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

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ɝɞɟ ɨɩɭɳɟɧɵ ɱɥɟɧɵ, ɫɨɞɟɪɠɚɳɢɟ ɜɵɫɲɢɟ ɫɬɟɩɟɧɢ (ɪ-ɪ

)

. ȿɫɥɢ ɨɛɨɡɧɚɱɢɬɶ

ɧɚɱɚɥɶɧɭɸ ɰɟɧɭ ɤɚɤ ɪ(0), ɬɨ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2.4.2) ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ

ȿɫɥɢ ɪɚɜɧɨɜɟɫɢɟ ɭɫɬɨɣɱɢɜɨ, ɬɨ ɩɪɢ tĺ+  ɢɦɟɟɬ ɦɟɫɬɨ ɪ(t)ĺ ɪ

. ɗɬɨ ɜɵɩɨɥɧɹɟɬɫɹ ɜ

ɬɨɦ ɢ ɬɨɥɶɤɨ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ

Ʉɨɝɞɚ ɤɪɢɜɚɹ ɩɪɟɞɥɨɠɟɧɢɹ ɢɦɟɟɬ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɧɚɤɥɨɧ, ɩɨɫɥɟɞɧɟɟ ɭɫɥɨɜɢɟ

ɜɵɩɨɥɧɹɟɬɫɹ. ȿɫɥɢ ɠɟ ɧɚɤɥɨɧ ɨɬɪɢɰɚɬɟɥɶɧɵɣ, ɨɧ ɞɨɥɠɟɧ ɛɵɬɶ ɦɟɧɟɟ ɤɪɭɬɵɦ, ɱɟɦ ɭ

ɤɪɢɜɨɣ ɫɩɪɨɫɚ. Ɍɚɤ ɱɬɨ ɟɫɥɢ ɜɵɩɨɥɧɟɧɵ ɭɫɥɨɜɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ, ɬɨ ɨɬɜɟɬ ɧɚ

ɩɨɫɬɚɜɥɟɧɧɵɣ ɜɨɩɪɨɫ ɩɨɥɭɱɟɧ: ɤɨɝɞɚ ɪɚɫɬɟɬ ɫɩɪɨɫ, ɞɨɥɠɧɚ ɪɚɫɬɢ ɰɟɧɚ. Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ, ɪɟɡɭɥɶɬɚɬɵ ɫɪɚɜɧɢɬɟɥɶɧɨɝɨ ɫɬɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɦɨɝɭɬ ɛɵɬɶ ɜɵɜɟɞɟɧɵ ɢɡ

ɭɫɥɨɜɢɣ ɭɫɬɨɣɱɢɜɨɫɬɢ.

ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɫɬɶ ɬɟɨɪɢɢ ȼɚɥɶɪɚɫɚ, ɜ ɬɟɨɪɢɢ ɧɨɪɦɚɥɶɧɨɣ ɰɟɧɵ Ɇɚɪɲɚɥɥɚ

ɤɨɥɢɱɟɫɬɜɨ ɩɪɟɞɥɨɠɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɜɟɥɢɱɢɧɨɣ ɫɚɦɨɪɟɝɭɥɢɪɭɟɦɨɣ. ȿɫɥɢ «ɰɟɧɚ

ɫɩɪɨɫɚ» ɩɪɟɜɵɲɚɟɬ «ɰɟɧɭ ɩɪɟɞɥɨɠɟɧɢɹ», ɤɨɥɢɱɟɫɬɜɨ ɩɪɟɞɥɨɠɟɧɢɹ ɛɭɞɟɬ

ɭɜɟɥɢɱɢɜɚɬɶɫɹ. ɋɨɯɪɚɧɢɜ ɨɛɨɡɧɚɱɟɧɢɹ (2.4.2) ɢ ɢɡɛɚɜɢɜɲɢɫɶ ɨɬ ɜɵɫɲɢɯ ɫɬɟɩɟɧɟɣ,

ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ

ɪɟɲɟɧɢɟɦ ɤɨɬɨɪɨɝɨ ɹɜɥɹɟɬɫɹ

Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɪɚɜɧɨɜɟɫɢɟ ɛɵɥɨ ɭɫɬɨɣɱɢɜɵɦ, ɩɨɬɪɟɛɭɟɦ, ɱɬɨɛɵ

ɬ. ɟ. ɧɚɤɥɨɧ ɤɪɢɜɨɣ ɩɪɟɞɥɨɠɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɦɟɧɶɲɟ, ɱɟɦ ɭ ɤɪɢɜɨɣ ɫɩɪɨɫɚ.

ȿɫɥɢ Dp < 0, ɢɦɟɟɦ (S

- D

)/S

> 0. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɥɨɜɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ

Ɇɚɪɲɚɥɥɚ ɬɪɟɛɭɸɬ, ɱɬɨɛɵ ɤɨɥɢɱɟɫɬɜɨ

Ɂɞɟɫɶ D

ɢ S

— ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɩɟɪɟɦɟɧɧɨɣ ɪ ɮɭɧɤɰɢɣ D ɢ S ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.—ɉɪɢɦ. ɪɟɞ.

ɩɪɟɞɥɨɠɟɧɢɹ ɭɜɟɥɢɱɢɜɚɥɨɫɶ ɜ ɥɸɛɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫɩɪɨɫ, ɬɨɝɞɚ ɤɚɤ

ɢɡɦɟɧɟɧɢɹ ɰɟɧɵ ɨɤɚɡɵɜɚɸɬɫɹ ɧɟɹɜɧɨ ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɡɧɚɤɚ, ɧɚɤɥɨɧɚ ɤɪɢɜɨɣ

ɩɪɟɞɥɨɠɟɧɢɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ ɢɧɮɨɪɦɚɰɢɢ ɨɛ ɭɫɥɨɜɢɹɯ ɭɫɬɨɣɱɢɜɨɫɬɢ ɦɵ

ɧɟɦɟɞɥɟɧɧɨ ɩɨɥɭɱɚɟɦ, ɱɬɨ ɪɨɫɬ ɫɩɪɨɫɚ ɩɪɢɜɟɞɟɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɩɪɨɢɡɜɨɞɫɬɜɚ

Ɍɚɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɭɫɥɨɜɢɹɦɢ ɭɫɬɨɣɱɢɜɨɫɬɢ ɢ ɪɟɡɭɥɶɬɚɬɚɦɢ

ɫɪɚɜɧɢɬɟɥɶɧɨɣ ɫɬɚɬɢɤɢ ɧɚɡɜɚɧɵ ɋɚɦɭɷɥɶɫɨɧɨɦ «ɩɪɢɧɰɢɩɨɦ ɫɨɨɬɜɟɬɫɬɜɢɹ».

ɉɪɟɞɩɨɥɚɝɚɥɨɫɶ, ɱɬɨ ɟɫɥɢ ɷɬɨɬ ɩɪɢɧɰɢɩ ɜɟɪɟɧ, ɬɨ ɦɟɬɨɞ ɫɪɚɜɧɟɧɢɣ ɫ ɪɚɜɧɨɜɟɫɢɟɦ

ɦɨɠɧɨ ɩɪɢɡɧɚɬɶ ɝɨɞɧɵɦ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɥɟɞɫɬɜɢɣ ɞɚɧɧɵɯ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɯ

ɢɡɦɟɧɟɧɢɣ. ȿɫɥɢ ɡɚɜɟɞɨɦɨ ɩɪɟɞɩɨɥɚɝɚɬɶ ɩɪɨɰɟɫɫ ɭɫɬɨɣɱɢɜɵɦ, ɬɨ ɦɚɥɵɟ ɢɡɦɟɧɟɧɢɹ

ɩɚɪɚɦɟɬɪɨɜ ɦɨɝɭɬ ɩɪɢɜɨɞɢɬɶ ɬɨɥɶɤɨ ɤ ɩɥɚɜɧɵɦ ɢɡɦɟɧɟɧɢɹɦ ɩɟɪɟɦɟɧɧɵɯ. ɇɢɤɚɤɢɯ

ɜɧɟɡɚɩɧɵɯ ɩɟɪɟɦɟɧ ɧɚɛɥɸɞɚɬɶɫɹ ɢ ɧɟ ɦɨɠɟɬ.

ɋɩɪɚɜɟɞɥɢɜɨɫɬɶ ɩɪɢɧɰɢɩɚ ɫɨɨɬɜɟɬɫɬɜɢɹ ɡɚɜɢɫɢɬ ɨɬ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ

ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɨɛ ɭɫɬɨɣɱɢɜɨɫɬɢ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ, ȼɚɠɧɨ ɢɫɫɥɟɞɨɜɚɬɶ, ɱɬɨ

ɩɪɨɢɡɨɣɞɟɬ, ɟɫɥɢ ɷɬɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨɫɥɚɛɢɬɶ.

2.5 ɇɟɭɫɬɨɣɱɢɜɨɫɬɶ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ

ɂɡ ɫɤɚɡɚɧɧɨɝɨ ɜɵɲɟ ɦɵ ɜɢɞɢɦ, ɱɬɨ ɝɢɩɨɬɟɡɚ ɭɫɬɨɣɱɢɜɨɫɬɢ ɜɚɠɧɚ, ɩɨɫɤɨɥɶɤɭ ɱɚɫɬɨ ɫ

ɟɟ ɩɨɦɨɳɶɸ ɦɨɠɧɨ ɩɨɥɭɱɚɬɶ ɨɫɦɵɫɥɟɧɧɵɟ ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɪɟɡɭɥɶɬɚɬɵ.

Ɇɧɨɝɨɱɢɫɥɟɧɧɵɟ ɭɞɚɱɧɵɟ ɩɪɢɥɨɠɟɧɢɹ ɩɪɢɧɰɢɩɚ ɫɨɨɬɜɟɬɫɬɜɢɹ ɤ ɪɚɡɥɢɱɧɵɦ

ɷɤɨɧɨɦɢɱɟɫɤɢɦ ɩɪɨɛɥɟɦɚɦ ɩɨɤɚɡɚɥɢ, ɱɬɨ ɷɬɨɬ ɦɟɬɨɞ ɞɨɜɨɥɶɧɨ ɩɨɥɟɡɟɧ. Ɉɞɧɚɤɨ

ɧɟɨɛɯɨɞɢɦɨ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɩɪɢɧɹɬɢɟ ɝɢɩɨɬɟɡɵ ɭɫɬɨɣɱɢɜɨɫɬɢ ɧɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ

ɷɤɨɧɨɦɢɫɬɚɦ ɧɭɠɧɨ ɨɬɜɟɪɝɧɭɬɶ ɮɚɤɬ ɧɚɥɢɱɢɹ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ, ɩɪɨɫɬɨ

ɨɬɧɨɲɟɧɢɟ ɤ

ɧɟɭɫɬɨɣɱɢɜɨɫɬɹɦ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ɞɨɥɠɧɨ ɛɵɬɶ ɢɡɦɟɧɟɧɨ.

ɂɡɦɟɧɟɧɢɟ ɩɨɡɢɰɢɢ ɛɨɥɶɲɢɧɫɬɜɚ ɷɤɨɧɨɦɢɫɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɢɩɨɬɟɡɵ

ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ɦɨɠɧɨ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ ɜɵɞɟɪɠɤɚɦɢ

ɢɡ ɜɨɫɩɨɦɢɧɚɧɢɣ ɋɚɦɭɷɥɶɫɨɧɚ ɤɚɤ ɤɥɚɫɫɢɤɚ-ɬɟɨɪɟɬɢɤɚ, ɨɬɧɨɫɹɳɢɯɫɹ ɤ 1932-37

ɝɨɞɚɦ: «ɩɨɫɤɨɥɶɤɭ, ɟɫɬɟɫɬɜɟɧɧɨ, ɬɟɨɪɟɬɢɤ ɪɚɜɧɨɜɟɫɢɹ ɫɬɪɟɦɢɥɫɹ ɪɚɫɫɦɚɬɪɢɜɚɬɶ

ɦɨɞɟɥɢ, ɜ ɤɨɬɨɪɵɯ ɩɪɨɰɟɫɫɵ ɭɫɬɪɟɦɥɹɸɬɫɹ ɤ ɟɞɢɧɫɬɜɟɧɧɨɦɭ ɩɨɥɨɠɟɧɢɸ ɧɟ-

ɡɚɜɢɫɢɦɨ ɨɬ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ...

ɱɟɫɬɧɨ ɝɨɜɨɪɹ, ɦɵ, ɬɟɨɪɟɬɢɤɢ, ɧɚɞɟɹɥɢɫɶ ɧɟ

ɜɜɨɞɢɬɶ ɹɜɥɟɧɢɹ ɝɢɫɬɟɪɟɡɢɫɚ ɜ ɧɚɲɭ ɦɨɞɟɥɶ..., ɧɨ ɜ ɪɟɚɥɶɧɨɫɬɢ ɦɵ ɧɟɹɜɧɨ

ɢɫɩɨɥɶɡɨɜɚɥɢ ɦɨɞɟɥɢ, ɫɨɞɟɪɠɚɳɢɟ ɝɢɫɬɟɪɟɡɢɫ:

ɂɫɩɚɧɢɹ ɧɟ ɦɨɝɥɚ ɛɵ ɨɫɬɚɜɚɬɶɫɹ ɩɪɟɠɧɟɣ ɩɨɫɥɟ Ʉɨɥɭɦɛɚ... ɨɱɟɜɢɞɧɨ, ɱɬɨ ɜ ɬɚɤɢɯ

ɦɨɞɟɥɹɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɜɜɟɞɟɧɢɹ ɜ ɫɢɫɬɟɦɭ ɨɩɪɟɞɟɥɟɧɧɨɝɨ

Ɂɞɟɫɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɜ ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ ɜɵɩɨɥɧɟɧɨ ɭɫɥɨɜɢɟ S

> D

. ɂɡ ɷɬɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ

ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ ɨɛɟɫɩɟɱɟɧɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɬɨɱɤɢ ɪɚɜɧɨɜɟɫɢɹ ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ D

< 0. — ɉɪɢɦ. ɪɟɞ.

ɪɚɡɛɚɥɚɧɫɢɪɭɸɳɟɝɨ ɮɚɤɬɨɪɚ Ɇ ɜɫɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɧɟ ɨɫɬɚɸɬɫɹ

ɧɟɢɡɦɟɧɧɵɦɢ...» (ɋɚɦɭɷɥɶɫɨɧ, 1972, ɫ. 540-1).

ɉɪɢɧɹɬɢɟ ɤɨɧɰɟɩɰɢɢ ɭɫɬɨɣɱɢɜɨɫɬɢ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ɛɵɥɨ ɜ

ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ ɨɛɭɫɥɨɜɥɟɧɨ ɪɚɡɜɢɬɢɟɦ ɟɫɬɟɫɬɜɟɧɧɵɯ ɧɚɭɤ, ɝɞɟ ɞɥɹ

ɩɪɨɜɟɞɟɧɢɹ ɨɫɦɵɫɥɟɧɧɨɝɨ ɚɧɚɥɢɡɚ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɬɪɟɛɨɜɚɥɚɫɶ ɢɯ

ɭɫɬɨɣɱɢɜɨɫɬɶ. Ⱦɥɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɧɚɭɤ ɷɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɞɟɫɤɪɢɩɬɢɜɧɵɟ ɦɨɞɟɥɢ

ɞɨɥɠɧɵ ɩɪɢɜɨɞɢɬɶ ɤ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɤɚɱɟɫɬɜɟɧɧɵɦ

ɪɟɡɭɥɶɬɚɬɚɦ, ɟɫɥɢ ɷɤɫɩɟɪɢɦɟɧɬ

ɩɨɜɬɨɪɹɟɬɫɹ ɩɪɢ ɦɚɥɵɯ ɢɡɦɟɧɟɧɢɹɯ ɭɫɥɨɜɢɣ. Ɍɚɤɨɟ ɨɬɧɨɲɟɧɢɟ ɤ ɪɟɚɥɶɧɨɫɬɢ ɜɵ-

ɬɟɤɚɥɨ ɢɡ ɩɨɬɪɟɛɧɨɫɬɢ ɜ ɬɨɦ, ɱɬɨɛɵ ɨɧɚ ɛɵɥɚ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɭɫɬɨɣɱɢɜɨɣ ɜ

ɧɟɤɨɬɨɪɨɦ ɫɬɪɭɤɬɭɪɧɨɦ ɫɦɵɫɥɟ. Ⱥ ɭɛɟɠɞɟɧɢɟ ɜ ɬɨɦ, ɱɬɨ ɦɚɥɵɟ ɢɡɦɟɧɟɧɢɹ

ɨɤɪɭɠɚɸɳɢɯ ɭɫɥɨɜɢɣ ɧɟ ɩɪɢɜɟɞɭɬ ɤ ɤɨɪɟɧɧɵɦ ɢ ɤɚɱɟɫɬɜɟɧɧɵɦ ɪɚɡɥɢɱɢɹɦ ɜ

ɩɨɜɟɞɟɧɢɢ ɪɟɚɥɶɧɨɣ ɫɢɫɬɟɦɵ, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɧɚɫɥɟɞɢɟ

ɦɟɯɚɧɢɫɬɢɱɟɫɤɢ

ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɝɨ 19-ɝɨ ɜɟɤɚ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɢɞɟɹɦɢ ɞɟɬɟɪɦɢɧɢɫɬɢɱɟɫɤɨɣ

ɦɟɯɚɧɢɤɢ, ɫɥɨɠɧɵɟ ɹɜɥɟɧɢɹ, ɤɨɬɨɪɵɟ ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɨɛɴɹɫɧɟɧɵ ɫ ɩɪɢɜɥɟɱɟɧɢɟɦ

ɨɛɵɱɧɵɯ ɦɨɞɟɥɟɣ, ɫɜɨɞɹɬɫɹ ɥɢɛɨ ɤ ɩɨɫɬɭɥɚɬɭ, ɱɬɨ ɩɨɞɨɛɧɵɟ ɹɜɥɟɧɢɹ ɧɟ ɩɨɞɥɟɠɚɬ

ɚɧɚɥɢɬɢɱɟɫɤɨɦɭ ɪɚɫɫɦɨɬɪɟɧɢɸ, ɥɢɛɨ ɤ ɭɬɜɟɪɠɞɟɧɢɸ, ɱɬɨ ɫɢɫɬɟɦɚ ɧɚɯɨɞɢɬɫɹ ɩɨɞ

ɜɨɡɞɟɣɫɬɜɢɟɦ ɱɢɫɬɨ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɜɥɢɹɧɢɣ. Ʉɚɤ ɫɥɟɞɫɬɜɢɟ, ɯɚɨɬɢɱɟɫɤɢɟ ɹɜɥɟɧɢɹ ɜ

ɷɜɨɥɸɰɢɨɧɧɵɯ ɫɢɫɬɟɦɚɯ ɬɪɚɤɬɭɸɬɫɹ

ɤɚɤ ɩɪɟɯɨɞɹɳɢɟ ɹɜɥɟɧɢɹ ɢɥɢ ɩɪɨɫɬɵɟ

ɜɨɡɦɭɳɟɧɢɹ ɞɨɥɝɨɜɪɟɦɟɧɧɨɣ ɪɚɜɧɨɜɟɫɧɨɣ ɷɜɨɥɸɰɢɢ.

ɇɵɧɟ ɷɬɚ ɬɨɱɤɚ ɡɪɟɧɢɹ ɧɚ ɭɫɬɨɣɱɢɜɨɫɬɶ ɩɪɟɬɟɪɩɟɥɚ ɢɡɦɟɧɟɧɢɹ. ɍɫɬɨɣɱɢɜɨɫɬɶ

ɛɨɥɟɟ ɧɟ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɜ ɧɚɭɤɟ ɚɩɪɢɨɪɧɨ. ɉɨɤɚɡɚɧɨ, ɱɬɨ ɦɚɥɵɟ ɫɞɜɢɝɢ ɩɚɪɚɦɟɬɪɨɜ

ɦɨɝɭɬ ɩɪɢɜɨɞɢɬɶ ɤ ɫɬɪɭɤɬɭɪɧɵɦ ɢɡɦɟɧɟɧɢɹɦ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ. Ɍɚɤɢɟ

ɫɬɪɭɤɬɭɪɧɵɟ ɢɡɦɟɧɟɧɢɹ ɜ ɷɜɨɥɸɰɢɨɧɧɵɯ ɫɢɫɬɟɦɚɯ ɹɜɥɹɸɬɫɹ ɧɟ ɢɫɤɥɸɱɢɬɟɥɶɧɵɦɢ,

ɚ, ɫɤɨɪɟɟ, ɨɛɳɢɦɢ ɫɥɭɱɚɹɦɢ. Ⱦɥɹ

ɧɟɥɢɧɟɣɧɵɯ ɧɟɭɫɬɨɣɱɢɜɵɯ ɫɢɫɬɟɦ ɯɚɪɚɤɬɟɪɧɵ

ɫɥɨɠɧɵɟ ɹɜɥɟɧɢɹ, ɬɚɤɢɟ, ɤɚɤ ɪɟɝɭɥɹɪɧɵɟ ɨɫɰɢɥɥɹɰɢɢ ɢ ɯɚɨɫ. Ⱦɚɠɟ ɜ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɩɪɨɫɬɵɯ ɧɟɥɢɧɟɣɧɵɯ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ ɦɨɠɟɬ ɧɚɛɥɸɞɚɬɶɫɹ ɫɩɨɧɬɚɧɧɨɟ

ɨɛɪɚɡɨɜɚɧɢɟ (ɢɡ ɯɚɨɫɚ) ɫɥɨɠɧɨ ɨɪɝɚɧɢɡɨɜɚɧɧɵɯ ɫɬɪɭɤɬɭɪ. Ȼɵɥɨ ɧɚɣɞɟɧɨ, ɱɬɨ ɫɥɨɠɧɨ

ɨɪɝɚɧɢɡɨɜɚɧɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ, ɜɪɟɦɟɧɧɵɟ ɢɥɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ

ɫɬɪɭɤɬɭɪɵ ɜɨɡɧɢɤɚɸɬ ɢɡ ɯɚɨɬɢɱɟɫɤɢɯ ɫɨɫɬɨɹɧɢɣ, ɢ ɜ ɬɚɤɢɯ ɫɚɦɨɨɪɝɚɧɢɡɭɸɳɢɯɫɹ

ɫɢɫɬɟɦɚɯ ɜɦɟɫɬɨ ɭɫɬɨɣɱɢɜɨɫɬɢ ɢ

ɝɚɪɦɨɧɢɢ ɦɵ ɨɛɧɚɪɭɠɢɜɚɟɦ ɷɜɨɥɸɰɢɨɧɧɵɟ

ɩɪɨɰɟɫɫɵ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɟɳɟ ɛɨɥɶɲɟɦɭ ɪɚɡɧɨɨɛɪɚɡɢɸ ɢ ɭɫɥɨɠɧɟɧɢɸ ɫɬɪɭɤɬɭɪ

(ɇɢɤɨɥɢɫ ɢ ɉɪɢɝɨɠɢɧ, 1977, ɏɚɤɟɧ, 1977, 1983).

Ɇɵ ɩɨɤɚɠɟɦ, ɱɬɨ ɷɬɢ ɢɞɟɢ ɦɨɝɭɬ ɛɵɬɶ ɩɪɢɥɨɠɟɧɵ ɢ ɤ ɷɤɨɧɨɦɢɤɟ. ȼ

ɫɨɜɪɟɦɟɧɧɵɯ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ ɧɚ ɩɨɜɟɫɬɤɟ ɞɧɹ ɨɤɚɡɚɥɢɫɶ ɦɟɞɥɟɧɧɵɟ

ɩɪɨɰɟɫɫɵ, ɫɦɟɧɹɸɳɢɟɫɹ ɪɟɡɤɢɦɢ, ɢɧɨɝɞɚ ɧɟɩɪɟɞɫɤɚɡɭɟɦɵɦɢ ɩɟɪɟɦɟɧɚɦɢ.

ɗɤɨɧɨɦɢɱɟɫɤɢɟ ɫɢɫɬɟɦɵ, ɬɚɤɢɟ, ɤɚɤ ɪɵɧɤɢ

ɬɪɭɞɚ, ɤɪɟɞɢɬɧɨ-ɞɟɧɟɠɧɵɟ ɪɵɧɤɢ,

ɭɪɛɚɧɢɫɬɢɱɟɫɤɢɟ ɫɢɫɬɟɦɵ, ɫɢɫɬɟɦɵ ɩɟɪɟɜɨɡɨɤ ɢ ɫɜɹɡɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɧɚɥɢɱɢɟɦ

ɯɚɨɫɚ.

ȼɫɟ ɷɬɢ ɯɨɪɨɲɨ ɧɚɛɥɸɞɚɟɦɵɟ, ɡɚɩɭɬɚɧɧɵɟ ɹɜɥɟɧɢɹ ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɚɞɟɤɜɚɬɧɨ

ɨɛɴɹɫɧɟɧɵ ɫɭɳɟɫɬɜɭɸɳɢɦɢ ɷɤɨɧɨɦɢɱɟɫɤɢɦɢ ɬɟɨɪɢɹɦɢ. Ɋɚɫɬɭɳɟɟ ɩɪɢɡɧɚɧɢɟ

ɡɧɚɱɟɧɢɹ ɩɨɞɨɛɧɵɯ ɧɟɪɟɝɭɥɹɪɧɨɫɬɟɣ — ɢɥɢ ɫɬɪɭɤɬɭɪɧɵɯ ɢɡɦɟɧɟɧɢɣ ɢ ɯɚɨɬɢɱɟɫɤɢɯ

ɹɜɥɟɧɢɣ — ɜɵɡɵɜɚɟɬ ɮɭɧɞɚɦɟɧɬɚɥɶɧɭɸ ɩɨɬɪɟɛɧɨɫɬɶ ɜ ɧɨɜɵɯ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢɞɟɹɯ ɢ

ɢɧɫɬɪɭɦɟɧɬɚɯ, ɤɨɬɨɪɵɟ ɦɨɝɥɢ ɛɵ ɩɨɡɜɨɥɢɬɶ ɩɪɨɜɨɞɢɬɶ ɢɫɫɥɟɞɨɜɚɧɢɹ ɡɚ ɝɪɚɧɢɰɚɦɢ

ɬɪɚɞɢɰɢɨɧɧɨɣ ɷɤɨɧɨɦɢɤɢ, ɛɚɡɢɪɭɸɳɟɣɫɹ ɧɚ ɬɟɨɪɢɢ ɨɩɬɢɦɢɡɚɰɢɢ, ɚɧɚɥɢɡɟ

ɭɫɬɨɣɱɢɜɨɫɬɢ ɢ

ɫɪɚɜɧɢɬɟɥɶɧɨɣ ɫɬɚɬɢɤɟ. ɋɢɧɟɪɝɟɬɢɱɟɫɤɚɹ ɷɤɨɧɨɦɢɤɚ ɩɪɟɞɨɫɬɚɜɥɹɟɬ

ɧɨɜɵɟ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɪɚɦɤɢ ɢ ɦɟɬɨɞɵ, ɫɩɨɫɨɛɧɵɟ ɭɞɨɜɥɟɬɜɨɪɢɬɶ ɷɬɢ ɩɨɬɪɟɛɧɨɫɬɢ.

ȼɨɨɞɭɲɟɜɥɟɧɧɵɟ ɫɨɜɪɟɦɟɧɧɵɦɢ ɪɚɛɨɬɚɦɢ ɦɚɬɟɦɚɬɢɤɨɜ ɢ ɩɪɟɞɫɬɚɜɢɬɟɥɟɣ

ɟɫɬɟɫɬɜɟɧɧɵɯ ɧɚɭɤ ɜ ɨɛɥɚɫɬɢ ɧɟɥɢɧɟɣɧɵɯ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ, ɧɟɤɨɬɨɪɵɟ

ɷɤɨɧɨɦɢɫɬɵ ɩɪɢɫɬɭɩɢɥɢ ɤ ɨɛɴɹɫɧɟɧɢɸ ɫɥɨɠɧɵɯ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ, ɜɜɨɞɹ ɜ

ɞɢɧɚɦɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɮɚɤɬɨɪɵ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ ɢ ɧɟɥɢɧɟɣɧɨɫɬɢ. ɗɬɢ ɢɫɫɥɟɞɨɜɚɧɢɹ

ɞɚɥɢ ɧɚɱɚɥɨ ɧɨɜɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ ɜ

ɚɧɚɥɢɡɟ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ.

3 ɗɥɟɦɟɧɬɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ

ɬɟɨɪɢɢ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ

ɉɪɢɛɥɢɠɟɧɢɟ ɤ ɛɨɥɟɟ ɝɥɭɛɨɤɨɦɭ ɩɨɧɢɦɚɧɢɸ ɨɫ-

ɧɨɜɧɵɯ ɩɪɢɧɰɢɩɨɜ ɮɢɡɢɤɢ ɫɜɹɡɚɧɨ ɫɨ ɜɫɟ ɛɨɥɟɟ

ɫɥɨɠɧɵɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɦɟɬɨɞɚɦɢ.

Ⱥɥɶɛɟɪɬ ɗɣɧɲɬɟɣɧ

Ɇɚɬɟɦɚɬɢɤɚ—ɫɥɭɠɚɧɤɚ ɫɨɜɪɟɦɟɧɧɨɣ ɧɚɭɤɢ. Ȼɟɡ ɟɟ ɭɱɚɫɬɢɹ ɜɪɹɞ ɥɢ ɨɤɚɡɚɥɢɫɶ ɛɵ

ɜɨɡɦɨɠɧɵ ɦɧɨɝɢɟ ɢɡ ɧɵɧɟɲɧɢɯ ɝɥɭɛɨɤɢɯ ɩɪɨɧɢɤɧɨɜɟɧɢɢ ɧɚɭɤɢ ɜ ɫɭɬɶ ɩɪɢɪɨɞɵ. ɋ

ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɦɚɬɟɦɚɬɢɤɚ ɠɢɜɟɬ ɫɜɨɟɣ ɫɨɛɫɬɜɟɧɧɨɣ ɠɢɡɧɶɸ. Ɍɪɭɞɵ ɇɶɸɬɨɧɚ,

Ʌɟɣɛɧɢɰɚ ɢ ɮɨɧ ɇɟɣɦɚɧɚ ɹɜɥɹɸɬ ɫɨɛɨɣ ɩɪɟɤɪɚɫɧɵɣ ɩɪɢɦɟɪ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ

ɦɚɬɟɦɚɬɢɤɨɣ ɢ ɞɪɭɝɢɦɢ ɧɚɭɤɚɦɢ.

Ɇɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɢ ɫɨɜɪɟɦɟɧɧɚɹ ɷɤɨɧɨɦɢɤɚ

ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɪɢɦɟɧɟɧɢɟɦ

ɦɚɬɟɦɚɬɢɤɢ ɤ ɫɚɦɵɦ ɪɚɡɧɵɦ ɫɜɨɢɦ ɩɪɨɛɥɟɦɚɦ. ɇɚɢɛɨɥɟɟ ɩɨɥɧɨɝɨ ɩɨɧɢɦɚɧɢɹ ɱɢɫɬɨ

ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɜɨɩɪɨɫɨɜ ɧɟɥɶɡɹ ɞɨɫɬɢɱɶ ɛɟɡ ɩɪɢɜɥɟɱɟɧɢɹ ɦɚɬɟɦɚɬɢɤɢ. ɇɟ ɩɪɢɛɟɝɚɹ

ɤ ɹɡɵɤɭ ɦɚɬɟɦɚɬɢɤɢ, ɛɵɥɨ ɛɵ ɬɪɭɞɧɨ ɨɛɴɹɫɧɢɬɶ ɩɨɧɹɬɢɹ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ

ɪɚɜɧɨɜɟɫɢɹ ɢ ɧɟɪɚɜɧɨɜɟɫɢɹ, ɭɫɬɨɣɱɢɜɨɫɬɢ ɢ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ, ɷɤɨɧɨɦɢɱɟɫɤɢ

ɭɫɬɨɣɱɢɜɵɯ ɫɨɫɬɨɹɧɢɣ ɢ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɯɚɨɫɚ.

ɂɫɬɨɪɢɹ ɩɪɢɦɟɧɟɧɢɹ ɦɚɬɟɦɚɬɢɤɢ ɜ ɷɤɨɧɨɦɢɤɟ ɬɚɤ ɠɟ ɫɬɚɪɚ,

ɤɚɤ ɢɫɬɨɪɢɹ ɫɚɦɨɣ

ɦɚɬɟɦɚɬɢɤɢ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɢɫɱɢɫɥɟɧɢɟ ɢɫɩɨɥɶɡɨɜɚɥɨɫɶ ɜ ɷɤɨɧɨɦɢɤɟ ɟɳɟ ɫ

ɧɚɱɚɥɚ ɞɟɜɹɬɧɚɞɰɚɬɨɝɨ ɜɟɤɚ (Ʉɭɪɧɨ, 1838). ɂɦɟɧɧɨ ɛɥɚɝɨɞɚɪɹ ɢɫɩɨɥɶɡɨɜɚɧɢɸ ɷɬɨɝɨ

ɚɩɩɚɪɚɬɚ ȼɚɥɶɪɚɫ (1874) ɢ ɉɚɪɟɬɨ (1908) ɫɮɨɪɦɭɥɢɪɨɜɚɥɢ ɬɟɨɪɢɸ ɨɛɳɟɝɨ ɷɤɨ-

ɧɨɦɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, ɤɨɬɨɪɚɹ ɜ ɩɟɪɢɨɞ ɜɬɨɪɨɣ ɦɢɪɨɜɨɣ ɜɨɣɧɵ ɞɨɫɬɢɝɥɚ ɫɜɨɟɣ

ɤɭɥɶɦɢɧɚɰɢɢ ɜ «ȼɟɥɢɱɢɧɟ ɢ ɤɚɩɢɬɚɥɟ» ɏɢɤɫɚ (1939) ɢ «Ɉɫɧɨɜɚɯ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ

ɚɧɚɥɢɡɚ» ɋɚɦɭɷɥɶɫɨɧɚ (1947). ɉɨɫɥɟ ɜɬɨɪɨɣ ɦɢɪɨɜɨɣ

ɜɨɣɧɵ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɜ

ɷɤɨɧɨɦɢɤɟ ɧɚɲɥɢ ɬɚɤɢɟ ɪɚɡɞɟɥɵ ɦɚɬɟɦɚɬɢɤɢ, ɤɚɤ ɜɵɩɭɤɥɵɣ ɚɧɚɥɢɡ, ɬɨɩɨɥɨɝɢɹ ɢ ɞɪ.

(ɇɢɤɚɣɞɨ, 1968, ɗɪɪɨɭ ɢ ɏɚɧ, 1971, Ɍɚɤɚɹɦɚ, 1985, Ɇɚɤ-Ʉɨɥɟɥɥ, 1985, Ⱥɧɞɟɪɫɨɧ,

ɗɪɪɨɭ ɢ ɉɚɣɧɫ, 1988). ȼ ɩɨɫɥɟɞɧɟɟ ɜɪɟɦɹ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɷɤɨɧɨɦɢɱɟɫɤɢɯ

ɷɜɨɥɸɰɢɨɧɧɵɯ ɩɪɨɰɟɫɫɨɜ ɜɫɟ ɲɢɪɟ ɫɬɚɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɟɨɪɢɸ ɤɚɬɚɫɬɪɨɮ ɢ ɬɟɨɪɢɸ

ɛɢɮɭɪɤɚɰɢɣ. ɉɪɟɞɫɬɚɜɥɹɟɬɫɹ, ɱɬɨ ɡɚɩɚɡɞɵɜɚɧɢɟ ɜɨ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɩɨɥɭɱɟɧɢɟɦ

ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɪɟɡɭɥɶɬɚɬɨɜ ɢ ɢɯ ɩɪɢɥɨɠɟɧɢɟɦ ɜ ɷɤɨɧɨɦɢɤɟ ɢɦɟɟɬ ɬɟɧɞɟɧɰɢɸ ɤ

ɫɨɤɪɚɳɟɧɢɸ — ɜ ɫɚɦɨɦ ɞɟɥɟ, ɩɪɟɠɞɟ; ɱɟɦ ɧɚɲɥɨ ɩɪɢɦɟɧɟɧɢɟ ɢɫɱɢɫɥɟɧɢɟ

ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ, ɩɪɨɲɥɨ ɨɤɨɥɨ ɩɨɥɭɬɨɪɚ ɫɬɨɥɟɬɢɣ; ɚ ɱɬɨɛɵ ɧɚɣɬɢ ɩɪɢɦɟɧɟɧɢɟ

ɬɟɨɪɢɢ ɤɚɬɚɫɬɪɨɮ ɢ ɬɟɨɪɢɢ ɛɢɮɭɪɤɚɰɢɣ, ɷɤɨɧɨɦɢɫɬɚɦ ɩɨɧɚɞɨɛɢɥɨɫɶ ɜɫɟɝɨ

ɧɟɫɤɨɥɶɤɨ ɥɟɬ.

ȼ ɷɬɨɣ ɝɥɚɜɟ ɨɛɫɭɠɞɚɸɬɫɹ ɧɟɤɨɬɨɪɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ, ɩɨɬɟɧɰɢɚɥɶɧɨ

ɩɨɥɟɡɧɵɟ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ

ɫɢɧɟɪɝɟɬɢɱɟɫɤɨɣ ɷɤɨɧɨɦɢɤɢ. Ɂɞɟɫɶ ɢɡɭɱɚɸɬɫɹ ɬɨɥɶɤɨ ɬɚɤɢɟ

ɞɢɧɚɦɢɱɟɫɤɢɟ ɫɢɫɬɟɦɵ, ɤɨɬɨɪɵɟ ɨɩɢɫɵɜɚɸɬɫɹ ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɦɢ

ɨɛɵɤɧɨɜɟɧɧɵɦɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ. Ⱦɪɭɝɢɟ ɬɢɩɵ ɞɢɧɚɦɢɱɟɫɤɢɯ

ɫɢɫɬɟɦ, ɫɜɹɡɚɧɧɵɟ ɫ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦɢ ɡɚɜɢɫɢɦɨɫɬɹɦɢ ɢ ɫɬɨɯɚɫɬɢɱɧɨɫɬɶɸ, ɛɭɞɭɬ

ɨɛɫɭɠɞɚɬɶɫɹ ɩɨɡɠɟ.

3.1 Ⱦɢɧɚɦɢɤɚ ɢ ɪɚɜɧɨɜɟɫɢɟ

Ɉɛɵɤɧɨɜɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜ ɬɟɨɪɢɢ

ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɞɢɧɚɦɢɤɢ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɞɢɧɚɦɢɱɟɫɤɢɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ

ɷɤɨɧɨɦɢɱɟɫɤɢɦɢ ɩɟɪɟɦɟɧɧɵɦɢ, ɬɚɤɢɦɢ, ɤɚɤ ɰɟɧɵ, ɡɚɪɚɛɨɬɧɚɹ ɩɥɚɬɚ ɢ ɤɚɩɢɬɚɥ,

ɨɩɢɫɵɜɚɸɬɫɹ ɫɢɫɬɟɦɚɦɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ. ɇɟɤɨɬɨɪɵɟ ɞɢɧɚɦɢɱɟɫɤɢɟ

ɡɚɞɚɱɢ ɩɪɢɜɨɞɹɬ ɤ (ɩɚɪɚɛɨɥɢɱɟɫɤɢɦ) ɭɪɚɜɧɟɧɢɹɦ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɧɨ ɷɬɢ

ɬɢɩɵ ɭɪɚɜɧɟɧɢɣ ɦɵ ɪɚɫɫɦɨɬɪɢɦ, ɤɨɝɞɚ ɛɭɞɟɦ ɢɡɭɱɚɬɶ ɩɪɨɛɥɟɦɵ ɮɨɪɦɢɪɨɜɚɧɢɹ

ɝɨɪɨɞɫɤɢɯ ɫɬɪɭɤɬɭɪ

. ȼ ɨɛɳɟɦ ɜɢɞɟ ɞɢɧɚɦɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɚ ɬɚɤ:

ɝɞɟ x: = x(r,t) — ɜɟɤɬɨɪ ɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ, r — ɪɚɫɫɬɨɹɧɢɟ, ƒ (ɯ) — ɧɟɥɢɧɟɣɧɚɹ

ɜɟɤɬɨɪ-ɮɭɧɤɰɢɹ ɨɬ ɯ, a D — ɦɚɬɪɢɰɚ ɞɢɮɮɭɡɢɢ. ɇɚɩɪɢɦɟɪ, ɜ ɭɩɪɨɳɟɧɧɨɣ ɦɨɞɟɥɢ

Ʉɟɣɧɫɚ, ɨ ɤɨɬɨɪɨɣ ɛɭɞɟɬ ɪɟɱɶ ɜ ɪɚɡɞ. 5.3, ɤɨɦɩɨɧɟɧɬɚɦɢ ɜɟɤɬɨɪɚ ɯ (ɩɪɢ D = 0)

ɹɜɥɹɸɬɫɹ ɧɚɰɢɨɧɚɥɶɧɵɣ ɞɨɯɨɞ ɢ ɫɬɚɜɤɚ ɩɪɨɰɟɧɬɚ. ȼ ɦɨɞɟɥɢ ɝɨɪɨɞɚ ɢɡ ɪɚɡɞ. 8.4

ɩɟɪɟɦɟɧɧɚɹ x(r, t

) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɥɨɬɧɨɫɬɶ ɧɚɫɟɥɟɧɢɹ ɢ ɡɟɦɟɥɶɧɭɸ ɪɟɧɬɭ, ɚ r

— ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɝɨɪɨɞɫɤɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɞɨ ɰɟɧɬɪɚɥɶɧɨɝɨ

ɞɟɥɨɜɨɝɨ ɪɚɣɨɧɚ (ɐȾɊ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɫɢɫɬɟɦɚ (3.1.1) ɦɨɠɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɥɹ

ɨɩɢɫɚɧɢɹ ɩɪɨɰɟɫɫɚ ɝɪɚɞɨɮɨɪɦɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɣ ɨɬɨɛɪɚɠɚɟɬɫɹ ɞɢɧɚɦɢɤɨɣ

ɩɟɪɟɦɟɧɧɨɣ ɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɩɪɟɧɟɛɪɟɠɟɦ ɞɢɮɮɭɡɢɨɧɧɵɦɢ

ɱɥɟɧɚɦɢ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɛɭɞɭɬ ɪɚɫɫɦɨɬɪɟɧɵ

ɨɬɞɟɥɶɧɨ

ɜ ɝɥ. 8.

Ȼɟɡ ɭɱɟɬɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɫɢɫɬɟɦɚ (3.1.1) ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɚ

ɤɚɤ

x=f(x). (3.1.2)

Ɇɵ ɞɚɞɢɦ ɛɟɝɥɵɣ ɨɱɟɪɤ ɧɟɤɨɬɨɪɵɯ ɦɟɬɨɞɨɜ ɚɧɚɥɢɡɚ ɬɚɤɢɯ ɭɪɚɜɧɟɧɢɣ.

Ɂɚ ɛɨɥɟɟ ɩɨɥɧɵɦ ɢɡɥɨɠɟɧɢɟɦ ɨɬɫɵɥɚɟɦ ɱɢɬɚɬɟɥɹ, ɧɚɩɪɢɦɟɪ, ɤ Ʉɨɞɞɢɧɝɬɨɧɭ ɢ

Ʌɟɜɢɧɫɨɧɭ (1955) ɢɥɢ ɤ ɑɭ ɢ ɏɟɣɥɭ (1982).

ɉɟɪɜɨɟ ɨɫɧɨɜɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɚɤɢɯ ɭɪɚɜɧɟɧɢɣ ɢɡɜɟɫɬɧɨ ɤɚɤ

ɬɟɨɪɟɦɚ ɉɢɤɚɪɚ-Ʉɨɲɢ-Ʌɢɩɲɢɰɚ, ɤɨɬɨɪɚɹ ɡɜɭɱɢɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

Ɍɟɨɪɟɦɚ 3.1.1. Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ

x= f(x,t).

ɉɭɫɬɶ ɮɭɧɤɰɢɢ f

(x, t) ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɹɦ Ʌɢɩɲɢɰɚ ɩɨ ɜɫɟɦ ɫɜɨɢɦ

ɩɟɪɟɦɟɧɧɵɦ (ɬ. ɟ. ɧɟɩɪɟɪɵɜɧɵ ɢ ɨɝɪɚɧɢɱɟɧɵ ɜ ɧɟɤɨɬɨɪɨɣ ɡɚɦɤɧɭɬɨɣ ɨɛɥɚɫɬɢ, ɢ ɞɥɹ

ɜɫɟɯ ɯ, ɯ' ɢɡ ɷɬɨɣ ɨɛɥɚɫɬɢ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɚɹ ɩɨɫɬɨɹɧɧɚɹ L, ɱɬɨ

() ()

−

′

≤−

′

kkii

xxLtxftxf ,, — ɉɪɢɦ. ɩɟɪɟɜ.). Ɍɨɝɞɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ t = t

ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɯ = x(t), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ

x(0).

Ȼɨɥɟɟ ɬɨɝɨ, ɷɬɨ ɪɟɲɟɧɢɟ ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ.

ȿɫɥɢ



= f(x,t,r),

ɝɞɟ r — ɩɚɪɚɦɟɬɪ, ɢ ɤɚɠɞɚɹ ɮɭɧɤɰɢɹ f

, ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ r

ɭɞɨɜɥɟɬɜɨɪɹɟɬ

ɭɫɥɨɜɢɸ Ʌɢɩɲɢɰɚ ɪɚɜɧɨɦɟɪɧɨ ɩɨ r ɢ ɹɜɥɹɟɬɫɹ ɩɨ r ɧɟɩɪɟɪɵɜɧɨɣ, ɬɨ ɩɪɟɞɵɞɭɳɢɟ

ɭɬɜɟɪɠɞɟɧɢɹ ɜɵɩɨɥɧɹɸɬɫɹ ɜɨ ɜɫɟɣ ɨɤɪɟɫɬɧɨɫɬɢ r

. ɂ ɫɜɟɪɯ ɬɨɝɨ, ɯ = x(t,r) ɹɜɥɹɟɬɫɹ

ɧɟɩɪɟɪɵɜɧɨɣ ɜ ɷɬɨɣ ɨɤɪɟɫɬɧɨɫɬɢ.

ɉɨɧɹɬɢɟ ɭɫɬɨɣɱɢɜɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.1.1. (ɍɫɬɨɣɱɢɜɨɫɬɶ.) Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ dx/dt = f(x, t). Ɋɟɲɟɧɢɟ

ɯ = u(t), ɨɩɪɟɞɟɥɟɧɧɨɟ ɧɚ [t

, ], ɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ ɡɚɞɚɧɧɨɝɨ ∈ > 0

ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ δ, ɱɬɨ ɟɫɥɢ ɢ*(t

) — ɩɪɨɢɡɜɨɥɶɧɵɣ ɜɟɤɬɨɪ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ

ɭɫɥɨɜɢɸ

ɬɨ ɪɟɲɟɧɢɟ x=u*(t) ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ x(t

) = u*(t

) ɫɭɳɟɫɬɜɭɟɬ ɧɚ [t

, ] ɢ

ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ

ɞɥɹ ɜɫɟɯ t ≥ t

Ɉɩɪɟɞɟɥɟɧɢɟ 3.1.2. (Ⱥɫɢɦɩɬɨɬɢɱɟɫɤɚɹ ɭɫɬɨɣɱɢɜɨɫɬɶ.) Ɋɟɲɟɧɢɟ u(t)

ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ (ɚ) ɨɧɨ ɭɫɬɨɣɱɢɜɨ ɢ (b) ɫɭɳɟɫɬɜɭɟɬ μ > 0, ɬɚɤɨɟ,

ɱɬɨ ɟɫɥɢ

ɬɨ

Ɋɟɲɟɧɢɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɨ ɜ ɰɟɥɨɦ, ɟɫɥɢ μ. ɦɨɠɟɬ ɛɵɬɶ ɜɵɛɪɚɧɨ

ɩɪɨɢɡɜɨɥɶɧɨ ɛɨɥɶɲɢɦ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.1.3. (ɇɟɭɫɬɨɣɱɢɜɨɫɬɶ.) Ɋɟɲɟɧɢɟ ɢ(t) ɧɟɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ ɞɥɹ

ɧɟɤɨɬɨɪɨɝɨ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ∈ ɢ ɥɸɛɨɝɨ δ > 0 ɫɭɳɟɫɬɜɭɟɬ

ɪɟɲɟɧɢɟ u*(t), ɬɚɤɨɟ, ɱɬɨ ɞɥɹ ɧɟɤɨɬɨɪɨɝɨ t > t

ɜɵɩɨɥɧɹɸɬɫɹ ɭɫɥɨɜɢɹ (ɚ)

ȼ ɩɨɜɫɟɞɧɟɜɧɨɣ ɠɢɡɧɢ ɦɨɠɧɨ ɧɚɣɬɢ ɦɧɨɠɟɫɬɜɨ ɩɪɢɦɟɪɨɜ, ɩɨɦɨɝɚɸɳɢɯ ɩɨɧɹɬɶ

ɫɭɬɶ ɹɜɥɟɧɢɹ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ. ɉɨɤɨɢɜɲɚɹɫɹ ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɠɢɞɤɨɫɬɶ, ɩɟɪɟɣɞɹ ɤ

ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɢɦ ɤɨɥɟɛɚɧɢɹɦ, ɬɟɦ ɫɚɦɵɦ ɩɟɪɟɯɨɞɢɬ ɨɬ ɫɬɚɪɨɝɨ ɫɨɫɬɨɹɧɢɹ

ɪɚɜɧɨɜɟɫɢɹ ɜ ɧɨɜɨɟ, ɬɟɪɹɹ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɜɨɸ ɭɫɬɨɣɱɢɜɨɫɬɶ. ȼ ɭɫɥɨɜɢɹɯ

ɮɢɡɢɱɟɫɤɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɤɨɝɞɚ ɦɵ ɢɡɦɟɧɹɟɦ ɨɩɪɟɞɟɥɟɧɧɵɟ ɭɫɥɨɜɢɹ, ɧɚɩɪɢɦɟɪ

ɜɯɨɞɧɭɸ ɦɨɳɧɨɫɬɶ, ɫɢɫɬɟɦɚ ɦɨɠɟɬ ɩɪɨɣɬɢ ɱɟɪɟɡ ɪɹɞ ɧɟɭɫɬɨɣɱɢɜɵɯ ɫɨɫɬɨɹɧɢɣ,

ɩɪɢɜɨɞɹɳɢɯ ɤ ɫɨɜɟɪɲɟɧɧɨ ɪɚɡɥɢɱɧɵɦ ɬɢɩɚɦ ɩɨɜɟɞɟɧɢɹ. ɋɥɨɠɧɨɟ ɧɟɭɫɬɨɣɱɢɜɨɟ

ɩɨɜɟɞɟɧɢɟ ɦɨɠɧɨ ɢɡɭɱɚɬɶ ɬɚɤɠɟ ɧɚ ɩɪɢɦɟɪɟ ɞɢɧɚɦɢɤɢ ɨɛɦɟɧɧɵɯ ɤɭɪɫɨɜ ɜɚɥɸɬ ɧɚ

ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɪɵɧɤɚɯ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.1.4. (Ɉɪɛɢɬɚɥɶɧɚɹ ɭɫɬɨɣɱɢɜɨɫɬɶ.) Ɋɟɲɟɧɢɟ u(t) ɚɜɬɨɧɨɦɧɨɣ

ɫɢɫɬɟɦɵ dx/dt = f(x) ɹɜɥɹɟɬɫɹ ɨɪɛɢɬɚɥɶɧɨ ɭɫɬɨɣɱɢɜɵɦ, ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ ɞɚɧɧɨɝɨ

∈ > 0

ɫɭɳɟɫɬɜɭɟɬ δ > 0, ɬɚɤɨɟ, ɱɬɨ ɟɫɥɢ

ɞɥɹ ɥɸɛɨɝɨ t ≥ t

ɉɨɧɹɬɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɢ ɨɪɛɢɬɚɥɶɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɧɟ ɫɥɟɞɭɟɬ ɩɭɬɚɬɶ. Ⱦɥɹ

ɢɥɥɸɫɬɪɚɰɢɢ ɪɚɡɥɢɱɢɹ ɧɚ ɪɢɫ. 3.1 ɩɪɢɜɟɞɟɧ ɫɥɟɞɭɸɳɢɣ ɩɪɢɦɟɪ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ

ɋ ɢ ɋ' — ɞɜɟ ɨɪɛɢɬɵ ɪɚɡɧɵɯ ɩɟɪɢɨɞɨɜ. ɏɨɬɹ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɧɢɦɢ ɨɫɬɚɟɬɫɹ ɜɫɟ

ɜɪɟɦɹ ɨɝɪɚɧɢɱɟɧɧɵɦ, ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɞɜɭɦɹ ɬɨɱɤɚɦɢ 1 ɢ 1′ ɧɚ ɷɬɢɯ ɨɪɛɢɬɚɯ

ɜɫɥɟɞɫɬɜɢɟ ɫɞɜɢɝɚ ɮɚɡ, ɩɨɪɨɠɞɟɧɧɨɝɨ ɪɚɡɧɨɫɬɶɸ ɩɟɪɢɨɞɨɜ, ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ

ɦɨɠɟɬ ɭɜɟɥɢɱɢɜɚɬɶɫɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɚɠɟ ɟɫɥɢ

ɬɨ

ɢ (b)

Ɋɢɫ. 3.1. ɋɪɚɜɧɟɧɢɟ ɩɨɧɹɬɢɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɢ ɨɪɛɢɬɚɥɶɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ.

ɨɪɛɢɬɚ ɋ ɨɪɛɢɬɚɥɶɧɨ ɭɫɬɨɣɱɢɜɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɟɣ ɪɟɲɟɧɢɟ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ

ɭɫɬɨɣɱɢɜɨ.

ȼɫɟ ɷɬɢ ɨɩɪɟɞɟɥɟɧɢɹ, ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɟ ɞɥɹ dx/dt = f(x,t), ɫɩɪɚɜɟɞɥɢɜɵ ɢ ɞɥɹ

ɚɜɬɨɧɨɦɧɵɯ ɫɢɫɬɟɦ, ɤɨɝɞɚ f(x,t) = f(x).

Ɍɨɱɤɚ ɪɚɜɧɨɜɟɫɢɹ ɚɜɬɨɧɨɦɧɨɣ ɫɢɫɬɟɦɵ x

ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɭɫɥɨɜɢɹ f (x

) = 0. ȼ

ɷɬɨɦ ɫɥɭɱɚɟ ɯ = x

ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ. Ɋɚɜɧɨɜɟɫɢɟ

ɹɜɥɹɟɬɫɹ (ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ) ɭɫɬɨɣɱɢɜɵɦ, ɟɫɥɢ ɪɟɲɟɧɢɟ ɯ = x

(ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ)

ɭɫɬɨɣɱɢɜɨ.

Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ ɷɬɢɯ ɩɨɧɹɬɢɣ ɪɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɨɫɬɚ ɋɨɥɨɭ.

ɗɬɚ ɦɨɞɟɥɶ ɢɝɪɚɟɬ ɜɚɠɧɭɸ ɪɨɥɶ ɜ ɧɟɨɤɥɚɫɫɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɨɫɬɚ.

Ɇɨɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ ɛɨɥɶɲɢɧɫɬɜɨ ɧɟɨɤɥɚɫɫɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɪɨɫɬɚ ɹɜɥɹɸɬɫɹ

ɪɚɫɲɢɪɟɧɢɹɦɢ ɢ ɨɛɨɛɳɟɧɢɹɦɢ ɩɢɨɧɟɪɫɤɢɯ ɪɚɛɨɬ ɋɨɥɨɭ (1956) ɢ ɋɜɚɧɚ (1956) (ɫɦ.

ɬɚɤɠɟ Ɂɚɧɝ, 1989).

ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɨɞɟɥɢ ɋɨɥɨɭ ɩɪɟɞɩɨɥɚɝɚɥ, ɱɬɨ ɟɫɬɶ ɬɨɥɶɤɨ ɨɞɢɧ ɜɢɞ ɬɨɜɚɪɚ

(

ɞɥɢɬɟɥɶɧɨɝɨ ɩɨɥɶɡɨɜɚɧɢɹ); ɪɵɧɤɢ ɫɛɵɬɚ ɩɪɨɞɭɤɰɢɢ ɪɚɛɨɬɚɸɬ ɛɟɫɩɟɪɟɛɨɣɧɨ;

ɩɪɟɞɥɨɠɟɧɢɟ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɮɚɤɬɨɪɨɜ ɧɟɷɥɚɫɬɢɱɧɨ (ɬ. ɟ. ɨɧɢ ɫɭɳɟɫɬɜɟɧɧɨ ɧɟ

ɩɨɧɢɠɚɸɬɫɹ ɢ ɧɟ ɩɨɜɵɲɚɸɬɫɹ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɰɟɧ), ɢ ɜɫɟ ɞɨɫɬɭɩɧɵɟ ɮɚɤɬɨɪɵ ɜ ɤɚɠ-

ɞɵɣ ɦɨɦɟɧɬ ɩɨɥɧɨɫɬɶɸ ɢɫɩɨɥɶɡɭɸɬɫɹ. ȼɫɟ ɫɛɟɪɟɠɟɧɢɹ ɝɪɚɠɞɚɧɚɦɢ ɞɨɛɪɨɜɨɥɶɧɨ

ɫɞɚɸɬɫɹ ɢ ɚɛɫɨɪɛɢɪɭɸɬɫɹ ɮɢɪɦɚɦɢ ɞɥɹ ɧɚɤɨɩɥɟɧɢɹ ɤɚɩɢɬɚɥɚ. ɋɭɳɟɫɬɜɭɸɬ ɞɜɚ

ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɮɚɤɬɨɪɚ: ɤɚɩɢɬɚɥ Ʉ ɢ ɬɪɭɞ L. Ɍɟɯɧɨɥɨɝɢɹ ɧɟ ɩɨɞɜɟɪɠɟɧɚ ɧɢɤɚɤɢɦ

ɢɡɦɟɧɟɧɢɹɦ. ɉɪɨɰɟɫɫ ɩɪɨɢɡɜɨɞɫɬɜɚ ɨɩɢɫɵɜɚɟɬɫɹ ɧɟɤɨɬɨɪɨɣ ɞɨɫɬɚɬɨɱɧɨ ɝɥɚɞɤɨɣ

ɮɭɧɤɰɢɟɣ

Y=F(K, L), (3.1.3)

ɝɞɟ Y — ɩɨɬɨɤ ɩɪɨɞɭɤɰɢɢ, ɡɚɜɢɫɹɳɢɣ ɨɬ ɤɨɧɤɪɟɬɧɵɯ ɡɧɚɱɟɧɢɣ Ʉ ɢ L.

ɉɪɨɢɡɜɨɞɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ F ɫɱɢɬɚɟɬɫɹ ɧɟɨɤɥɚɫɫɢɱɟɫɤɨɣ, ɟɫɥɢ ɨɧɚ ɭɞɨɜɥɟɬɜɨɪɹɟɬ

ɫɥɟɞɭɸɳɢɦ ɭɫɥɨɜɢɹɦ: (1) F(K, L) ɧɟɨɬɪɢɰɚɬɟɥɶɧɚ, ɟɫɥɢ Ʉ ɢ L ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ; (2)

F(0,0) = 0; (3)

ɩɪɢɪɨɫɬɵ

ɮɭɧɤɰɢɢ F

ɢ F

ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ; (4) ɫɭɳɟɫɬɜɭɸɬ ɜɬɨɪɵɟ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ

ɮɭɧɤɰɢɢ F ɩɨ Ʉ ɢ L; (5) ɮɭɧɤɰɢɹ ɨɞɧɨɪɨɞɧɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ: F(rK, rL) == rF(K,L)

ɞɥɹ ɜɫɟɯ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɯ r; (6) ɮɭɧɤɰɢɹ ɫɬɪɨɝɨ ɤɜɚɡɢɜɨɝɧɭɬɚ.

ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ L ɷɤɡɨɝɟɧɧɨ ɜɨɡɪɚɫɬɚɟɬ ɫ ɉɨɫɬɨɹɧɧɵɦ ɬɟɦɩɨɦ ɪɨɫɬɚ ɩ:

L = L

exp(nt).

ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ ɬɚɤɠɟ, ɱɬɨ ɩɨɫɬɨɹɧɧɚɹ ɞɨɥɹ s ɨɛɳɟɝɨ ɨɛɴɟɦɚ ɩɪɨɢɡɜɨɞɫɬɜɚ ɢɞɟɬ

ɧɚ ɫɛɟɪɟɠɟɧɢɟ ɢ, ɜɵɩɚɞɚɹ ɢɡ ɫɮɟɪɵ ɩɨɬɪɟɛɥɟɧɢɹ, ɞɨɛɚɜɥɹɟɬɫɹ ɤ ɫɭɦɦɚɪɧɨɦɭ

ɤɚɩɢɬɚɥɭ. ȿɫɥɢ ɩɪɟɧɟɛɪɟɱɶ ɩɪɨɰɟɫɫɨɦ ɨɛɟɫɰɟɧɢɜɚɧɢɹ ɤɚɩɢɬɚɥɚ, ɬɨ ɢɦɟɟɦ

dK/dt = sY,

Ʉ(0) > 0. ȼ ɫɥɭɱɚɟ ɧɟɨɤɥɚɫɫɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ ɩɪɢɯɨɞɢɦ ɤ ɫɨɨɬɧɨɲɟɧɢɸ

ɝɞɟ k = K/L, f(k)= F(K,L)/L = F(k, 1). Ɏɭɧɤɰɢɹ f(k) ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ:

f(0) =0, f '(k) > 0

ɩɪɢ k≥0; f "(k)< 0 ɩɪɢ k≥0.

ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ (3.1.4) ɦɨɠɧɨ ɞɨɤɚɡɚɬɶ. ɏɨɪɨɲɨ ɢɡɜɟɫɬɧɨ,

ɱɬɨ ɜ ɦɨɞɟɥɢ ɋɨɥɨɭ ɩɨɫɥɟ ɨɩɪɟɞɟɥɟɧɢɹ ɞɢɧɚɦɢɤɢ ɨɛɴɟɦɚ ɤɚɩɢɬɚɥɚ ɧɚ ɞɭɲɭ

ɧɚɫɟɥɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɪɚɫɫɱɢɬɚɧɚ ɞɢɧɚɦɢɤɚ ɜɫɟɯ ɨɫɬɚɥɶɧɵɯ ɩɟɪɟɦɟɧɧɵɯ — Ʉ, Y,

ɩɨɬɪɟɛɥɟɧɢɹ, ɧɚɤɨɩɥɟɧɢɹ, ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɵ, ɫɭɦɦɵ ɚɪɟɧɞɧɵɯ ɩɥɚɬɟɠɟɣ.

Ɍɟɨɪɟɦɚ 3.1.2. (ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɪɚɜɧɨɜɟɫɢɹ.) ȿɫɥɢ ɩ ɢ s ɭɞɨɜɥɟɬɜɨɪɹɸɬ

ɧɟɪɚɜɟɧɫɬɜɭ

ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ k

, ɬɚɤɨɟ, ɱɬɨ sf(k

)/n = k

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ ɦɨɠɧɨ ɧɚɣɬɢ, ɧɚɩɪɢɦɟɪ ɭ Ʉɭɩɦɚɧɫɚ (1965). Ɏɚɡɨɜɭɸ

ɞɢɚɝɪɚɦɦɭ ɦɨɞɟɥɢ ɋɨɥɨɭ ɩɪɢ ɭɫɥɨɜɢɢ (3.1.5) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɫɯɟɦɚɬɢɱɟɫɤɢ, ɤɚɤ

ɧɚ ɪɢɫ. 3.2.

ɋɥɟɞɭɸɳɚɹ ɧɢɠɟ ɬɟɨɪɟɦɚ ɞɨɤɚɡɵɜɚɟɬ ɭɫɬɨɣɱɢɜɨɫɬɶ ɪɚɜɧɨɜɟɫɢɹ ɜ ɦɨɞɟɥɢ ɋɨɥɨɭ.

Ɍɟɨɪɟɦɚ 3.1.3. (ɍɫɬɨɣɱɢɜɨɫɬɶ ɪɚɜɧɨɜɟɫɢɹ.) ɋɢɫɬɟɦɚ ɋɨɥɨɭ ɝɥɨɛɚɥɶɧɨ ɭɫɬɨɣɱɢɜɚ

(

ɗɪɪɨɭ ɢ Ƚɭɪɜɢɰ, 1958). Ȼɨɥɟɟ ɬɨɝɨ, ɜ ɨɛɥɚɫɬɢ k > 0 ɪɚɜɧɨɜɟɫɢɟ ɹɜɥɹɟɬɫɹ

ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɵɦ.

Ⱥɫɢɦɩɬɨɬɢɱɟɫɤɚɹ ɭɫɬɨɣɱɢɜɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɞɨɤɚɡɚɧɚ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ

Ʌɹɩɭɧɨɜɚ V = U

, ɝɞɟ U = k — k

(Ȼɭɪɦɟɣɫɬɟɪ ɢ Ⱦɨɛɟɥɥ, 1970). Ⱦɢɧɚɦɢɤɭ

ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɚɡɜɢɬɢɹ ɦɨɠɧɨ ɨɩɢɫɚɬɶ