
2
2
22 2
2=
= + (1.15)
Henon - Heiles ), ,
, .
, ,
. ,
( ) .
.
1.1.6 .
, ,
, .
.
11
, ( 1.3)
.
.
,
.
. Boltzmann Gibbs
( 1.3),
, [123].
.
1.3 () , . . () . .
12
1.2 .
.
, t ,
.
,
,
, [5, 50, 123].
, (stable focus) (stable node) .
.
()
.
. ,
.
.
.
,
.
.
.
( ,
), .
1.2.1 , ,
. ,
,
13
1.4 .
.
. 1.4
.
.
,
.
.
1.2.2
.
, ,
.
, -, (
14
1.5 . . .
).
( 1.5).
.
, (, )
, .
1.2.3 () r (r2)
r
,
.
,
.
, (quasi-periodic),
15
1.6 . .
( 1.6).
.
, 2/24 2/365 .
16
1.3
1.3.1 , [96]. , , , , . x(t), t
.
(), t
( ).
.
1.3.2 () x(t)
()= {[x(t+)-][x(t)-]} (3.1)
,
(autocorrelation function).
, [96].
()
() = ()(0) (3.2)
x(t) x(t+) .
, (0)=1 |()| 1.
.
17
=
=
+= kN
t
kN
tk
xtx
xtxxktxr
1
2
1
])([
])(][)([ (3.3)
k = 0,1,2,...m, x x(t). m
/4.
ck
=
+=kN
tk xtxxktxN
c1
])(][)([1 (3.4)
rk = cc
k
0 (3.5)
rk
k
.
rk = 0 k
. rk
. , rk
, rk
k.
rk
k .
1.3.3
, [96].
- ( ),
{x(t)} = Cov{x(t+),x(t)} = () (3.6)
18
.
,
.
1.3.4 .
, ,
[95].
F(), [96].
(k) F()
( ) = k e dFi k ( ) (3.7)
[-, ]. F()
(-, ) F()
.
f()
fdF
d( )
( )
= (3.8)
f()d
(, +d)
f() .
(3.7) (3.8)
( ) = k e f( )di k (3.9)
(3.9)
=
=k
kiekf
)(21)( (3.10)
Fourier (k).
19
+
=
==
2
1
2
1
2sin2cos1)(N
tt
N
ttj N
jtxN
jtxN
I
(3.11)
, .
=
=
=1
)1(
1)(Nk
Nk
kikj
jecI
(3.12)
,
Vr[I()]=2/2,
, .
+=
=
M
kkkj M
jkcwcwf1
00 )2cos(2
21)(
(3.13)
wk (Tukey, Parzen, )
.
1.4 - 1.4.1
.
,
,
. 1960
, (). 1963 E. Lorenz
,
,
. (
) 3- ,
,
20
.
Lorenz ,
.
.
,
. Poincare
...
,
. ()
() .
. ,
60
.
()
Lorenz,
.
,
.
. 60
.
,
.
,
,
.
21
,
[123, 138]. ,
()
()
.
,
()
.
, ,
, ,
.
, ,
.
.
.
.
,
.
Lyapunov (
) .
,
. ,
. .
,
.
.
22
.
,
.
o (fractal) ,
, [123].
.
. .
,
.
, ,
,
( )
. ,
,
,
.
.
23
1.4.2 Lyapunov1.4.2.1 Lyapunov
.
Lyapunov. n n Lyapunov,
. Lyapunov
. Lyapunov
(
)
. Lyapunov
.
( )
.
Lyapunov.
,
.
Lyapunov
.
:
) (-,-,-)
) (0,-,-)
) (0,0,-)
) (+,0,-)
Lyapunov
.
V(0) V
V(t)=V(0)e C t (4.1)
24
C=1+2+...+n Lyapunov.
V
V0 t, 1+2+...+n2>...>n.
Lyapunov.
Lyapunov
. 1 ,
1, 2 .
d- d
, [1, 5, 54, 123, 135].
1.4.2.3 Lyapunov
ddtx
F x c= ( , ) , xRn (4.3)
25
x(t) c .
x(t)
n-
ddt
= DF x( )(t) (4.4)
DF(x(t)) nXn [Fi(x(t))]/xj], i,j=1,2,...,n.
(4.4)
(t)=(t,0)(0) (4.5)
(t,0)
ddtA
A= DF x( )(t) (4.6)
(0,0)=, .
Lyapunov
= limt {[(t)TA(t)]} (4.7)
(t)T . ej t j,
j=1,2...n Lyapunov, [1, 5, 54, 123, 135].
1.4.3
(fractal) , ,
. ,
,
.
,
.
26
1.4.3.1 S Rn. ()
( n)
. ()
d
N ()=d
(4.8)
(4.8)
dN A
= +log log
log( ( )
(()
log() ) (4.9)
D0 (4.9) 0
DN
0 = lim( ( ))
( 0
)log
log (4.10)
D0
.
D0 , [1, 123].
1.4.3.2
.
S () n-
ios i ,
Pi=Ni /N.
DP Pi i
i
N
1 0
1=
=
limlog
log
)
)
(
( (4.11)
P Pi ii
N
log=
1
()
,
.
27
, Pi=1/N i, D1=D0. Pi
( ) D1
1.5
1.5.1
, ( )
.
, ,
, ( )
, [50, 123].
,
,
.
.
,
Lyapunov
.
,
,
.
( ),
.
, ,
, . ,
,
.
, ,
,
29
,
.
,
, .
,
Lyapunov.
.
.
.
, (
) ( )
.
.
1.5.2 .
,
.
.
.
.
.
.
.
30
,
.
. , Lorenz
.
[123, 138].
.
.
1.5.3
,
,
.
, , Lyapunov
, [1, 123].
, ,
.
.
,
, .
.
,
.
, [96].
.
31
.
, ()
.
(SVD).
1.5.4
.
,
.
, ,
.
,
.
.
k1 ,
( Henon) ..
, [74].
,
, [123].
( )
, [96].
32
1.5.5
, .
.
,
,
.
,
.
[1, 74, 96, 123].
1.5.6
x(i)=x(ti) ( ti)
s(t)=ft(s0) ,
Ruelle Takens,
[99], Witney [134], d-
() Rm m 2d+1,
: Rm.
x(t)=(s(t))
.
x(t)=Gt(xo)
Rm
G t(x0)=(s) f t (s0)-1(x). (5.1)
33
,
f t(s0) ( s0) G t(x0)
Rm,
, Lyapunov .
.
x(i)=h(s(ti)), h
,
x(i)=[x(i),x(i+), ..., x(i-(m-1))] (5.2)
(delay time).
,
.
1.5.6.1 Takens Ruelle
.
m
,
.
.
)
. x(n+jT), x(n+(j+1)T)
x(n) ,
.
34
.
k 0.5
1/e.
.
, [1, 123].
)
.
Shannon ,
[1, 38, 107, 108].
S {si, i=1,2...}
S
S (S)
))logP((P)( iii
ssSH = (5.3) P(si) S si n(si)/nT,
n(si) si nT
.
S
Q.
. (S,Q) Q={x(i)} S={x(i+)},
x(i), x(i+)
ti ti+. Q
S
ISQ =H(S)-H(S/Q)=H(S)+H(Q)-H(Q,S) (5.4)
))(P(log))((P))(P(log))((P( 2)(
2)(
ixixixixIixix
=)
35
+)( )(
2 )(),(P(log))(),(P(ix ix
ixixixix (5.5)
(0, max), Imax = I(0)
H(x). {Qx(i)} {Sx(i+)}
.
.
I()= Imax,
=0.
()
,
.
,
.
1.5.6.2 Takens - Ruelle [99]
Whitney to 1936 [134]
.
Rm
m D,
Rm.
m,
m
.
m>
2D, m .
Lorenz D=2.06
m=5 . Lorenz
R3,
m=5 m0=3.
36
m, m>2D +1,
.
m0.
m
.
m0
, .
.
) -
m0
D ( m0
).
D m0.
)]d[ln(]d[lnlim
rC(r)D
0r= (5.6)
C(r ) .
r
C( r ) rd r 0 (5.7)
,
= =
=N
i
N
jji
nNmC
1 1
)()(()1(
2),( xxrr (5.8)
() =1 > 0 ()= 0
D
d M, .
D .
)
m0.
,
.
.
,
m0.
x(j) x(i) m.
222 .... ++++= 1))(m(jx1))(m(ixx(j)(i)(i,j)rm x
m m+1 222 )) +++=+ m(jxm(ix(i,j)r(i,j)r m1m (5.9)
TmRr
m(jxm(ix>
++ )) (5.10)
x(j), x(i) , [1]. RT
[10, 50].
m m0
.
)
38
,
m0 , [23,
35].
X
xx
x
=
+ +
+
=
x t x t x t mx t x t x t m
x t x t x t mN N N
( ), ( ),...( ( ( )( ), ( ),...( ( ( )
( ), ( ),...( ( ( ).
1 1 1
2 2 2
11
1
. . ... .
))
)
1T
2T
NT
(5.11)
x(ti)
. xiT
Rm. Rm
Rm
.
{ci i=1,2,...m} Rm
s X ciT
iT= i=1,2,...m0 (5.12)
siT RN.
SVD si ci
(stucture matrix) XXT
(covariance matrix)
XXTsi= i2 si ci= i
2 ci (5.13)
si, ci i X,
SVD X
=SCT
39
S=[s1,s2,...sm] , C=[c1,c2, ...cm] =[1,2,...m] 12...m.
SVD (stucture
matrix) (covariance
matrix). m0 ( m0m)
rankX=rankXXT=rankXTX=m0.
m0
m-m0
.
1.5.7 Lyapunov Lyapunov ,
. w(i),
x(i)
w(i+n)=DG n(x(i))w(i) (5.14)
DG G x(i). DG n
x(i)
minS min1k
(i n) (i)A A j i j
2
j 1
k
i i
= + =
w A w (5.15)
k x(i) k
wj, j=1,...k, ||wj||<
x(i) tp=nt (propagator time). wj
Ai DG n x(i).
Lyapunov
,
jp
i ji
i 1
N1Nt
log==
A e (5.16)
e ji x(i)
e ji n+ = Ai e j
i tp, [47, 55, 139].
Lyapunov
Wolf, [135]. x(1) , L(1)
. tp=nt
40
L(1) L(1),
x(1+kn), k=1,...,M M .
Lyapunov
LMtp k
M
max ln==
11
L'(k + 1)L(k) (5.17)
1.6 .
x(t)
,
x(t) .
,
G: x(i)x(i+1) ( G ).
G , GT
, , [1, 123].
G ,
x(i+T)
x =GT(x(i),) .
GT x(i),
(local), , ,
.
,
( )
( ) , [1, 123, 130].
.
, NRMSE (normalized root mean square error)
41
[ ]
[ ]NRMSE
x k x k
x k x
k
M
k
M=
=
=
( ) ( )
( )
2
1
2
1
(6.1)
() , x
. NRMSE 0 ,
NRMSE >1 ()
, .
CC (correlation coefficient)
(, )
CCx k x x k x
x k x x k x
k
M
k
M
k
M=
=
==
[ ( ) ][ ( ) ]
[ ( ) ] [ ( ) ]
1
2 2
11
(6.2)
x , x .
CC [-1,1]. o CC 1 ,
.
( ) .
1.6.1 GT
q.
q, m , pqT
. ,
, ,
,
.
42
pqT =+x pi T q
Ti( )x xiRm,
Voltera-Wiener [92] q m
+ = + + + + + + +
+ + +x i T a a x i a i m a x i a x i x i
a x i mm m m
Mq
( ) ( ) ... ( ( ) ) ... ( ) ( ) ( ) ...( ( ) )
0 1 12
211
(6.3)
M=(m+q)!/(m!q!).
1.6.2 ,
,
.
, x(i) GT
k {x(i(1)),...x(i(k)}
x(i).
1.6.2.1 (Local weighted averaging)
,
T .
{x(i(1)),...x(i(k)} x(i) () ,
x(i). x(i+T)
{x(i(1+)),...x(i(k+)}
x(i) k , [1, 112].
1.6.2.2
GT ,
+ = +x i T iT( ) ( )a a x0 .
x(i), k ,
k>m m . m+1
43
{0, }, k m+1
, OLS (ordinary least
squeres).
+x i T( ) , [26, 36].
1.6.2.3
OLS (ordinary least squeres),
. ,
,
.
OLS (Principal Components
Regression (PCR)), [138, 140]. PCR
q m principal components,
PCR(q).
44