Занг В.Б. Синергетическая экономика. Время и перемены в нелинейной экономической теории

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ɝɞɟ ɯ = (ɯ

, ɯ

)

ɩɪɢɱɟɦ ɯ ɢɡɦɟɧɹɟɬɫɹ ɜ ɩɪɟɞɟɥɚɯ U ⊂ R

, ɚ f ɢ J — ɞɨɫɬɚɬɨɱɧɨ ɝɥɚɞɤɢɟ

ɮɭɧɤɰɢɢ ɨɬ ɯ. ȼɫɟɫɬɨɪɨɧɧɟɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ ɫɢɫɬɟɦ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ

ɩɪɨɜɟɞɟɧɨ Ƀɟ ɢ ɞɪ. (1986) (ɫɦ. ɬɚɤɠɟ Ȼɚɭɬɢɧ, Ʌɟɨɧɬɨɜɢɱ, 1990, ɫɧɨɫɤɚ ɜ ɪɚɡɞ. 3.6. — Ɋɟɞ.).

Ɍɨɱɤɚ ɯ* ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɩɪɟɞɟɥɶɧɚɹ ɬɨɱɤɚ ɯ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɚɹ

ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ, ɱɬɨ lim W

(x) = ɯ* ɩɪɢ tĺ, ɝɞɟ W

(x) — ɩɨɬɨɤ ɫɢɫɬɟɦɵ.

Ɍɟɨɪɟɦɚ 5.2.1. (ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɩ.) ɇɟɩɭɫɬɨɟ ɤɨɦɩɚɤɬɧɨɟ ɩɪɟɞɟɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ

ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ Ʌ

, ɤɨɬɨɪɨɟ ɧɟ ɫɨɞɟɪɠɢɬ ɬɨɱɟɤ

ɪɚɜɧɨɜɟɫɢɹ, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɡɚɦɤɧɭɬɭɸ ɨɪɛɢɬɭ.

ɋɥɟɞɭɸɳɚɹ ɧɢɠɟ ɬɟɨɪɟɦɚ ɹɜɥɹɟɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɬɟɨɪɟɦɵ 5.2.1.

Ɍɟɨɪɟɦɚ 5.2.2. Ɉɛɥɚɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɚɹ ɡɚɦɤɧɭɬɨɣ ɬɪɚɟɤɬɨɪɢɟɣ ɧɟɩɪɟɪɵɜɧɨ

ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ R

, ɞɨɥɠɧɚ ɫɨɞɟɪɠɚɬɶ ɬɨɱɤɭ ɪɚɜɧɨɜɟɫɢɹ dx/dt =

0. Ȼɨɥɟɟ ɬɨɝɨ, ɟɫɥɢ ɬɪɚɟɤɬɨɪɢɹ ɩɪɢɧɚɞɥɟɠɢɬ ɡɚɦɤɧɭɬɨɦɭ ɨɝɪɚɧɢɱɟɧɨɦɭ ɩɨɞɦɧɨɠɟɫɬɜɭ D ɜ U,

ɬɨ ɟɟ ɩɪɟɞɟɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ L(x) ɧɟɩɭɫɬɨ, ɡɚɦɤɧɭɬɨ ɢ ɫɜɹɡɧɨ.

ɋɦɵɫɥ ɷɬɢɯ ɬɟɨɪɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɟɫɥɢ ɜ ɦɧɨɠɟɫɬɜɟ U ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɧɟɤɨɬɨɪɨɟ

ɩɨɞɦɧɨɠɟɫɬɜɨ D ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨ ɩɪɟɞɟɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ L(x) ɧɟɩɭɫɬɨ,

ɹɜɥɹɟɬɫɹ ɤɨɦɩɚɤɬɨɦ

ɢ ɧɟ ɫɨɞɟɪɠɢɬ ɬɨɱɟɤ ɪɚɜɧɨɜɟɫɢɹ, ɬɨ ɷɬɨ ɩɪɟɞɟɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ

ɡɚɦɤɧɭɬɭɸ ɨɪɛɢɬɭ, ɨɤɪɭɠɚɸɳɭɸ ɬɨɱɤɭ ɪɚɜɧɨɜɟɫɢɹ. ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɩɟɪɟɱɢɫɥɟɧɧɵɟ

ɜɵɲɟ ɬɟɨɪɟɦɵ ɧɟ ɢɫɤɥɸɱɚɸɬ ɜɨɡɦɨɠɧɨɫɬɢ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɧɟɫɤɨɥɶɤɢɯ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ.

ɉɭɫɬɶ D — ɨɞɧɨɫɜɹɡɧɚɹ ɨɛɥɚɫɬɶ ɜ U. ɂɦɟɟɦ ɫɥɟɞɭɸɳɭɸ ɬɟɨɪɟɦɭ:

Ɍɟɨɪɟɦɚ 5.2.3. (Ȼɟɧɞɢɤɫɨɧ.) ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ f ɢ J ɜ (5.2.1) ɨɛɥɚɞɚɸɬ ɜ D

ɧɟɩɪɟɪɵɜɧɵɦɢ

ɩɟɪɜɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ. ȿɫɥɢ ɫɭɦɦɚ (∂f/∂x

+∂J/∂ɯ

) ɧɟ ɦɟɧɹɟɬ ɡɧɚɤɚ ɜɨ ɜɫɟɣ ɨɛɥɚɫɬɢ D, ɬɨ

ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ (5.2.1), ɰɟɥɢɤɨɦ ɥɟɠɚɳɟɝɨ ɜ D, ɧɟ ɫɭɳɟɫɬɜɭɟɬ.

Ɍɟɨɪɟɦɚ Ȼɟɧɞɢɤɫɨɧɚ ɮɨɪɦɭɥɢɪɭɟɬ ɭɫɥɨɜɢɟ ɨɬɫɭɬɫɬɜɢɹ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ ɜ ɨɛɥɚɫɬɢ D.

Ɍɟɨɪɟɦɚ 5.2.4. (Ⱦɟ Ȼɚɝɝɢɫ.) ɉɭɫɬɶ ɫɢɫɬɟɦɚ ɫɬɪɭɤɬɭɪɧɨ ɭɫɬɨɣɱɢɜɚ. Ɍɨɝɞɚ ɨɧɚ ɢɦɟɟɬ ɜ D

ɥɢɲɶ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ, ɤɨɬɨɪɵɟ ɩɨɩɟɪɟɦɟɧɧɨ ɭɫɬɨɣɱɢɜɵ ɢ ɧɟɭɫɬɨɣɱɢɜɵ ɜ

ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɦ ɫɦɵɫɥɟ

ɋɦ. ɫɧɨɫɤɢ ɜ ɪɚɡɞ. 3.2 ɢ 3.6. — ɉɪɢɦ. ɪɟɞ.

ɋɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɩɪɢɥɨɠɟɧɢɣ ɬɟɨɪɟɦɵ ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɧɚ ɤ ɷɤɨɧɨɦɢɤɟ (ɫɦ.

ɒɢɧɚɫɢ, 1982, ɋɟɦɦɥɟɪ, 1985, 1986). ȼ ɱɚɫɬɧɨɫɬɢ, ɑɚɧɝ ɢ ɋɦɢɬ (1971) ɩɪɢɜɟɥɢ ɩɪɢɥɨɠɟɧɢɟ

ɬɟɨɪɟɦɵ ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɧɚ ɤ ɦɨɞɟɥɢ Ʉɚɥɞɨɪɚ (1940) ɞɟɥɨɜɨɝɨ ɰɢɤɥɚ. Ɇɨɞɟɥɶ ɑɚɧɝɚ-

ɋɦɢɬɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ

ɝɞɟ Y, Ʉ, S ɢ I(Y, K) ɨɛɨɡɧɚɱɚɸɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɟɚɥɶɧɵɣ ɞɨɯɨɞ, ɤɚɩɢɬɚɥ, ɮɭɧɤɰɢɸ

ɩɨɬɪɟɛɥɟɧɢɹ ɢ ɮɭɧɤɰɢɸ ɱɢɫɬɵɯ ɢɧɜɟɫɬɢɰɢɣ. Ȼɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ S

< 0 ɢ °I

«>°S

ȼ ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɪɚɜɧɨ α(S

- S

). ȼ ɫɥɭɱɚɟ

ɧɟɫɟɞɥɨɜɨɣ ɬɨɱɤɢ ɷɬɚ ɜɟɥɢɱɢɧɚ ɞɨɥɠɧɚ ɛɵɬɶ ɩɨɥɨɠɢɬɟɥɶɧɨɣ. ȿɫɥɢ ɬɪɟɛɨɜɚɬɶ ɧɟɭɫɬɨɣɱɢɜɨɫɬɶ

ɫɨɫɬɨɹɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, ɬɨ ɫɭɦɦɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ, ɪɚɜɧɚɹ α(I

— S

) + I

, ɞɨɥɠɧɚ ɛɵɬɶ

ɫɬɪɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɣ. ɑɚɧɝ ɢ ɋɦɢɬ ɞɨɤɚɡɚɥɢ ɫɥɟɞɭɸɳɭɸ ɬɟɨɪɟɦɭ:

Ɍɟɨɪɟɦɚ 5.2.5. (ɑɚɧɝ ɢ ɋɦɢɬ.) ȿɫɥɢ ɫɢɫɬɟɦɚ (5.2.2) ɨɩɪɟɞɟɥɟɧɚ ɜ

R ɢ ɨɛɥɚɞɚɟɬ

ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ:

(i) I

< S

< 0, I

> 0, S

> 0;

(ii) ɜ ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ (K

, Y

) ɢɦɟɟɬ ɦɟɫɬɨ α(I

— S

) + I

> 0 ɢ S

> S

;

(iii) ɩɪɨɢɡɜɨɞɧɚɹ dK/dt = 0 ɩɟɪɟɫɟɤɚɟɬ Ʉ-ɨɫɶ ɜ ɤɨɧɟɱɧɨɣ ɬɨɱɤɟ K(0) > 0;

(iv) dY/dt = 0 ɩɟɪɟɫɟɤɚɟɬ Y-ɨɫɶ ɜ ɤɨɧɟɱɧɨɣ ɬɨɱɤɟ Y

> Y

ɢ +∞=

→

Y 0

lim ;

(v) ɫɢɫɬɟɦɚ ɫɬɪɭɤɬɭɪɧɨ ɭɫɬɨɣɱɢɜɚ,

ɬɨ ɤɚɠɞɚɹ ɬɪɚɟɤɬɨɪɢɹ ɥɢɛɨ ɹɜɥɹɟɬɫɹ ɩɪɟɞɟɥɶɧɵɦ ɰɢɤɥɨɦ, ɥɢɛɨ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ

ɩɪɟɞɟɥɶɧɨɦɭ ɰɢɤɥɭ.

ɗɬɚ ɬɟɨɪɟɦɚ ɢɞɟɧɬɢɱɧɚ ɬɟɨɪɟɦɟ ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɧɚ. Ɇɟɧɟɟ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɦɨɞɟɥɶ

Ʉɚɥɞɨɪɚ 1940 ɝ. ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɟɪɜɭɸ ɷɧɞɨɝɟɧɧɭɸ ɦɨɞɟɥɶ ɞɟɥɨɜɨɝɨ ɰɢɤɥɚ, ɯɨɬɹ ɟɟ

ɨɬɥɢɱɢɟ ɨɬ ɲɢɪɨɤɨ ɢɡɜɟɫɬɧɨɣ ɦɨɞɟɥɢ, ɑɚɧɝɚ ɢ ɋɦɢɬɚ ɜɟɫɶɦɚ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɢ ɧɨɫɢɬ ɮɨɪ-

ɦɚɥɶɧɵɣ

ɯɚɪɚɤɬɟɪ.

Ɍɟɨɪɟɦɚ 5.2.5 ɮɨɪɦɭɥɢɪɭɟɬ ɭɫɥɨɜɢɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɩɪɟɞɟɥɶɧɨɝɨ ɰɢɤɥɚ. Ɉ ɟɝɨ

ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɧɢɱɟɝɨ ɧɟ ɝɨɜɨɪɢɬɫɹ, ɬɚɤ ɤɚɤ ɬɟɨɪɟɦɚ ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɧɚ ɞɨɤɚɡɵɜɚɟɬ

ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɨɞɧɨɝɨ ɩɪɟɞɟɥɶɧɨɝɨ ɰɢɤɥɚ, ɬ. ɟ. ɜɩɨɥɧɟ ɜɨɡɦɨɠɧɚ ɫɢɬɭɚɰɢɹ,

ɤɨɝɞɚ ɫɭɳɟɫɬɜɭɸɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɟɫɤɨɥɶɤɨ ɰɢɤɥɨɜ, ɩɨɩɟɪɟɦɟɧɧɨ ɭɫɬɨɣɱɢɜɵɯ ɢ

ɧɟɭɫɬɨɣɱɢɜɵɯ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɬɨ, ɧɚ ɤɚɤɨɣ ɩɪɟɞɟɥɶɧɵɣ ɰɢɤɥ

ɩɨɩɚɞɟɬ ɢ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɫɢɫɬɟɦɚ, ɡɚɜɢɫɢɬ ɨɬ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ, ɧɚɥɨɠɟɧɧɵɯ ɧɚ

ɩɟɪɟɦɟɧɧɵɟ.

ɉɨɩɵɬɤɚ ɪɚɡɪɟɲɢɬɶ ɩɪɨɛɥɟɦɭ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɞɥɹ ɦɨɞɟɥɢ ɑɚɧɝɚ-ɋɦɢɬɚ ɫ ɩɨɦɨɳɶɸ

ɬɟɨɪɟɦɵ Ʌɟɜɢɧɫɨɧɚ-ɋɦɢɬɚ ɛɵɥɚ ɩɪɟɞɩɪɢɧɹɬɚ ɧɟɞɚɜɧɨ Ʌɨɪɟɧɰɟɦ (1986). ɏɨɬɹ ɟɝɨ ɪɟɡɭɥɶɬɚɬ

ɧɨɫɢɬ ɞɨɜɨɥɶɧɨ ɱɚɫɬɧɵɣ ɯɚɪɚɤɬɟɪ, ɬɟɦ ɧɟ ɦɟɧɟɟ ɨɧ ɜɟɫɶɦɚ ɜɚɠɟɧ, ɩɨɫɤɨɥɶɤɭ ɜɨɩɪɨɫ ɨ

ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɩɪɟɞɟɥɶɧɨɝɨ ɰɢɤɥɚ ɜ ɧɟɥɢɧɟɣɧɵɯ ɦɨɞɟɥɹɯ

ɞɟɥɨɜɵɯ ɰɢɤɥɨɜ ɪɟɞɤɨ

ɩɨɞɧɢɦɚɟɬɫɹ ɜ ɥɢɬɟɪɚɬɭɪɟ.

ɉɪɟɠɞɟ ɱɟɦ ɫɨɨɛɳɢɬɶ ɪɟɡɭɥɶɬɚɬ Ʌɨɪɟɧɰɚ, ɨɫɬɚɧɨɜɢɦɫɹ ɧɚ ɬɟɨɪɟɦɟ Ʌɟɜɢɧɫɨɧɚ-ɋɦɢɬɚ.

Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ ɨɛɨɛɳɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ Ʌɶɟɧɚɪɚ

ɢɥɢ ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɟɦɭ ɭɪɚɜɧɟɧɢɟ

Ɍɟɨɪɟɦɚ 5.2.6. (Ʌɟɜɢɧɫɨɧ ɢ ɋɦɢɬ, 1942.) ɍɪɚɜɧɟɧɢɟ (5.2.3) ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ,

ɟɫɥɢ ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɭɫɥɨɜɢɹ:

i) f' ɢ J ɩɪɢɧɚɞɥɟɠɚɬ ɋ

;

ii) f'(x) < 0 ɞɥɹ ɯ ∈ (-x

, x

), ɝɞɟ x

, x

> 0; f'(x) ≥ 0 ɞɥɹ ɨɫɬɚɥɶɧɵɯ ɯ;

iii) ɯJ (ɯ) > 0 ɞɥɹ ɥɸɛɨɝɨ ɯ, ɧɟ ɪɚɜɧɨɝɨ ɧɭɥɸ;

iv)

∞==

∞→∞→

)(lim)(lim xGxF

, ɝɞɟ

v) G(-x

) = G(x

ɉɪɢɦɟɧɢɦ ɷɬɭ ɬɟɨɪɟɦɭ ɤ (5.2.2). Ɍɟɩɟɪɶ (5.2.2) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ

Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɭɪɚɜɧɟɧɢɣ ɪɟɝɭɥɢɪɭɟɦɨɝɨ ɩɨɬɪɟɛɢɬɟɥɶɫɤɨɝɨ ɪɵɧɤɚ ɩɨ ɜɪɟɦɟɧɢ

ɩɪɢɜɨɞɢɬ ɤ ɭɪɚɜɧɟɧɢɸ

ɍɪɚɜɧɟɧɢɟ (5.2.4) ɧɟɥɶɡɹ ɨɬɧɟɫɬɢ ɤ ɬɢɩɭ Ʌɶɟɧɚɪɚ. ɑɬɨɛɵ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɟɨɪɟɦɭ 5.2.6,

ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɡɦɟɧɟɧɢɟ ɤɚɩɢɬɚɥɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɮɭɧɤɰɢɟɣ ɧɚɤɨɩɥɟɧɢɹ, ɬ.ɟ. dK/dt

= S, ɝɞɟ S = S(Y), ɉɪɟɞɩɨɥɨɠɢɦ ɬɚɤɠɟ, ɱɬɨ ɜɵɪɚɠɟɧɢɟ (I

- S

), ɨɛɨɡɧɚɱɟɧɧɨɟ ɱɟɪɟɡ W(Y), ɚ

ɬɚɤɠɟ I

ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɟɥɢɱɢɧɵ ɧɚɤɨɩɥɟɧɢɹ ɤɚɩɢɬɚɥɚ. Ɍɨɝɞɚ (5.2.4) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɤɚɤ

Ʌɨɪɟɧɰ (1986) ɩɨɤɚɡɚɥ, ɱɬɨ ɢ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɞɪɭɝɢɯ, ɜɩɨɥɧɟ ɩɪɢɟɦɥɟɦɵɯ

ɩɪɟɞɩɨɥɨɠɟɧɢɹɯ, ɞɥɹ ɮɭɧɤɰɢɣ ɢɧɜɟɫɬɢɰɢɣ ɢ ɧɚɤɨɩɥɟɧɢɹ, ɫɢɦɦɟɬɪɢɱɧɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ Y,

ɩɟɪɢɨɞɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (5.2.5) ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɞɧɨɡɧɚɱɧɨ.

5.2.2 Ɍɟɨɪɟɦɚ ɏɨɩɮɚ ɨ ɛɢɮɭɪɤɚɰɢɹɯ

ɉɨɫɤɨɥɶɤɭ ɬɟɨɪɟɦɚ ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɧɚ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɥɢɲɶ ɧɚ ɫɢɫɬɟɦɵ ɜɬɨɪɨɝɨ

ɩɨɪɹɞɤɚ, ɟɟ ɩɪɢɦɟɧɟɧɢɟ ɜ ɷɤɨɧɨɦɢɤɟ ɞɨɜɨɥɶɧɨ ɨɝɪɚɧɢɱɟɧɧɨ. Ȼɨɥɶɲɢɧɫɬɜɨ ɷɤɨɧɨɦɢɱɟɫɤɢɯ

ɫɢɫɬɟɦ ɢɦɟɸɬ ɝɨɪɚɡɞɨ ɛɨɥɶɲɢɣ ɩɨɪɹɞɨɤ, ɬɚɤ ɱɬɨ ɠɟɥɚɬɟɥɶɧɨ ɢɦɟɬɶ ɜ ɚɪɫɟɧɚɥɟ ɚɧɚɥɢɬɢɱɟɫɤɢɟ

ɦɟɬɨɞɵ, ɩɪɢɝɨɞɧɵɟ ɞɥɹ ɫɢɫɬɟɦ ɜɵɫɨɤɢɯ ɩɨɪɹɞɤɨɜ. ȼɟɫɶɦɚ ɩɨɥɟɡɧɨɣ ɫ ɷɬɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ

ɨɤɚɡɵɜɚɟɬɫɹ ɛɢɮɭɪɤɚɰɢɨɧɧɚɹ ɬɟɨɪɟɦɚ ɏɨɩɮɚ.

ɇɭɠɧɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɬɟɨɪɟɦɚ ɉɭɚɧɤɚɪɟ

-Ⱥɧɞɪɨɧɨɜɚ-ɏɨɩɮɚ, ɧɚɡɵɜɚɟɦɚɹ ɨɛɵɱɧɨ

ɛɢɮɭɪɤɚɰɢɨɧɧɨɣ ɬɟɨɪɟɦɨɣ ɏɨɩɮɚ, ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɦ ɪɟɡɭɥɶɬɚɬɨɦ ɬɟɨɪɢɢ

ɛɢɮɭɪɤɚɰɢɣ. Ɍɟɪɦɢɧɨɦ «ɛɢɮɭɪɤɚɰɢɹ ɏɨɩɮɚ» ɧɚɡɵɜɚɸɬ ɹɜɥɟɧɢɟ ɪɨɠɞɟɧɢɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ

ɨɪɛɢɬɵ ɢɡ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɷɜɨɥɸɰɢɨɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɢɡɦɟɧɟɧɢɢ

ɛɢɮɭɪɤɚɰɢɨɧɧɨɝɨ ɩɚɪɚɦɟɬɪɚ. Ȼɢɮɭɪɤɚɰɢɨɧɧɚɹ ɬɟɨɪɟɦɚ ɏɨɩɮɚ ɮɨɪɦɭɥɢɪɭɟɬ ɞɨɫɬɚɬɨɱɧɵɟ

ɭɫɥɨɜɢɹ ɬɚɤɨɝɨ ɩɨɜɟɞɟɧɢɹ. ȼɨɡɦɨɠɧɨ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɩɪɢɧɰɢɩɚ ɦɚɤɫɢɦɭɦɚ ɉɨɧɬɪɹɝɢɧɚ, ɜ

ɦɚɬɟɦɚɬɢɤɟ ɛɨɥɶɲɟ ɧɟɬ ɬɟɨɪɟɦɵ, ɤɨɬɨɪɚɹ

ɢɦɟɥɚ ɛɵ ɫɬɨɥɶ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ. Ʉɚɤ ɧɚɦ

ɤɚɠɟɬɫɹ, ɩɪɢɱɢɧɵ ɷɬɨɣ ɩɨɩɭɥɹɪɧɨɫɬɢ ɤɪɨɸɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ:

(i) ɍɫɥɨɜɢɹ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɛɢɮɭɪɤɚɰɢɢ ɏɨɩɮɚ ɥɟɝɤɨ ɨɛɧɚɪɭɠɢɬɶ;

(ii) Ɍɟɨɪɟɦɚ ɩɪɢɦɟɧɢɦɚ ɞɥɹ ɥɸɛɵɯ ɪɚɡɦɟɪɧɨɫɬɟɣ ɢ ɩɪɨɫɬɪɚɧɫɬɜ;

(iii) Ȼɢɮɭɪɤɚɰɢɢ ɏɨɩɮɚ — ɟɞɢɧɫɬɜɟɧɧɵɣ ɜɢɞ ɧɟɫɬɚɰɢɨɧɚɪɧɨɝɨ ɩɨɜɟɞɟɧɢɹ, ɤɨɬɨɪɵɣ

ɯɨɪɨɲɨ ɩɨɧɹɬ ɜ ɪɚɦɤɚɯ ɬɟɨɪɢɢ ɛɢɮɭɪɤɚɰɢɣ;

(iv) Ɍɟɨɪɟɦɚ ɞɚɟɬ ɹɪɤɢɣ ɩɪɢɦɟɪ ɪɚɡɥɢɱɢɹ ɦɟɠɞɭ ɥɢɧɟɣɧɵɦɢ ɢ ɧɟɥɢɧɟɣɧɵɦɢ

ɹɜɥɟɧɢɹɦɢ;

ɢ, ɤɪɨɦɟ ɬɨɝɨ,

(v) ɉɪɢɜɨɞɢɬ ɜ ɫɢɫɬɟɦɭ ɩɪɨɰɟɫɫ ɧɚɯɨɠɞɟɧɢɹ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɨɪɛɢɬ «ɜ ɰɟɥɨɦ».

ɉɪɢɜɟɞɟɦ ɫɬɚɧɞɚɪɬɧɭɸ ɮɨɪɦɭɥɢɪɨɜɤɭ ɛɢɮɭɪɤɚɰɢɨɧɧɨɣ ɬɟɨɪɟɦɵ ɏɨɩɮɚ. Ɋɚɫɫɦɨɬɪɢɦ

ɚɜɬɨɧɨɦɧɭɸ ɫɢɫɬɟɦɭ

ɝɞɟ f : R

× R ĺ R

ɤɥɚɫɫɚ ɋ

෱

, ɚ r — ɛɢɮɭɪɤɚɰɢɨɧɧɵɣ ɩɚɪɚɦɟɬɪ, ɩɪɢɱɟɦ ɯ = 0 ɹɜɥɹɟɬɫɹ

ɬɨɱɤɨɣ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɵ ɞɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ r.

ɏɨɩɮ (1942), ɚ ɡɚɬɟɦ Ɇɚɪɫɞɟɧ ɢ Ɇɚɤ-Ʉɪɚɤɟɧ (1976) ɩɨɤɚɡɚɥɢ

, ɱɬɨ ɟɫɥɢ ɮɭɧɤɰɢɹ f

ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɧɟɤɨɬɨɪɵɦ ɭɫɥɨɜɢɹɦ, ɞɥɹ ɫɢɫɬɟɦɵ (5.2.6) ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ

ɨɞɧɨɩɚɪɚɦɟɬɪɢɱɟɫɤɨɟ ɫɟɦɟɣɫɬɜɨ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɬɪɚɟɤɬɨɪɢɣ, ɢɫɯɨɞɹɳɢɯ ɢɡ ɬɨɱɤɢ (x, r) =

(0, 0). ɉɭɫɬɶ Ⱥ(r) — əɤɨɛɢɚɧ ɪɚɡɦɟɪɧɨɫɬɢ ɩ × ɩ ɮɭɧɤɰɢɢ f ɧɚ ɫɬɚɰɢɨɧɚɪɧɨɦ ɪɟɲɟɧɢɢ. ɉɟɪɜɨɟ

ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɏɨɩɮɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ

(i) Ⱥ(0) ɢɦɟɟɬ ɩɚɪɭ ɧɟɧɭɥɟɜɵɯ ɱɢɫɬɨ ɦɧɢɦɵɯ ɩɪɨɫɬɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ

±i z

(ii) Ⱥ(0) ɧɟ ɢɦɟɟɬ ɞɪɭɝɢɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɧɚ ɦɧɢɦɨɣ ɨɫɢ.

(5.2.7)

Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ (ii) ɦɨɠɧɨ ɨɫɥɚɛɢɬɶ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɟɪɢɨɞɢɱɟɫɤɢɟ

ɨɪɛɢɬɵ ɫɭɳɟɫɬɜɭɸɬ ɢ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ Ⱥ(0) ɢɦɟɟɬ ɧɚ ɦɧɢɦɨɣ ɨɫɢ ɢ ɞɪɭɝɢɟ ɫɨɛɫɬɜɟɧɧɵɟ

ɡɧɚɱɟɧɢɹ, ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɧɢ ɨɞɧɨ ɢɡ ɧɢɯ ɧɟ ɹɜɥɹɟɬɫɹ ɰɟɥɵɦ ɤɪɚɬɧɵɦ ɡɧɚɱɟɧɢɸ ±i z

. Ʉɪɨɦɟ

ɬɨɝɨ, ɦɵ ɬɪɟɛɭɟɦ, ɱɬɨɛɵ Ⱥ(r) ɢɦɟɥɚ ɩɪɨɫɬɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɢɞɚ z

(r) ±i z

(r), ɝɞɟ z

(0)

= 0, z

(0) = z

, ɩɪɢɱɟɦ z

- ɝɥɚɞɤɢɟ ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɧɧɨɣ r. ɗɬɨ ɫɥɟɞɭɟɬ ɢɡ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ

ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ Ⱥ(r) ɞɟɣɫɬɜɢɬɟɥɶɧɵ ɢ ɝɥɚɞɤɨ ɡɚɜɢɫɹɬ ɨɬ r, ɚ ɱɢɫɬɨ ɦɧɢɦɵɟ ɫɨɛɫɬɜɟɧɧɵɟ

ɡɧɚɱɟɧɢɹ ɹɜɥɹɸɬɫɹ ɩɪɨɫɬɵɦɢ.

ȼɬɨɪɨɟ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɏɨɩɮɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ

'(0) ɧɟ ɪɚɜɧɨ ɧɭɥɸ,

(5.2.8)

ɬɨ ɟɫɬɶ ɩɪɢ ɩɟɪɟɯɨɞɟ r ɱɟɪɟɡ ɧɭɥɶ ɦɧɢɦɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɚɬɪɢɰɵ Ⱥ(r)

ɩɟɪɟɫɟɤɚɸɬ ɦɧɢɦɭɸ ɨɫɶ ɫ ɧɟɧɭɥɟɜɨɣ ɫɤɨɪɨɫɬɶɸ (ɫɦ. ɪɢɫ. 5.2).

ȼ ɬɟɨɪɟɦɟ ɏɨɩɮɚ ɨ ɛɢɮɭɪɤɚɰɢɹɯ ɭɬɜɟɪɠɞɚɟɬɫɹ, ɱɬɨ ɟɫɥɢ ɜɵɩɨɥɧɟɧɵ ɭɫɥɨɜɢɹ (5.2.7) ɢ

(5.2.8), ɬɨ ɭɪɚɜɧɟɧɢɟ dx/dt = f(x, r) ɨɛɥɚɞɚɟɬ ɨɞɧɨɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦ ɫɟɦɟɣɫɬɜɨɦ ɩɟɪɢɨɞɢɱɟɫɤɢɯ

ɪɟɲɟɧɢɣ.

Cɦ. ɬɚɤɠɟ Ⱥ. Ⱥ. Ⱥɧɞɪɨɧɨɜ «ɉɪɢɦɟɧɟɧɢɟ ɬɟɨɪɢɢ ɉɭɚɧɤɚɪɟ ɨ «ɬɨɱɤɚɯ ɛɢɮɭɪɤɚɰɢɢ»

ɢ «ɫɦɟɧɟ ɭɫɬɨɣɱɢɜɨɫɬɢ» ɤ ɩɪɨɫɬɟɣɲɢɦ ɚɜɬɨɤɨɥɟɛɚɬɟɥɶɧɵɦ ɫɢɫɬɟɦɚɦ». ɋ. R. Ac.

Sci., Paris, 189, 15 (1929), ɫ. 559-561, ɚ ɬɚɤɠɟ ɤɨɦɦɟɧɬɚɪɢɢ ɜ ɤɧɢɝɟ ȼ. ɂ.

Ⱥɪɧɨɥɶɞɚ (ɫɧɨɫɤɚ ɜ ɪɚɡɞ. 3.2). — ɉɪɢɦ. ɪɟɞ.

Ⱦɥɹ ɩɨɥɧɨɝɨ ɚɧɚɥɢɡɚ ɩɨɜɟɞɟɧɢɹ ɫɢɫɬɟɦɵ ɜɚɠɧɨ ɡɧɚɬɶ ɬɚɤɠɟ, ɹɜɥɹɟɬɫɹ ɥɢ ɛɢɮɭɪɤɚɰɢɹ ɫɭɛ-

ɢɥɢ ɫɭɩɟɪɤɪɢɬɢɱɟɫɤɨɣ ɢ ɭɫɥɨɜɢɹ ɛɢɮɭɪɤɚɰɢɨɧɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɰɢɤɥɨɜ. ȼɫɟ ɷɬɢ ɜɨɩɪɨɫɵ

ɪɚɫɫɦɨɬɪɟɧɵ ɜ ɬɚɤɢɯ ɤɧɢɝɚɯ, ɤɚɤ, ɧɚɩɪɢɦɟɪ, Ɇɚɪɫɞɟɧ ɢ Ɇɚɤ-Ʉɪɚɤɟɧ (1976); Ƀɨɫɫ ɢ Ⱦɠɨɡɟɮ

(1980), ɑɭ ɢ ɏɟɣɥ (1982) ɢɥɢ Ƚɨɥɭɛɢɰɤɢɣ ɢ ɒɟɮɮɟɪ (1984)

ɉɪɢɜɟɞɟɦ ɫɥɟɞɭɸɳɢɣ ɩɪɢɦɟɪ ɛɢɮɭɪɤɚɰɢɢ ɏɨɩɮɚ. Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ

ɇɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ (0,0) ɹɜɥɹɟɬɫɹ ɟɟ ɬɨɱɤɨɣ ɪɚɜɧɨɜɟɫɢɹ. Ʌɢɧɟɣɧɵɣ ɚɧɚɥɢɡ ɷɬɨɣ ɫɢɫɬɟɦɵ

ɩɨɡɜɨɥɹɟɬ ɫɞɟɥɚɬɶ ɫɥɟɞɭɸɳɢɟ ɡɚɤɥɸɱɟɧɢɹ. Ⱦɥɹ r < 0 ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɟɲɟɧɢɟ ɹɜɥɹɟɬɫɹ

ɭɫɬɨɣɱɢɜɵɦ, ɬɨɝɞɚ ɤɚɤ ɞɥɹ r > 0 ɨɧɨ ɧɟɭɫɬɨɣɱɢɜɨ. ɉɪɢ r = 0 ɫɨɫɬɨɹɧɢɟ ɧɟɣɬɪɚɥɶɧɨ. Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ, ɛɢɮɭɪɤɚɰɢɸ ɏɨɩɮɚ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɩɪɢ ɜɨɡɪɚɫɬɚɧɢɢ r ɨɬ ɧɭɥɹ ɜ ɫɬɨɪɨɧɭ

ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ. Ƚɨɥɭɛɢɰɤɢɣ ɢ ɒɟɮɮɟɪ (1984) ɩɨɤɚɡɚɥɢ, ɱɬɨ

ɮɚɡɨɜɵɣ ɩɨɪɬɪɟɬ ɷɬɨɣ

ɫɢɫɬɟɦɵ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɜɢɞɟ, ɢɡɨɛɪɚɠɟɧɧɨɦ ɧɚ ɪɢɫ. 5.3. ɉɪɢ r > 0 ɭ ɫɢɫɬɟɦɵ

(5.2.6) ɟɫɬɶ ɜ ɬɨɱɧɨɫɬɢ ɨɞɧɨ ɩɟɪɢɨɞɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ. Ȼɨɥɟɟ ɬɨɝɨ, ɷɬɨ ɪɟɲɟɧɢɟ ɭɫɬɨɣɱɢɜɨ ɜ

ɬɨɦ ɫɦɵɫɥɟ, ɱɬɨ ɜɫɟ ɨɪɛɢɬɵ, ɧɚɯɨɞɹɳɢɟɫɹ ɜ ɟɝɨ ɨɤɪɟɫɬɧɨɫɬɢ, ɤ ɧɟɦɭ ɫɬɪɟɦɹɬɫɹ. Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ,

ɋɦ. ɬɚɤɠɟ ɭɩɨɦɢɧɚɜɲɢɟɫɹ ɪɚɧɟɟ ɤɧɢɝɢ Ȼɚɭɬɢɧɚ ɢ Ʌɟɨɧɬɨɜɢɱ (1990) ɢ Ⱥɪɧɨɥɶɞɚ (1984). — ɉɪɢɦ.

ɪɟɞ.

ɭɫɬɨɣɱɢɜɨɟ ɩɪɢ r < 0 ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɟɲɟɧɢɟ x = 0 ɫ ɪɨɫɬɨɦ r ɬɟɪɹɟɬ ɭɫɬɨɣɱɢɜɨɫɬɶ, ɢ

ɩɪɨɢɫɯɨɞɢɬ ɡɚɪɨɠɞɟɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ, ɭɫɬɨɣɱɢɜɵɯ ɩɪɢ r > 0.

Ʉɭɛɢɱɟɫɤɢɟ ɱɥɟɧɵ ɫɢɫɬɟɦɵ (5.2.9) ɧɚɩɪɚɜɥɹɸɬ ɯ ɜɨɜɧɭɬɪɶ ɤɪɭɝɨɜ °ɯ° = const. ȼ ɫɥɭɱɚɟ

ɛɨɥɶɲɢɯ |x| ɷɬɢ ɱɥɟɧɵ ɩɪɟɨɛɥɚɞɚɸɬ, ɫɬɹɝɢɜɚɹ ɨɪɛɢɬɵ ɤ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ,

ɤɨɝɞɚ °ɯ° ɦɚɥɨ, ɞɨɦɢɧɢɪɭɸɬ ɥɢɧɟɣɧɵɟ ɱɥɟɧɵ, ɢ ɟɫɥɢ r > 0, ɥɢɧɟɣɧɵɟ ɱɥɟɧɵ ɜɵɬɚɥɤɢɜɚɸɬ

ɨɪɛɢɬɵ ɢɡ ɨɤɪɟɫɬɧɨɫɬɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ ɹɜɥɹɟɬɫɹ

ɪɟɡɭɥɶɬɚɬɨɦ ɤɨɧɤɭɪɟɧɰɢɢ ɷɬɢɯ ɫɢɥ

ɂɦɟɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɫɬɚɬɟɣ, ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɫɬɟɩɟɧɢ ɨɛɨɛɳɚɸɳɢɯ ɛɢɮɭɪɤɚɰɢɨɧɧɭɸ

ɬɟɨɪɟɦɭ ɏɨɩɮɚ (ɜ ɱɚɫɬɧɨɫɬɢ, ɧɚ ɫɥɭɱɚɣ ɫɢɫɬɟɦ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɨɝɨ ɩɨɪɹɞɤɚ), ɚ ɬɚɤɠɟ ɪɹɞ

ɤɧɢɝ, ɜ ɤɨɬɨɪɵɯ ɪɚɫɫɦɨɬɪɟɧɵ ɩɪɢɥɨɠɟɧɢɹ ɷɬɨɣ ɬɟɨɪɟɦɵ ɜ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ ɧɚɭɤɢ (ɫɦ.,

ɧɚɩɪɢɦɟɪ, Ɇɚɪɫɞɟɧ ɢ Ɇɚɤ-Ʉɪɚɤɟɧ, 1976, Ƚɭɷɥ ɢ Ɋɟɫɥɟɪ, 1979, ɏɷɫɫɚɪɞ, Ʉɚɡɚɪɢɧɨɜ ɢ ȼɷɧ,

1981). ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ

ɨɧɚ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɢ ɜ ɷɤɨɧɨɦɢɤɟ (ɫɦ.; ɧɚɩɪɢɦɟɪ, Ȼɟɧɯɚɛɢɛ

ɢ Ɇɢɣɚɨ, 1981, Ȼɟɧɯɚɛɢɛ ɢ ɇɢɲɢɦɭɪɚ, 1986, Ɂɚɧɝ, 1988d, 1989b, 1990ɚ) — ɷɬɢɦ ɩɪɢɦɟɪɚɦ

ɩɨɫɜɹɳɟɧɚ ɨɫɬɚɥɶɧɚɹ ɱɚɫɬɶ ɝɥɚɜɵ.

5.3 ɍɩɪɨɳɟɧɧɚɹ ɦɨɞɟɥɶ ɞɟɥɨɜɨɝɨ ɰɢɤɥɚ Ʉɟɣɧɫɚ

Ɋɚɫɫɦɨɬɪɢɦ ɞɢɧɚɦɢɱɟɫɤɭɸ ɷɤɨɧɨɦɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ, ɤɨɬɨɪɚɹ ɩɪɟɞɥɨɠɟɧɚ Ʉɟɣɧɫɨɦ ɜ ɟɝɨ

«Ɉɛɳɟɣ ɬɟɨɪɢɢ». ɍɩɪɨɳɟɧɧɚɹ ɦɨɞɟɥɶ ɞɟɥɨɜɨɝɨ ɰɢɤɥɚ, ɫɨɝɥɚɫɧɨ Ʉɟɣɧɫɭ, ɨɩɢɫɵɜɚɟɬɫɹ

ɭɪɚɜɧɟɧɢɹɦɢ

Ɂɞɟɫɶ ɜɫɟ ɩɚɪɚɦɟɬɪɵ ɢ ɩɟɪɟɦɟɧɧɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵ ɢ ɨɡɧɚɱɚɸɬ

Y — ɧɚɰɢɨɧɚɥɶɧɵɣ ɞɨɯɨɞ;

R — ɩɪɨɰɟɧɬɧɭɸ ɫɬɚɜɤɭ;

I(Y, R) — ɮɭɧɤɰɢɸ ɫɩɪɨɫɚ ɧɚ ɢɧɜɟɫɬɢɰɢɢ (I

> 0, I

< 0);

S(Y, R) — ɮɭɧɤɰɢɸ ɫɛɟɪɟɠɟɧɢɣ (S

> 0, S

> 0);

L(Y, R) — ɫɭɦɦɚɪɧɵɣ ɫɩɪɨɫ ɧɚ ɞɟɧɶɝɢ (L

> 0, L

< 0);

— ɩɪɟɞɥɨɠɟɧɢɟ ɞɟɧɟɝ (ɮɢɤɫɢɪɨɜɚɧɧɚɹ ɜɟɥɢɱɢɧɚ);

α,

— ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ ɭɫɬɚɧɨɜɥɟɧɢɹ

ɗɬɚ ɫɢɫɬɟɦɚ ɨɬɪɚɠɚɟɬ ɬɨɬ ɩɪɨɫɬɨɣ ɮɚɤɬ, ɱɬɨ ɩɪɟɜɵɲɟɧɢɟ ɫɩɪɨɫɚ ɧɚ ɢɧɜɟɫɬɢɰɢɢ ɧɚɞ

ɫɛɟɪɟɠɟɧɢɹɦɢ ɩɪɢɜɨɞɢɬ ɤ ɜɨɡɪɚɫɬɚɧɢɸ ɞɨɯɨɞɚ, ɢ ɧɚɨɛɨɪɨɬ; ɢ ɱɬɨ ɟɫɥɢ ɫɩɪɨɫ ɧɚ ɞɟɧɶɝɢ ɜɵɲɟ,

ɱɟɦ ɢɯ ɩɪɟɞɥɨɠɟɧɢɟ, ɬɨ ɫɬɚɜɤɚ ɩɪɨɰɟɧɬɚ ɩɪɢɛɵɥɢ ɪɚɫɬɟɬ.

ɍɫɥɨɜɢɹ, ɧɚɥɚɝɚɟɦɵɟ ɧɚ ɜɯɨɞɹɳɢɟ ɜ ɫɢɫɬɟɦɭ ɮɭɧɤɰɢɢ ɢ ɢɯ ɩɪɨɢɡɜɨɞɧɵɟ (I

> 0, I

< 0, S

> 0, S

> 0, L

< 0), ɨɡɧɚɱɚɸɬ, ɱɬɨ ɢɧɜɟɫɬɢɰɢɢ ɧɚɯɨɞɹɬɫɹ ɜ ɩɪɹɦɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ

ɨɛɴɟɦɚ ɜɵɩɭɫɤɚ ɩɪɨɞɭɤɰɢɢ ɢ ɜ ɨɛɪɚɬɧɨɣ ɨɬ ɩɪɨɰɟɧɬɧɨɣ ɫɬɚɜɤɢ. ɗɬɨ ɡɧɚɱɢɬ ɬɚɤɠɟ, ɱɬɨ ɪɨɫɬ

ɧɚɰɢɨɧɚɥɶɧɨɝɨ ɞɨɯɨɞɚ ɢɥɢ ɩɪɨɰɟɧɬɧɨɣ ɫɬɚɜɤɢ ɛɭɞɟɬ ɩɨɛɭɠɞɚɬɶ ɧɚɫɟɥɟɧɢɟ ɤ ɛɨɥɶɲɢɦ

ɫɛɟɪɟɠɟɧɢɹɦ, ɚ ɩɪɢ ɭɫɥɨɜɢɢ ɪɨɫɬɚ ɩɪɨɢɡɜɨɞɫɬɜɚ ɩɪɨɞɭɤɰɢɢ ɢɥɢ ɭɦɟɧɶɲɟɧɢɹ ɩɪɨɰɟɧɬɧɨɣ

ɫɬɚɜɤɢ ɫɩɪɨɫ ɧɚ ɞɟɧɶɝɢ ɜɨɡɪɚɫɬɚɟɬ.

ȼ ɬɚɤɨɣ ɦɨɞɟɥɢ

ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ (Y

, R

);

ɤɨɬɨɪɨɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɟɪɟɫɟɱɟɧɢɟɦ ɤɪɢɜɵɯ L(Y, R) = L

ɢ F(Y, R) = 0. Ɋɚɫɫɦɨɬɪɟɧɢɟ ɫɢɫɬɟɦɵ

ɞɨɫɬɚɬɨɱɧɨ ɨɝɪɚɧɢɱɢɬɶ ɥɨɤɚɥɶɧɨɣ ɨɛɥɚɫɬɶɸ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɛɥɢɡɢ ɪɚɜɧɨɜɟɫɢɹ.

ɇɚɥɢɱɢɟ ɰɢɤɥɨɜ ɜ ɷɬɨɣ ɦɨɞɟɥɢ ɩɟɪɜɵɦ ɩɪɟɞɩɨɥɨɠɢɥ Ɍɨɪɪɟ (1977). Ɇɵ ɩɨɜɬɨɪɢɦ ɡɞɟɫɶ

ɟɝɨ ɚɧɚɥɢɡ.

ɑɬɨɛɵ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɛɢɮɭɪɤɚɰɢɨɧɧɨɣ ɬɟɨɪɟɦɨɣ ɏɨɩɮɚ, ɦɵ ɞɨɥɠɧɵ ɨɩɪɟɞɟɥɢɬɶ

ɭɫɥɨɜɢɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɩɚɪɵ ɱɢɫɬɨ ɦɧɢɦɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɜɵɹɫɧɢɬɶ, ɤɨɝɞɚ

ɪɚɜɧɨɜɟɫɢɟ ɬɟɪɹɟɬ ɭɫɬɨɣɱɢɜɨɫɬɶ. Ȼɥɚɝɨɞɚɪɹ Ɍɨɪɪɟ (1977) ɦɵ ɡɧɚɟɦ,

ɱɬɨ ɷɬɢ ɭɫɥɨɜɢɹ ɜɵ-

ɩɨɥɧɹɸɬɫɹ, ɟɫɥɢ ɜ ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ ɢɦɟɟɬ ɦɟɫɬɨ

ɉɨɫɤɨɥɶɤɭ α ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɥɸɛɵɟ ɡɧɚɱɟɧɢɹ ɢɡ R

, ɧɚɣɞɟɬɫɹ ɢ ɬɚɤɨɟ ɡɧɚɱɟɧɢɟ, ɩɪɢ

ɤɨɬɨɪɨɦ ɜɵɩɨɥɧɹɟɬɫɹ ɩɟɪɜɨɟ ɪɚɜɟɧɫɬɜɨ ɢɡ (5.3.2). Ⱥ ɩɨɫɤɨɥɶɤɭ F

= I

- S

, ɭɫɥɨɜɢɟ F

> 0

ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɢ ɡɚɞɚɧɧɨɦ ɨɛɴɟɦɟ ɩɪɨɢɡɜɨɞɫɬɜɚ ɩɪɟɞɟɥɶɧɵɟ ɢɧɜɟɫɬɢɰɢɢ ɜ

Ʉɨɷɮɮɢɰɢɟɧɬɵ α ɢ β ɜ ɞɚɧɧɨɦ ɤɨɧɬɟɤɫɬɟ ɧɚɡɵɜɚɸɬ ɬɚɤɠɟ ɩɚɪɚɦɟɬɪɚɦɢ ɚɞɚɩɬɚɰɢɢ ɢɥɢ

ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɪɟɚɤɰɢɢ. ɉɨɫɥɟɞɧɟɟ, ɧɚ ɧɚɲ ɜɡɝɥɹɞ, ɧɚɢɛɨɥɟɟ ɭɞɚɱɧɨ, ɩɨɫɤɨɥɶɤɭ ɷɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ

ɨɩɪɟɞɟɥɹɸɬ ɪɟɡɭɥɶɬɢɪɭɸɳɭɸ ɪɟɚɤɰɢɸ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɚɝɟɧɬɨɜ ɧɚ ɨɬɤɥɨɧɟɧɢɟ ɫɢɫɬɟɦɵ ɨɬ ɫɨɫɬɨɹɧɢɹ

ɪɚɜɧɨɜɟɫɢɹ. — ɉɪɢɦ. ɪɟɞ.

ɩɪɨɢɡɜɨɞɫɬɜɨ ɜɵɲɟ ɩɪɟɞɟɥɶɧɵɯ ɫɛɟɪɟɠɟɧɢɣ

. Ȼɨɥɟɟ ɪɚɡɜɟɪɧɭɬɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ

ɫɨɨɬɧɨɲɟɧɢɣ (5.3.2) ɟɫɬɶ ɭ Ɍɨɪɪɟ (1977).

Ɍɟɨɪɟɦɚ 5.3.1. ɉɭɫɬɶ (5.3.2) ɜɵɩɨɥɧɹɸɬɫɹ. Ɍɨɝɞɚ ɜ ɫɢɫɬɟɦɟ (5.3.1) ɫɭɳɟɫɬɜɭɸɬ

ɩɪɟɞɟɥɶɧɵɟ ɰɢɤɥɵ ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ (ɍɨ,Ⱦɨ) (ɛɢɮɭɪɤɚɰɢɹ ɏɨɩɮɚ). Ʉɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ

ɛɢɮɭɪɤɚɰɢɨɧɧɨɝɨ ɩɚɪɚɦɟɬɪɚ α ɪɚɜɧɨ α

. Ɋɨɠɞɟɧɧɵɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɛɢɮɭɪɤɚɰɢɢ ɰɢɤɥɵ ɩɟɪɢɨɞɚ

2π/

(

) ɩɪɢɛɥɢɠɟɧɧɨ ɨɩɢɫɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ

ɝɞɟ z

= {α

— F

)}

1/2

- ɩɚɪɚɦɟɬɪ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɚɦɩɥɢɬɭɞɟ, ɢ

ɝɞɟ ɯ

—ɤɨɧɫɬɚɧɬɵ. Ȼɨɥɟɟ ɬɨɝɨ, ɟɫɥɢ ɯ

ɩɨɥɨɠɢɬɟɥɶɧɨ, ɬɨ ɩɟɪɢɨɞɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ

ɛɭɞɟɬ ɭɫɬɨɣɱɢɜɵɦ, ɜ ɫɥɭɱɚɟ ɨɬɪɢɰɚɬɟɥɶɧɨɝɨ ɯ

— ɧɟɭɫɬɨɣɱɢɜɵɦ.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɟɨɪɟɦɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɦɟɬɨɞɨɦ Ƀɨɫɫɚ ɢ Ⱦɠɨɡɟɮɚ

(1980). ɉɨɫɤɨɥɶɤɭ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ ɞɨɤɚɡɚɧɨ Ɍɨɪɪɟ (1977), ɧɚɦ ɧɭɠɧɨ

ɩɨɤɚɡɚɬɶ ɡɞɟɫɶ ɬɨɥɶɤɨ ɦɟɬɨɞ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɚɫɱɟɬɚ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ.

ɑɬɨɛɵ ɡɚɩɢɫɚɬɶ ɢɫɯɨɞɧɭɸ ɫɢɫɬɟɦɭ ɜ ɥɨɤɚɥɶɧɨɦ ɜɢɞɟ, ɜɜɟɞɟɦ ɧɨɜɵɟ ɮɭɧɤɰɢɢ U

= Y – Y

= R – R

, ɝɞɟ ɮɭɧɤɰɢɢ Y ɢ R ɭɞɨɜɥɟɬɜɨɪɹɸɬ (5.3.1). Ȼɭɤɜɨɣ ɯ ɨɛɨɡɧɚɱɟɧɵ ɦɚɥɵɟ ɨɬɤɥɨɧɟɧɢɹ

ɩɚɪɚɦɟɬɪɚ α ɨɬ ɡɧɚɱɟɧɢɹ α

, ɬ.ɟ. ɯ = α — α

. Ɍɨɝɞɚ (5.3.1) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɤɚɤ

ɝɞɟ U = (U

, U

)

— ɹɤɨɛɢɚɧ ɜ ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ

ɢ N - ɱɥɟɧɵ, ɤɜɚɞɪɚɬɢɱɧɵɟ ɩɨ U. ɉɚɪɚ ɢɫɤɨɦɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ

ɍɫɥɨɜɢɟ F

> 0 ɨɡɧɚɱɚɟɬ, ɱɬɨ ɡɞɟɫɶ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɜɵɪɨɠɞɟɧɧɵɣ ɫɥɭɱɚɣ ɦɨɞɟɥɢ Ʉɟɣɧɫɚ, ɤɨɝɞɚ IS-

ɤɪɢɜɚɹ ɦɨɧɨɬɨɧɧɨ ɜɨɡɪɚɫɬɚɟɬ — ɉɪɢɦ. ɪɟɞ.

ɹɤɨɛɢɚɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɜɵɪɚɠɟɧɢɹ

ɝɞɟ α = α

+ ɯ. ȼ ɬɨɱɤɟ ɯ = 0 ɜ ɫɥɭɱɚɟ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɣ (5.3.2) ɢɦɟɟɦ ɩɚɪɭ ɱɢɫɬɨ

ɦɧɢɦɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ±iz

. ȿɫɥɢ ɨɛɨɡɧɚɱɢɬɶ ɱɟɪɟɡ z(x) ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ,

ɤɨɬɨɪɨɟ ɩɪɢ ɯ = 0 ɪɚɜɧɨ iz

, ɬɨ Rɟ[z

(0)] ɧɟ ɛɭɞɟɬ ɪɚɜɧɵɦ ɧɭɥɸ, ɱɬɨ ɨɛɭɫɥɨɜɥɢɜɚɟɬ ɩɨɬɟɪɸ

ɭɫɬɨɣɱɢɜɨɫɬɢ ɪɚɜɧɨɜɟɫɢɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɜɵɜɟɥɢ ɭɫɥɨɜɢɟ ɛɢɮɭɪɤɚɰɢɢ ɏɨɩɮɚ.

ɑɬɨɛɵ ɩɨɥɭɱɢɬɶ ɬɨɱɧɭɸ ɮɨɪɦɭɥɭ ɞɥɹ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ, ɧɚɣɞɟɦ ɢɡ

ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ ɏ ɢ ɫɨɩɪɹɠɟɧɧɵɣ ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ X*, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ

ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ z(x), ɤɨɬɨɪɵɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɹɦ ¢X, X*² = 1 ɢ ¢

Õ , X*²= 0, ɝɞɟ

(,) — ɷɪɦɢɬɨɜɚ ɛɢɥɢɧɟɣɧɚɹ ɮɨɪɦɚ ɜ ɋ

. ɂɦɟɟɦ

Ɍɚɤ ɤɚɤ ɏ ɢ

ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ, U ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɜɢɞɟ ɢɯ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ

ɝɞɟ ɮɭɧɤɰɢɹ

(t) ɩɨɤɚ ɧɟɢɡɜɟɫɬɧɚ. ɉɨɞɫɬɚɜɥɹɹ (5.3.9) ɜ (5.3.5), ɭɦɧɨɠɚɹ ɩɨɥɭɱɟɧɧɵɟ

ɮɭɧɤɰɢɢ ɧɚ

ɢ ɫɤɥɚɞɵɜɚɹ ɡɚɬɟɦ ɩɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ, ɩɨɥɭɱɢɦ

ɝɞɟ r

— ɧɟɤɢɟ ɤɨɦɩɥɟɤɫɧɵɟ ɱɢɫɥɚ, ɤɨɬɨɪɵɟ ɦɵ ɡɞɟɫɶ ɜɵɩɢɫɵɜɚɬɶ ɧɟ ɛɭɞɟɦ. Ʉɚɤ ɩɨɤɚɡɚɧɨ

ɜ ɤɧɢɝɟ Ƀɨɫɫɚ ɢ Ⱦɠɨɡɟɮɚ, ɪɟɲɟɧɢɹ (5.3.10) ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɜ ɜɢɞɟ ɪɹɞɨɜ