Динамический хаос в механических и оптических системах

  • Published on
    05-Apr-2017

  • View
    219

  • Download
    6

Embed Size (px)

Transcript

  • 262

    . . , . .

    , -

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

    , [16]. - , , -, , , , - , , - .

    - , - , , . - -, [5]. - - . , , . , - - .

    , - . - , .

    , - (. 1).

    , - x. , - , ( )t= N d E , d , E . , -

  • 263

    ( ) sin .I F t+ = (1)

    I -, F(t) = E(t)d. .

    . .

    . 1 , -

    . :

    0 sin cos( )I F t+ = , (2)

    , F 0 = E0d, E0 . (2)

    . , :

    sin cos ,f t+ = (3)

    I= ,

    02

    Ff

    I= . (4)

    , (3)

    , - . , , (0) /2, , - (3)

    22 1

    fA =

    +.

  • 264

    ,

  • 265

    . 2

    . 3

    , . 4. f. - f, .

  • 266

    , , - . - , .

    . 4 . 4 .

    f , , . f .

    5. . 4 , , .

    6. , , - . - . , , f, ( ) . : , ( ). -, , , , . , -

  • 267

    2 2 00

    ( ) cos(2 ) ( )n nn

    t A nt t t== + + (5)

    ( = 1) , ,

    2 12 10

    ( ) cos ( 2 1) / 2nnn

    t A n t ++=

    = + + (6)

    , . .

    7. , - , , - , - , , , ( - ), [7].

    , . , , , . , -. , , (- ). - [8, 9].

    . - :

    m = mm , (7)

    2

    2

    MH

    I= , M :

    M

    i

    =

    . -

    , m, :

    22 2

    2mm Bm

    I = = , (8)

    1

    exp( )2

    m im= , m = 0, 1, 2, ... (9)

    ( I

    B2

    = ).

  • 268

    , , -. , , . , , . - . (9)

    1

    sin( )

    sm m = , 1

    cos( )

    cm m = . (10)

    .

    , , , , - .

    V(t) = E0 dcos cos(t). (11)

    , V(t), - , , x (. 1). (10) .

    , - , ( -) . , . - |m|=1. n- , . - m , (V(t))n. n

    |m|=1, 3,... n, (12)

    n:

    |m|=0, 2, 4,... n . (13)

    , - , n- -

    Em Em = n.

  • 269

    ,

    B((|m|+m) 2 m2) = n,

    Bm(2|m|+1) = n. (14)

    , m=0 m=1 - l ,

    B=l. (15)

    (14)

    lm(2|m|+1)=n. (16)

    (12), (13) , m n . l l(2|m|+1) , m n . , l n, - . l , n , , m . , l , m m - .

    , - , - . - , .

    , . - - , , , . , , - , , . [10]. - , . - , , , - .

    . , . - , -

  • 270

    . , - . , - . , - :

    m = B(2|m|+1). (17)

    , . x (. 1). , , y. - , , - y.

    , (15). m

    m = l(2|m|+1).

    , ,

    y, , , , l.

    . , y, dy. - (5) (6), , , , , . y - , -, , .

    , , (15), l , - l . , , .

    , m = 0 m = 1, l = 1 (. 5, a). , , - . , (4) - f . - -, . 5, , l=2. . - (. 5, ) , , , - .

  • 271

    (. 5, ) , l=3 - (. 5, ).

    ) ) ) ) )

    . 5. , ,

    - . - , , , . - , - E0, - , F0 = E0d [10]. , F0 B, - .

    , , - fl, (4):

    2 2

    20 012

    2l

    F l F lf f l

    IB B= = = . (18)

    , -

    (18), f1 .

    (18) , l

    f . , - , , -, ,

    f . . . 6.

  • 272

    . 6.

    , , -

    . 4, . , - . . 6 , - f = 70. , , - , .

    .6, - f1, . -, 0,5. , - F0 B F0 = B/4, - .

    - , - . k . (17),

    = (2|m|+1)B=/k. (19)

    , , B. , - m = 0 m = 2 2 = 4B, - (19)

  • 273

    = /2, , , .

    1. Berge P., Pomeau Y., Vidal Ch. Order within Chaos. Wiley, New York, 1986. 2. Schuster H. G. Deterministic Chaos. VCH, Weinheim, 1988. 3. Briggs K. Simple experiments in chaotic dynamics // Am. J. Phys. 1987. V. 55. P. 1083

    1089. 4. Croquette V., Poitou C. Cascade of period doubling bifurcations and large stochasticity

    in the motions of a compass // J. Phys. Lett. 1981. V. 42. P. 537539. 5. Weltner K., Esperidiao S., Andrade R. F. S. Demonstration of different forms of the bent

    tuning curve with a mechanical oscillator // Am. J. Phys. 1994. V. 62(1). P. 5659. 6. Ballico M. J., Sawley M. L., Skiff F. The bipolar motor: a simple demonstration of de-

    terministic chaos // Am. J. Phys. 1990. V. 58. P. 5861. 7. . . . ., 1971. 8. . . ., 1981. 9. Papouek D., Aliev M. R. Molecular vibrational-rotational spectra. Prague, 1983. 10. ., . . ., 1978.

    A. Kondratiev, A. Liaptsev

    DYNAMIC CHAOS IN MECHANICAL AND OPTICAL SYSTEMS

    A theoretical investigation of one-dimensional rotator under external har-monic oscillating field is carried out. The classical model of a rotator is described by nonlinear differential equation which is solved numerically. The solutions depend on the parameters of the model, particularly on the field strength. The solutions can have regular periodic structure corresponding to the rotation or vibration of the ro-tator, or a nonperiodic chaotic structure. The character of solutions is explained by comparison of the classical and quantum mechanical description of the model. The peculiarities of the classical solution may be explained with the help of the corre-spondence principle as a consequence of multiphoton transitions in the quantum mechanical model.

Recommended

View more >