Chaos Book 1

44
 ΜΕΡΟΣ Α ΘΕΩΡΙΑ 1

description

Chaos Book 1

Transcript of Chaos Book 1

  • 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