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

Подождите немного. Документ загружается.

ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɫɬɚɪɬɭɹ ɢɡ ɥɸɛɨɣ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ, ɷɤɨɧɨɦɢɤɚ ɜɫɟɝɞɚ

ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɢɬɫɹ ɤ ɟɞɢɧɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ ɫɨɨɬɧɨɲɟɧɢɹ «ɤɚɩɢɬɚɥ/ɬɪɭɞ» ɧɚ

ɛɨɥɶɲɢɯ ɜɪɟɦɟɧɚɯ. Ȼɨɥɟɟ ɬɨɝɨ, ɜɞɨɥɶ ɪɚɜɧɨɜɟɫɧɨɣ ɬɪɚɟɤɬɨɪɢɢ ɪɨɫɬɚ ɤɚɩɢɬɚɥ

ɜɨɡɪɚɫɬɚɟɬ ɫ ɬɨɣ ɠɟ ɫɤɨɪɨɫɬɶɸ, ɱɬɨ ɪɚɫɬɟɬ ɱɢɫɥɟɧɧɨɫɬɶ ɧɚɫɟɥɟɧɢɹ — ɷɬɨ ɩɪɨɫɬɨɟ ɢ

ɤɪɚɫɢɜɨɟ ɫɥɟɞɫɬɜɢɟ ɦɨɞɟɥɢ ɪɨɫɬɚ ɋɨɥɨɭ. ɂɫɬɨɪɢɱɟɫɤɢɣ ɨɛɡɨɪ ɪɚɡɜɢɬɢɹ ɷɬɨɣ ɦɨɞɟɥɢ

ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɇɨɛɟɥɟɜɫɤɨɣ ɥɟɤɰɢɢ ɋɨɥɨɭ 1987 ɝɨɞɚ (ɋɨɥɨɭ, 1988).

3.2 Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ

ɫɢɫɬɟɦ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ

ȿɫɥɢ f(ɯ) = Ⱥɯ, ɝɞɟ Ⱥ — ɩɨɫɬɨɹɧɧɚɹ ɱɢɫɥɨɜɚɹ ɦɚɬɪɢɰɚ, ɬɨ ɫɢɫɬɟɦɚ dx/dt = Ⱥɯ

ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɨɣ ɚɜɬɨɧɨɦɧɨɣ ɫɢɫɬɟɦɨɣ. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɟɞɢɧɫɬɜɟɧɧɚɹ ɬɨɱɤɚ

ɪɚɜɧɨɜɟɫɢɹ ɯ = 0 ɭɫɬɨɣɱɢɜɚ, ɟɫɥɢ ɞɥɹ ɤɚɠɞɨɝɨ ɫɨɛɫɬɜɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɦɚɬɪɢɰɵ Ⱥ

ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ Re (z) ≤ 0, ɩɪɢɱɟɦ ɩɪɢ Re (z) = 0 ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ z ɛɭɞɟɬ

ɩɪɨɫɬɵɦ. Ȼɨɥɟɟ ɬɨɝɨ, ɪɚɜɧɨɜɟɫɢɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɨ ɜ ɬɨɦ ɢ ɬɨɥɶɤɨ ɬɨɦ

ɫɥɭɱɚɟ, ɟɫɥɢ ɞɥɹ ɤɚɠɞɨɝɨ z ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ Re (z) <0.

ɑɬɨɛɵ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ ɧɟɤɨɬɨɪɵɟ ɤɨɧɰɟɩɰɢɢ, ɪɚɡɜɢɬɵɟ ɤ ɧɚɫɬɨɹɳɟɦɭ

ɜɪɟɦɟɧɢ, ɪɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɭɸ ɫɢɫɬɟɦɭ ɞɥɹ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ

ɉɨɫɤɨɥɶɤɭ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɚ, ɦɵ ɢɳɟɦ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ

ɝɞɟ ɢ ɢ v — ɤɨɧɫɬɚɧɬɵ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ

zu=au+bv, zv = cu+dv. (3.2.2)

ɋɢɫɬɟɦɚ (3.2.2) ɪɚɡɪɟɲɢɦɚ, ɤɨɝɞɚ |A - zI| = 0, ɝɞɟ A — ɦɚɬɪɢɰɚ ɫɢɫɬɟɦɵ (3.2.1), ɬ.ɟ.

ɤɨɝɞɚ z ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɦɚɬɪɢɰɵ, ɚ (ɢ, v)

— ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ

ɫɨɛɫɬɜɟɧɧɵɦ ɜɟɤɬɨɪɨɦ. ɋɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ

ɭɪɚɜɧɟɧɢɹ

ɝɞɟ

T =

ɚ + d, W= ad-bc,

ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɥɟɞ ɢ ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ Ⱥ.

ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɟ (3.2.3) ɨɛɥɚɞɚɟɬ ɞɜɭɦɹ ɪɚɡɥɢɱɧɵɦɢ ɤɨɪɧɹɦɢ z

ɢ z

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (3.2.1) ɢɦɟɟɬ ɜɢɞ

ɝɞɟ c

, c

ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ Θ

, Θ

ɹɜɥɹɸɬɫɹ

ɤɨɪɧɹɦɢ ɭɪɚɜɧɟɧɢɹ

ɂɡ ɷɬɢɯ ɜɵɪɚɠɟɧɢɣ ɥɟɝɤɨ ɭɫɬɚɧɨɜɢɬɶ, ɱɬɨ ɤɚɱɟɫɬɜɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɟɲɟɧɢɹ

ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɸɬɫɹ ɬɢɩɨɦ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ. ɉɨɫɤɨɥɶɤɭ ɫɨɛɫɬɜɟɧɧɵɟ

ɡɧɚɱɟɧɢɹ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɤɜɚɞɪɚɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ (3.2.3), ɢɦɟɸɳɟɦɭ ɤɨɪɧɢ z

ɢ z

ɦɨɠɧɨ ɜɵɞɟɥɢɬɶ ɫɥɟɞɭɸɳɢɟ ɫɥɭɱɚɢ (ɫɦ., ɧɚɩɪɢɦɟɪ, Ȼɪɢɬɬɟɧ, 1986)

ɉɭɫɬɶ z

ɢ z

ɞɟɣɫɬɜɢɬɟɥɶɧɵ ɢ ɪɚɡɥɢɱɧɵ ɢ ɢɦɟɸɬ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɡɧɚɤ. Ɍɨɝɞɚ

ɪɚɜɧɨɜɟɫɢɟ ɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ ɨɧɢ ɨɬɪɢɰɚɬɟɥɶɧɵ, ɢ ɧɟɭɫɬɨɣɱɢɜɨ — ɟɫɥɢ

ɩɨɥɨɠɢɬɟɥɶɧɵ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɬɨɱɤɚ ɪɚɜɧɨɜɟɫɢɹ ɧɚɡɵɜɚɟɬɫɹ ɭɫɬɨɣɱɢɜɵɦ ɢɥɢ

ɧɟɭɫɬɨɣɱɢɜɵɦ ɭɡɥɨɦ. Ɍɪɚɟɤɬɨɪɢɢ ɢɦɟɸɬ ɮɨɪɦɭ, ɩɨɤɚɡɚɧɧɭɸ ɧɚ ɪɢɫ. 3.3.

II)

ȿɫɥɢ z

ɢ z

— ɞɟɣɫɬɜɢɬɟɥɶɧɵ ɢ ɢɦɟɸɬ ɪɚɡɧɵɟ ɡɧɚɤɢ, ɪɚɜɧɨɜɟɫɢɟ ɧɚɡɵɜɚɟɬɫɹ

ɫɟɞɥɨɜɨɣ ɬɨɱɤɨɣ ɢɥɢ ɫɟɞɥɨɦ. ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɬɪɚɟɤɬɨɪɢɢ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 3.4.

III)

ɉɭɫɬɶ z

ɢ z

ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɵ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ Re(z

) = Re (z

) = Re (z).

Ȼɥɚɝɨɞɚɪɹ ɦɧɢɦɨɣ ɱɚɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɬɪɚɟɤɬɨɪɢɢ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ

ɛɭɞɭɬ ɨɯɜɚɬɵɜɚɬɶ ɬɨɱɤɭ ɪɚɜɧɨɜɟɫɢɹ. ȿɫɥɢ Re (z) < 0, ɨɧɢ ɛɭɞɭɬ ɞɜɢɝɚɬɶɫɹ ɩɨ

ɫɩɢɪɚɥɢ ɤ

ɋɦ. ɬɚɤɠɟ Ⱥɪɧɨɥɶɞ ȼ. ɂ. Ɉɛɵɤɧɨɜɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ, Ɇ.:, ɇɚɭɤɚ, ɢɡɞ. 3, 1984,

272 ɫ. — ɉɪɢɦ. ɪɟɞ.

Ɋɢɫ.3.3. Ⱦɜɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵ ɢ ɢɦɟɸɬ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɡɧɚɤ, (ɚ)

ɭɫɬɨɣɱɢɜɵɣ ɭɡɟɥ, (b) ɧɟɭɫɬɨɣɱɢɜɵɣ ɭɡɟɥ.

ɪɚɜɧɨɜɟɫɢɸ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.5ɚ, ɚ ɜ ɫɥɭɱɚɟ Re(z) > 0 — ɜ ɧɚɩɪɚɜɥɟɧɢɢ,

ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɪɚɜɧɨɜɟɫɢɸ, ɤɚɤ ɧɚ ɪɢɫ. 3.5b. Ɍɚɤɨɟ ɪɚɜɧɨɜɟɫɢɟ ɧɚɡɵɜɚɟɬɫɹ

ɭɫɬɨɣɱɢɜɵɦ ɥɢɛɨ ɧɟɭɫɬɨɣɱɢɜɵɦ ɮɨɤɭɫɨɦ. ȿɫɥɢ Re(z) = 0, ɢɦɟɟɦ ɩɪɟɞɟɥɶɧɵɣ ɫɥɭɱɚɣ

—

ɰɟɧɬɪ, ɜɨɤɪɭɝ ɤɨɬɨɪɨɝɨ ɬɪɚɟɤɬɨɪɢɢ ɡɚɦɵɤɚɸɬɫɹ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.6.

IV)

Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɞɜɭɯ ɪɚɜɧɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɱɢɫɟɥ. Ȼɟɡ ɩɨɬɟɪɢ

ɨɛɳɧɨɫɬɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɚ = d ɢ bc = 0. ȿɫɥɢ ɷɬɨ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ,

ɢɫɯɨɞɧɭɸ ɫɢɫɬɟɦɭ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɤ ɞɚɧɧɨɣ ɮɨɪɦɟ ɥɢɧɟɣɧɵɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ

ɢɥɢ ɩɨɜɨɪɨɬɨɦ ɨɫɟɣ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ. ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ ɢɦɟɸɬɫɹ ɞɜɟ

ɜɨɡɦɨɠɧɨɫɬɢ. ɉɟɪɜɚɹ b = ɫ = 0. ɗɬɨɬ ɫɥɭɱɚɣ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ. 3.7. ȼɬɨɪɚɹ ɜɨɡɦɨɠ-

ɧɨɫɬɶ — b (ɢɥɢ ɫ) ɪɚɜɧɨ ɧɭɥɸ (ɪɢɫ. 3.8). Ɉɛɚ ɪɚɜɧɨɜɟɫɢɹ ɨɬɧɨɫɹɬɫɹ ɤ ɬɢɩɭ ɭɡɟɥ.

ɉɨɫɥɟɞɧɹɹ ɜɨɡɦɨɠɧɨɫɬɶ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɨɞɧɨ (ɢɥɢ ɨɛɚ) ɫɨɛɫɬɜɟɧɧɵɯ

ɡɧɚɱɟɧɢɹ ɪɚɜɧɵ ɧɭɥɸ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɚɬɪɢɰɚ Ⱥ ɫɢɧɝɭɥɹɪɧɚ,

ɫɭɳɟɫɬɜɭɟɬ ɧɟɬɪɢɜɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ Ⱥ(x, ɭ)

= 0, ɢ ɜɫɟ ɬɨɱɤɢ q(x,y)

ɹɜɥɹɸɬɫɹ

ɪɚɜɧɨɜɟɫɢɹɦɢ (ɝɞɟ q —ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɤɨɧɫɬɚɧɬɚ). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ

ɧɭɥɶ ɧɟ ɹɜɥɹɟɬɫɹ ɛɨɥɶɲɟ ɢɡɨɥɢɪɨɜɚɧɧɨɣ ɨɫɨɛɨɣ ɬɨɱɤɨɣ.

ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɷɬɢɯ ɫɥɭɱɚɟɜ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɡɧɚɱɟɧɢɹɦɢ Ɍ

ɢ W ɞɚɧɚ ɜ ɤɧɢɝɟ ɇɢɤɨɥɢɫɚ ɢ ɉɪɢɝɨɠɢɧɚ (1977).

ɉɨɞɨɛɧɵɦ ɨɛɪɚɡɨɦ ɦɨɝɭɬ ɛɵɬɶ ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɧɵ ɢ ɫɢɫɬɟɦɵ ɛɨɥɶɲɢɯ

ɪɚɡɦɟɪɧɨɫɬɟɣ.

3.3 ɉɪɢɧɰɢɩ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɨ ɥɢɧɟɣɧɨɦɭ

ɩɪɢɛɥɢɠɟɧɢɸ

ɉɨɫɤɨɥɶɤɭ, ɤɚɤ ɩɪɚɜɢɥɨ, ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ ɭɫɬɨɣɱɢɜɨɫɬɶ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ

ɨɛɵɱɧɨ ɛɵɜɚɟɬ ɭɞɨɛɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɬɚɤɢɯ ɭɪɚɜɧɟɧɢɣ ɜ ɨɤɪɟɫɬɧɨɫɬɢ

ɨɬɞɟɥɶɧɨɝɨ «ɢɫɯɨɞɧɨɝɨ» ɫɨɫɬɨɹɧɢɹ (ɧɚɩɪɢɦɟɪ, ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ

). Ɇɵ ɦɨɠɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɪɨɢɡɜɨɥɶɧɨɟ ɪɟɲɟɧɢɟ ɯ(t) ɤɚɤ ɧɟɤɨɬɨɪɨɟ

ɧɟɜɨɡɦɭɳɟɧɧɨɟ ɪɟɲɟɧɢɟ u(t), ɤɨɬɨɪɨɟ ɩɨɞɜɟɪɝɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨ ɞɟɣɫɬɜɭɸɳɟɦɭ

ɜɧɟɲɧɟɦɭ ɜɨɡɦɭɳɟɧɢɸ ɢɥɢ ɜɧɭɬɪɟɧɧɟɣ ɮɥɭɤɬɭɚɰɢɢ X(t). ɗɬɨ ɜɵɡɵɜɚɟɬ ɫɞɜɢɝ

ɪɟɲɟɧɢɹ u(t) ɤ ɧɨɜɨɦɭ ɪɟɲɟɧɢɸ

ɯ(t) = u(t) + X(t) (3.3.2)

ɋɨɨɬɧɨɲɟɧɢɟ (3.3.2) ɨɩɪɟɞɟɥɹɟɬ ɧɨɜɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ

ɫ ɰɟɧɬɪɨɦ ɧɟ ɜ ɬɨɱɤɟ (0,...,0), ɚ ɜ ɧɨɜɨɣ ɬɨɱɤɟ ɢ. Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɢ — ɷɬɨ

ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɚɜɧɨɜɟɫɧɨɟ ɪɟɲɟɧɢɟ ɯ

. ɂɫɯɨɞɧɭɸ ɫɢɫɬɟɦɭ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ

J(X)=AX+N(X), (3.3.3)

ɝɞɟ Ⱥ — ɦɚɬɪɢɰɚ, N(X) = ɨ(ɏ) ɩɪɢ ɏ ĺ 0, (ɬ.ɟ. ɩɪɢ ⏐ɏ⏐ĺ 0 ɜɵɩɨɥɧɹɟɬɫɹ

|N (X)| / |X|

ĺ 0). ɋɥɚɝɚɟɦɨɟ N(X) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɧɟɥɢɧɟɣɧɵɣ ɱɥɟɧ ɭɪɚɜɧɟɧɢɹ.

Ɂɞɟɫɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ J(ɏ) ɞɨɫɬɚɬɨɱɧɨ ɝɥɚɞɤɚɹ ɮɭɧɤɰɢɹ, ɞɨɩɭɫɤɚɸɳɚɹ ɬɚɤɨɟ

ɩɪɟɞɫɬɚɜɥɟɧɢɟ, ɢ ,J(0) = 0.

ɍɫɬɨɣɱɢɜɨɫɬɶ ɢɫɯɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ x

ɫɜɹɡɚɧɚ ɬɟɩɟɪɶ ɫ

ɭɫɬɨɣɱɢɜɨɫɬɶɸ «ɬɪɢɜɢɚɥɶɧɨɝɨ ɪɟɲɟɧɢɹ» ɏ = 0. ɋɥɟɞɭɸɳɢɣ ɧɢɠɟ ɲɢɪɨɤɨ

ɢɡɜɟɫɬɧɵɣ ɪɟɡɭɥɶɬɚɬ ɢɥɥɸɫɬɪɢɪɭɟɬ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɭɫɬɨɣɱɢɜɨɫɬɶɸ ɧɟɥɢɧɟɣɧɨɣ

ɫɢɫɬɟɦɵ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɟɣ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ.

Ɍɟɨɪɟɦɚ 3.3.1. Ʌɢɧɟɚɪɢɡɨɜɚɧɧɚɹ ɫɢɫɬɟɦɚ, ɩɨɥɭɱɟɧɧɚɹ ɢɡ (3.3.3), ɢɦɟɟɬ ɜɢɞ

ȿɫɥɢ ɦɚɬɪɢɰɚ Ⱥ ɬɚɤɨɜɚ, ɱɬɨ (ɚ) ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢɦɟɸɬ ɨɬɪɢɰɚɬɟɥɶɧɭɸ

ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɱɚɫɬɶ, ɢɥɢ (ɛ) ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɨɞɧɨ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɟɬ

ɩɨɥɨɠɢɬɟɥɶɧɭɸ ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɱɚɫɬɶ,

ɬɨ ɜ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ ɭɫɬɨɣɱɢɜɨɫɬɶ ɬɪɢɜɢɚɥɶɧɨɝɨ ɪɟɲɟɧɢɹ

ɧɟɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ (3.3.3) ɢɦɟɟɬ ɬɨɬ ɠɟ ɯɚɪɚɤɬɟɪ, ɱɬɨ ɢ ɭɫɬɨɣɱɢɜɨɫɬɶ

ɬɪɢɜɢɚɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ (3.3.4). Ɉɞɧɚɤɨ ɟɫɥɢ

ɥɢɧɟɚɪɢɡɨɜɚɧɧɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɨɞɧɨ ɢɥɢ ɛɨɥɟɟ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɫ ɧɭɥɟɜɨɣ

ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɱɚɫɬɶɸ ɢ ɧɟ ɢɦɟɟɬ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɫ ɩɨɥɨɠɢɬɟɥɶɧɨɣ

ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɱɚɫɬɶɸ, ɬɨ ɧɟɥɢɧɟɣɧɚɹ ɫɢɫɬɟɦɚ ɛɭɞɟɬ ɭɫɬɨɣɱɢɜɚ, ɧɟɭɫɬɨɣɱɢɜɚ ɢɥɢ

ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɚ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɜɨɢɯ ɧɟɥɢɧɟɣɧɵɯ ɱɥɟɧɨɜ.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ ɦɨɠɧɨ ɧɚɣɬɢ, ɧɚɩɪɢɦɟɪ, ɭ Ʉɨɞɞɢɧɝɬɨɧɚ ɢ Ʌɟɜɢɧɫɨɧɚ

(1955)

ɢ ɭ ɗɪɪɨɭɫɦɢɬɚ ɢ ɉɥɟɣɫɚ (1982).

Ɇɵ ɧɟ ɛɭɞɟɦ ɨɫɬɚɧɚɜɥɢɜɚɬɶɫɹ ɧɚ ɭɫɥɨɜɢɹɯ ɭɫɬɨɣɱɢɜɨɫɬɢ ɥɢɧɟɣɧɵɯ ɫɢɫɬɟɦ, ɬɚɤ

ɤɚɤ ɢɯ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɥɸɛɨɦ ɫɬɚɧɞɚɪɬɧɨɦ ɭɱɟɛɧɢɤɟ ɩɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ

ɭɪɚɜɧɟɧɢɹɦ.

ɑɬɨɛɵ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ ɩɪɢɥɨɠɟɧɢɟ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɯ ɜɵɲɟ ɤɨɧɰɟɩɰɢɣ ɤ

ɷɤɨɧɨɦɢɤɟ, ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɨɫɬɚ, ɭɱɢɬɵɜɚɸɳɭɸ ɞɟɧɟɠɧɨɟ

ɨɛɪɚɳɟɧɢɟ. ɉɨɫɤɨɥɶɤɭ ɷɬɚ ɦɨɞɟɥɶ ɛɭɞɟɬ ɜɫɬɪɟɱɚɬɶɫɹ ɜ ɤɧɢɝɟ ɢ ɞɚɥɶɲɟ, ɦɵ ɞɚɞɢɦ

ɡɞɟɫɶ ɟɟ ɩɨɞɪɨɛɧɵɣ ɚɧɚɥɢɡ (ɫɦ. Ɂɚɧɝ, 1989).

ɇɢɠɟ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ ɢɞɟɧɬɢɱɧɚ ɬɨɣ, ɱɬɨ

ɢɫɩɨɥɶɡɨɜɚɧɚ ɜ ɦɨɞɟɥɢ ɋɨɥɨɭ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ ɬɚɤɠɟ, ɱɬɨ ɛɥɚɝɨɫɨɫɬɨɹɧɢɟ ɧɚɫɟɥɟɧɢɹ

ɦɨɠɟɬ ɨɛɟɫɩɟɱɢɜɚɬɶɫɹ ɧɟɫɤɨɥɶɤɢɦɢ ɜɡɚɢɦɨɢɫɤɥɸɱɚɸɳɢɦɢ ɩɭɬɹɦɢ. Ⱦɟɧɶɝɢ,

ɛɟɫɩɥɚɬɧɨ ɝɟɧɟɪɢɪɭɟɦɵɟ (ɜɜɨɞɢɦɵɟ ɜ ɨɛɨɪɨɬ) ɩɪɚɜɢɬɟɥɶɫɬɜɨɦ, ɫɥɭɠɚɬ ɦɟɪɨɣ.

Ⱦɟɧɶɝɢ ɬɪɟɛɭɸɬɫɹ ɞɥɹ ɩɪɨɜɟɞɟɧɢɹ ɫɞɟɥɨɤ ɢ ɢɧɜɟɫɬɢɰɢɢ. ɋɩɪɨɫ ɧɚ ɞɟɧɶɝɢ ɡɚɜɢɫɢɬ ɨɬ

ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɨɯɨɞɨɜ ɢ ɛɥɚɝɨɫɨɫɬɨɹɧɢɹ ɧɚɫɟɥɟɧɢɹ. Ɉɞɧɚɤɨ ɞɥɹ ɩɪɨɫɬɨɬɵ ɦɵ

ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɞɟɧɟɠɧɵɣ ɫɩɪɨɫ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɞɨɯɨɞɚ

ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ, ɛɥɚɝɨɫɨɫɬɨɹɧɢɹ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ ɢ ɩɪɢɛɵɥɢ, ɨɠɢɞɚɟɦɨɣ ɩɪɢ

ɞɚɧɧɨɦ ɜɥɨɠɟɧɢɢ ɤɚɩɢɬɚɥɚ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɞɟɧɟɠɧɵɣ ɪɵɧɨɤ ɜɫɟɝɞɚ ɧɚɯɨɞɢɬɫɹ

ɜ ɪɚɜɧɨɜɟɫɢɢ, ɬ. ɟ. ɫɩɪɨɫ ɧɚ ɞɟɧɶɝɢ ɜɫɟɝɞɚ ɪɚɜɟɧ ɩɪɟɞɥɨɠɟɧɢɸ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ

ɬɚɤɠɟ, ɱɬɨ ɮɭɧɤɰɢɹ ɫɩɪɨɫɚ ɧɚ ɞɟɧɶɝɢ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:

ɝɞɟ ɬ (=M/L, ɝɞɟ Ɇ — ɨɛɴɟɦ ɞɟɧɟɠɧɵɯ ɡɚɩɚɫɨɜ, a L — ɬɪɭɞɨɜɵɟ ɪɟɫɭɪɫɵ) — ɷɬɨ

ɨɛɴɟɦ ɞɟɧɟɠɧɵɯ ɡɚɩɚɫɨɜ, ɩɪɢɯɨɞɹɳɢɣɫɹ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ, G — ɧɟɩɪɟɪɵɜɧɚɹ

ɮɭɧɤɰɢɹ ɫɜɨɢɯ ɚɪɝɭɦɟɧɬɨɜ, ɭ — ɩɪɨɢɡɜɨɞɫɬɜɨ ɩɪɨɞɭɤɰɢɢ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ,

w (= pk +

ɬ, ɝɞɟ ɪ — ɰɟɧɚ) — ɷɬɨ ɛɥɚɝɨɫɨɫɬɨɹɧɢɟ ɜ ɞɟɧɟɠɧɨɦ ɷɤɜɢɜɚɥɟɧɬɟ . ɧɚ ɞɭɲɭ

ɧɚɫɟɥɟɧɢɹ, ɚ r — ɨɠɢɞɚɟɦɵɣ ɩɪɢɬɨɤ ɞɟɧɟɝ ɧɚ ɤɚɩɢɬɚɥ (r = f'(k) — d+ E[dp/dt/p], ɝɞɟ k

—

ɤɚɩɢɬɚɥ, ɩɪɢɯɨɞɹɳɢɣɫɹ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ, d — ɫɤɨɪɨɫɬɶ ɚɦɨɪɬɢɡɚɰɢɢ, E[dp/dt/p]

—

ɨɠɢɞɚɟɦɚɹ ɫɤɨɪɨɫɬɶ ɢɧɮɥɹɰɢɢ). ɋɥɟɞɭɹ ɬɪɚɞɢɰɢɹɦ ɤɟɣɧɫɢɚɧɫɬɜɚ, ɫɱɢɬɚɟɬɫɹ,

ɱɬɨ ɞɟɧɶɝɢ ɩɪɟɞɧɚɡɧɚɱɟɧɵ ɞɥɹ ɭɞɨɜɥɟɬɜɨɪɟɧɢɟ ɨɩɟɪɚɰɢɨɧɧɨɝɨ ɢ ɫɩɟɤɭɥɹɬɢɜɧɨɝɨ

ɫɩɪɨɫɚ, ɚ ɮɭɧɤɰɢɹ G/p = J (k,r) ɧɟ ɹɜɥɹɟɬɫɹ ɨɞɧɨɪɨɞɧɨɣ ɩɨ k.

Ɋɟɚɥɶɧɨɟ ɛɥɚɝɨɫɨɫɬɨɹɧɢɟ W ɢ ɪɟɚɥɶɧɵɣ ɪɚɫɩɨɥɚɝɚɟɦɵɣ ɞɨɯɨɞ Y

ɨɩɪɟɞɟɥɹɸɬɫɹ

ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɤɚɤ

W=K+M/p,

Yd = F(K, L) - dK + d(M/p)/dt.

ɉɨɫɤɨɥɶɤɭ F(K, L) = ɋ + dK + dK/dt, ɝɞɟ ɋ —ɩɨɬɪɟɛɥɟɧɢɟ, ɢɦɟɟɦ Y

= dW/dt + ɋ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɚɥɶɧɵɣ ɱɢɫɬɵɣ ɪɚɫɩɨɥɚɝɚɟɦɵɣ ɞɨɯɨɞ ɪɚɜɟɧ ɢɡɦɟɧɟɧɢɸ ɪɟɚɥɶɧɨɝɨ

ɛɥɚɝɨɫɨɫɬɨɹɧɢɹ ɩɥɸɫ ɪɟɚɥɶɧɨɟ ɩɨɬɪɟɛɥɟɧɢɟ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɪɟɚɥɶɧɨɟ

ɩɨɬɪɟɛɥɟɧɢɟ ɫɨɫɬɚɜɥɹɟɬ ɜɫɟɝɞɚ ɮɢɤɫɢɪɨɜɚɧɧɭɸ ɞɨɥɸ ɨɬ ɪɟɚɥɶɧɨɝɨ ɱɢɫɬɨɝɨ ɞɨɯɨɞɚ

ɋ = cY

, 0 < ɫ < 1, ɝɞɟ ɫ — ɩɪɟɞɟɥɶɧɚɹ ɫɤɥɨɧɧɨɫɬɶ ɤ ɩɨɬɪɟɛɥɟɧɢɸ. ɇɚ ɨɫɧɨɜɟ ɷɬɢɯ

ɩɪɟɞɩɨɥɨɠɟɧɢɣ ɩɨɥɭɱɚɟɦ dW/dt = sY

, ɝɞɟ s = (1 — ɫ). Ʉɨɥɢɱɟɫɬɜɨ ɞɟɧɟɝ ɜ ɪɟɚɥɶɧɵɯ

ɰɟɧɚɯ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɯ = M/pL. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɬɢɦɢ

ɩɪɟɞɩɨɥɨɠɟɧɢɹɦɢ ɢɦɟɟɦ

ɝɞɟ ɩ — ɮɢɤɫɢɪɨɜɚɧɧɚɹ ɫɤɨɪɨɫɬɶ ɪɨɫɬɚ ɧɚɫɟɥɟɧɢɹ, ɚ z — ɩɨɫɬɨɹɧɧɚɹ ɫɤɨɪɨɫɬɶ ɪɨɫɬɚ

ɧɨɦɢɧɚɥɶɧɵɯ ɞɟɧɟɠɧɵɯ ɧɚɤɨɩɥɟɧɢɣ. ɉɚɪɚɦɟɬɪ z ɮɢɤɫɢɪɭɟɬɫɹ ɩɪɚɜɢɬɟɥɶɫɬɜɨɦ.

Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɡɚɞɚɬɶ ɞɢɧɚɦɢɤɭ ɪɨɫɬɚ ɰɟɧ, ɞɚɜɚɣɬɟ ɫɞɟɥɚɟɦ ɧɚɢɜɧɨɟ

ɩɪɟɞɩɨɥɨɠɟɧɢɟ, ɱɬɨ ɢɧɮɥɹɰɢɨɧɧɵɟ ɨɠɢɞɚɧɢɹ ɜɫɟɝɞɚ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɟɚɥɶɧɨɣ

ɢɧɮɥɹɰɢɢ E[dp/dt/p] = dp/dt/p. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ dp/dt = p[r - f'(k) + d]. ɋ ɞɪɭɝɨɣ

ɫɬɨɪɨɧɵ, ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ (3.3.5), ɦɵ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ z ɤɚɤ ɮɭɧɤɰɢɸ k ɢ x:

r = u(k,

ɯ), ɝɞɟ ɢ

> 0, ɢ

< 0. ȼ ɪɟɡɭɥɶɬɚɬɟ ɞɢɧɚɦɢɤɚ ɰɟɧ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ

ɫ ɭɱɟɬɨɦ ɤɨɬɨɪɨɝɨ (3.3.6) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɤɚɤ

ɝɞɟ v = z - d- ɩ.

Ⱦɥɹ ɤɚɩɢɬɚɥɚ ɧɟɫɥɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫɥɟɞɭɸɳɟɟ ɭɪɚɜɧɟɧɢɟ:

ɋɢɫɬɟɦɚ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɭɪɚɜɧɟɧɢɣ (3.3.7) ɢ (3.3..8), ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɦɨɞɟɥɢ Ɍɨɛɢɧɚ

(

ɫɦ. Ɂɚɧɝ, 1989). ɂɡ ɫɤɚɡɚɧɧɨɝɨ ɫɥɟɞɭɟɬ, ɱɬɨ ɧɚɲɟ ɪɚɫɫɦɨɬɪɟɧɢɟ ɨɝɪɚɧɢɱɟɧɨ ɥɢɲɶ

ɬɨɣ ɨɛɥɚɫɬɶɸ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɨɜ ɡɚɞɚɱɢ, ɤɨɬɨɪɵɟ ɢɦɟɸɬ ɫɦɵɫɥ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ

ɷɤɨɧɨɦɢɤɢ. ɗɬɨ, ɤɪɨɦɟ ɩɪɨɱɟɝɨ, ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɢ ɧɚɞɥɟɠɚɳɢɯ ɭɫɥɨɜɢɹɯ ɦɨɠɟɬ

ɛɵɬɶ ɝɚɪɚɧɬɢɪɨɜɚɧɨ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ

(

Ȼɭɪɦɟɣɫɬɟɪ ɢ Ⱦɨɛɟɥɥ, 1970). ȼ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɰɟɧ

ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɨɠɢɬɟɥɶɧɨɣ, ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɢɥɢ ɧɭɥɟɜɨɣ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɡɧɚɤɚ

(r —

ɩ). Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɪɚɜɧɨɜɟɫɧɨɟ ɡɧɚɱɟɧɢɟ k ɜ ɫɥɭɱɚɟ ɨɬɫɭɬɫɬɜɢɹ ɞɟɧɟɝ

(

ɯ = 0) ɛɨɥɶɲɟ, ɱɟɦ ɪɚɜɧɨɜɟɫɧɨɟ ɡɧɚɱɟɧɢɟ ɨɬɧɨɲɟɧɢɹ «ɤɚɩɢɬɚɥ/ɬɪɭɞ» ɩɪɢ ɧɚɥɢɱɢɢ

ɞɟɧɟɝ. Ɍɨɱɤɚ ɪɚɜɧɨɜɟɫɢɹ ɹɜɥɹɟɬɫɹ ɫɟɞɥɨɦ. ɂɥɥɸɫɬɪɚɰɢɟɣ ɩɨɜɟɞɟɧɢɹ ɫɢɫɬɟɦɵ ɜɛɥɢɡɢ

ɪɚɜɧɨɜɟɫɢɹ ɦɨɠɟɬ ɩɨɫɥɭɠɢɬɶ ɪɢɫ. 3.9.

ɗɬɢ ɜɵɜɨɞɵ ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɩɪɢɦɟɧɹɹ ɨɩɢɫɚɧɧɵɟ ɜɵɲɟ ɚɧɚɥɢɬɢɱɟɫɤɢɟ

ɪɟɡɭɥɶɬɚɬɵ.

3.4 ɉɪɹɦɨɣ ɦɟɬɨɞ Ʌɹɩɭɧɨɜɚ

ɂɫɫɥɟɞɨɜɚɧɢɟ ɭɫɬɨɣɱɢɜɨɫɬɢ, ɨɫɧɨɜɚɧɧɨɟ ɧɚ ɫɢɫɬɟɦɟ ɥɢɧɟɚɪɢɡɨɜɚɧɧɵɯ ɜ

ɨɤɪɟɫɬɧɨɫɬɢ ɪɚɜɧɨɜɟɫɢɹ ɭɪɚɜɧɟɧɢɣ, ɩɪɢɜɨɞɢɬ ɤ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɩɪɹɦɨɝɨ

ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɷɬɨɣ ɫɢɫɬɟɦɵ. Ⱦɥɹ ɫɢɫɬɟɦ, ɫɨɞɟɪɠɚɳɢɯ ɦɧɨɠɟɫɬɜɨ ɷɤɨɧɨɦɢɱɟɫɤɢɯ

ɩɟɪɟɦɟɧɧɵɯ, ɷɬɨ ɡɚɱɚɫɬɭɸ ɧɟ ɥɭɱɲɢɣ ɜɵɯɨɞ ɢɡ ɩɨɥɨɠɟɧɢɹ, ɨɫɨɛɟɧɧɨ ɤɨɝɞɚ

ɭɩɨɦɹɧɭɬɨɟ ɪɚɜɧɨɜɟɫɢɟ ɹɜɧɨ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɢ/ɢɥɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ

ɩɟɪɟɦɟɧɧɵɯ.

ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɫɬɶ ɷɬɨɦɭ ɦɟɬɨɞɭ ɦɟɬɨɞ, ɢɡɜɟɫɬɧɵɣ ɤɚɤ ɜɬɨɪɨɣ (ɢɥɢ ɩɪɹɦɨɣ)

ɦɟɬɨɞ Ʌɹɩɭɧɨɜɚ, ɨɛɟɫɩɟɱɢɜɚɟɬ ɧɚɫ ɭɫɥɨɜɢɹɦɢ ɭɫɬɨɣɱɢɜɨɫɬɢ, ɤɨɬɨɪɵɟ (ɚ) ɧɟ ɬɪɟɛɭɸɬ

ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ, (ɛ) ɩɪɢɥɨɠɢɦɵ ɤ ɪɟɲɟɧɢɹɦ ɥɸɛɨɝɨ

ɬɢɩɚ, ɜɤɥɸɱɚɹ ɹɜɧɨ ɡɚɜɢɫɹɳɢɟ ɨɬ ɜɪɟɦɟɧɢ ɢ/ɢɥɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ, ɢ

ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɪɢɦɟɧɢɦɵ ɤ ɧɟɥɢɧɟɣɧɵɦ ɫɢɫɬɟɦɚɦ, ɩɨɞɨɛɧɵɦ (3.1.1). ɋɥɟɞɭɟɬ

ɡɚɦɟɬɢɬɶ, ɱɬɨ ɞɥɹ ɫɥɭɱɚɹ, ɤɨɝɞɚ ɧɟɥɢɧɟɣɧɵɟ ɱɥɟɧɵ ɫɭɳɟɫɬɜɟɧɧɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ

ɪɚɜɧɨɜɟɫɢɹ, ɦɵ ɪɚɫɩɨɥɚɝɚɟɦ ɨɱɟɧɶ ɧɟɛɨɥɶɲɢɦ ɱɢɫɥɨɦ ɬɟɨɪɟɦ ɨɛɳɟɝɨ ɯɚɪɚɤɬɟɪɚ, ɬɚɤ

ɱɬɨ ɨɛɵɱɧɨ ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɠɞɭɸ ɧɨɜɭɸ ɫɢɫɬɟɦɭ ɨɬɞɟɥɶɧɨ. ɂɦɟɧɧɨ

ɩɨɷɬɨɦɭ ɬɟɨɪɟɦɚ Ʌɹɩɭɧɨɜɚ ɜ ɬɟɨɪɢɢ ɭɫɬɨɣɱɢɜɨɫɬɢ ɢɝɪɚɟɬ ɨɫɨɛɟɧɧɨ ɜɚɠɧɭɸ ɪɨɥɶ.

ɗɬɨɬ ɦɟɬɨɞ ɧɚɯɨɞɢɬ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɢ ɜ ɷɤɨɧɨɦɢɤɟ (ɫɦ., ɧɚɩɪɢɦɟɪ, ɗɪɪɨɭ ɢ

ɏɚɧ, 1971).

Ɉɩɪɟɞɟɥɟɧɢɟ 3.4.1. (Ɏɭɧɤɰɢɹ Ʌɹɩɭɧɨɜɚ.) Ɏɭɧɤɰɢɹ V : R

ĺ R ɧɚɡɵɜɚɟɬɫɹ

ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɨɣ, ɟɫɥɢ (ɚ) V(0) = 0 ɢ (ɛ) V > 0 ɜɨ ɜɫɟɯ ɨɫɬɚɥɶɧɵɯ ɬɨɱɤɚɯ

ɢɡ ɧɟɤɨɬɨɪɨɣ ɨɬɤɪɵɬɨɣ ɨɛɥɚɫɬɢ G∈R

ɫɨɞɟɪɠɚɳɟɣ ɧɭɥɟɜɭɸ ɬɨɱɤɭ. Ⱦɥɹ ɥɸɛɨɝɨ

ɪɟɲɟɧɢɹ ɯ = x(t) ɭɪɚɜɧɟɧɢɹ dx/dt == f(x) ɮɭɧɤɰɢɹ V(x) = V(x(t)) ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ t,

ɢ ɟɟ ɩɨɥɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ

ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ V. Ɏɭɧɤɰɢɟɣ Ʌɹɩɭɧɨɜɚ V : R

ĺ R ɫɢɫɬɟɦɵ

ɭɪɚɜɧɟɧɢɣ dx/dt = f(x) ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɭɧɤɰɢɹ,

ɨɛɥɚɞɚɸɳɚɹ ɧɟɩɪɟɪɵɜɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ, ɬɚɤɢɦɢ, ɱɬɨ dV/dt ≤ 0 ɧɚ G ɞɥɹ ɥɸɛɨɝɨ

ɪɟɲɟɧɢɹ ɯ ɫɢɫɬɟɦɵ dx/dt = f(x).

Ɍɟɨɪɟɦɚ 3.4.1. ȿɫɥɢ ɞɥɹ ɫɢɫɬɟɦɵ (3.3.1) ɩɪɢ f(0) = 0 ɫɭɳɟɫɬɜɭɟɬ ɮɭɧɤɰɢɹ

Ʌɹɩɭɧɨɜɚ, ɬɨ ɪɚɜɧɨɜɟɫɢɟ ɜ ɧɭɥɟ ɹɜɥɹɟɬɫɹ ɭɫɬɨɣɱɢɜɵɦ.

Ɍɟɨɪɟɦɚ 3.4.2. ȿɫɥɢ ɮɭɧɤɰɢɹ Ʌɹɩɭɧɨɜɚ ɫɭɳɟɫɬɜɭɟɬ ɢ - dV/dt ɩɨɥɨɠɢɬɟɥɶɧɨ

ɨɩɪɟɞɟɥɟɧɚ, ɬɨ ɧɭɥɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜ.

Ɍɟɨɪɟɦɚ 3.4.3. Ɋɚɜɧɨɜɟɫɢɟ x = 0 ɧɟɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɩɨɥɨɠɢɬɟɥɶɧɨ

ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɭɧɤɰɢɹ V ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ dx/dt = f(x), ɤɨɬɨɪɚɹ ɜ ɧɭɥɟ ɨɛɪɚɳɚɟɬɫɹ

ɜ ɧɭɥɶ (V(0) = 0), ɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ ɨɛɥɚɞɚɟɬ ɤɨɧɟɱɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɢ

ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɨɨɬɧɨɲɟɧɢɸ V(dV/dt) > 0.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɷɬɨɣ ɬɟɨɪɟɦɵ ɦɨɠɧɨ ɧɚɣɬɢ ɭ Ʉɨɞɞɢɧɝɬɨɧɚ ɢ Ʌɟɜɢɧɫɨɧɚ (1955)

ɢɥɢ ɗɪɪɨɭɫɦɢɬɚ ɢ ɉɥɟɣɫɚ (1982).

ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ

ɝɞɟ h ɧɟɩɪɟɪɵɜɧɚ ɜɛɥɢɡɢ ɧɭɥɹ ɢ h(0, 0) = 0. ȼ ɧɭɥɟ ɥɢɧɟɚɪɢɡɨɜɚɧɧɚɹ ɫɢɫɬɟɦɚ

ɨɛɥɚɞɚɟɬ ɨɫɨɛɨɣ ɬɨɱɤɨɣ ɬɢɩɚ ɰɟɧɬɪ. Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ V(x, ɭ) = ɯ

+ ɭ

. ɗɬɚ

ɮɭɧɤɰɢɹ ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɚ, ɢ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɧɨɲɟɧɢɟ dV/dt = 2Vh(x,y).

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ h ɹɜɥɹɟɬɫɹ ɨɬɪɢɰɚɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɨɣ ɮɭɧɤɰɢɟɣ, ɬɨ ɧɭɥɶ ɜ

ɧɟɤɨɬɨɪɨɣ ɫɜɨɟɣ ɨɤɪɟɫɬɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɵɦ, ɚ ɟɫɥɢ

ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɨɣ — ɬɨ ɧɭɥɶ ɧɟɭɫɬɨɣɱɢɜ. ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɧɨɫɢɬ

ɧɟɥɨɤɚɥɶɧɵɣ ɯɚɪɚɤɬɟɪ.

ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɹɦɵɦ ɦɟɬɨɞɨɦ Ʌɹɩɭɧɨɜɚ ɞɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɭɫɬɨɣɱɢɜɨɫɬɢ

ɩɪɨɰɟɫɫɚ ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɨɝɨ ɭɫɬɚɧɨɜɥɟɧɢɹ ɰɟɧ ɜ ɦɨɞɟɥɢ ɗɪɪɨɭ-Ⱦɟɛɪɷ. ɋɥɟɞɭɸɳɢɣ

ɧɢɠɟ ɩɪɢɦɟɪ ɨɫɧɨɜɚɧ ɩɪɟɢɦɭɳɟɫɬɜɟɧɧɨ ɧɚ ɪɚɛɨɬɟ ɏɚɧɚ (1982).

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɷɤɨɧɨɦɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɪɚɛɨɬɚɟɬ ɧɚ ɩɪɨɢɡɜɨɞɫɬɜɨ N ɜɢɞɨɜ

ɬɨɜɚɪɨɜ ɢ ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ ɇ ɞɨɦɚɲɧɢɯ ɯɨɡɹɣɫɬɜ ɢ F ɮɢɪɦ-ɩɪɨɢɡɜɨɞɢɬɟɥɟɣ.

Ɉɩɪɟɞɟɥɢɦ ɯ

∈ R

ɤɚɤ ɜɟɤɬɨɪ ɬɨɪɝɨɜɨɝɨ ɫɚɥɶɞɨ ɯɨɡɹɣɫɬɜɚ h;

xx.

Ⱥɧɚɥɨɝɢɱɧɨ, ɭ

∈ R

ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɟɤɬɨɪ ɚɤɬɢɜɧɨɫɬɢ ɮɢɪɦɵ f, ɝɞɟ

ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɦɩɨɧɟɧɬɵ ɨɡɧɚɱɚɸɬ ɜɵɩɭɫɤ ɩɪɨɞɭɤɰɢɢ, ɚ ɨɬɪɢɰɚɬɟɥɶɧɵɟ —

ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɪɟɫɭɪɫɨɜ:

yy .

ɉɭɫɬɶ z — ɜɟɤɬɨɪ ɫɨɜɨɤɭɩɧɨɝɨ ɢɡɛɵɬɤɚ ɫɩɪɨɫɚ, a s — ɜɟɤɬɨɪ ɫɨɜɨɤɭɩɧɨɝɨ ɢɡɛɵɬɤɚ

ɩɪɟɞɥɨɠɟɧɢɹ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɤɚɤ

z = x-y = -s. (3.4.2)

ɉɭɫɬɶ ɞɥɹ ɭɞɨɛɫɬɜɚ Z, X, Y ɢ S ɨɛɨɡɧɚɱɚɸɬ ɜɟɤɬɨɪɵ z, ɯ, ɭ ɢ s ɛɟɡ ɩɟɪɜɨɣ

ɤɨɦɩɨɧɟɧɬɵ. ɉɭɫɬɶ ɪ ∈

. — ɜɟɤɬɨɪ ɰɟɧ, ɚ Ɋ — ɜɟɤɬɨɪ, ɪɚɜɧɵɣ (l/p

)p ɛɟɡ ɩɟɪɜɨɣ

ɤɨɦɩɨɧɟɧɬɵ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ p

> 0. ȼɤɥɚɞ h-ɨɝɨ ɯɨɡɹɣɫɬɜɚ ɜ ɩɨɬɪɟɛɥɟɧɢɟ

ɡɚɩɢɲɟɦ ɤɚɤ w

∈

. Ɉɩɪɟɞɟɥɢɦ

Ɇɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɷɤɨɧɨɦɢɤɢ, ɨɛɥɚɞɚɸɳɢɟ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɦɢ

ɮɭɧɤɰɢɹɦɢ ɢɡɛɵɬɨɱɧɨɝɨ ɫɩɪɨɫɚ (ɩɪɟɞɥɨɠɟɧɢɹ). ɏɨɪɨɲɨ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɜ ɫɥɭɱɚɟ

ɪɚɰɢɨɧɚɥɶɧɨɝɨ ɩɨɜɟɞɟɧɢɹ ɩɨɬɪɟɛɢɬɟɥɟɣ ɢ ɩɪɨɢɡɜɨɞɢɬɟɥɟɣ ɢɡɛɵɬɨɱɧɵɟ

ɩɪɟɞɥɨɠɟɧɢɟ ɢ ɫɩɪɨɫ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤ ɮɭɧɤɰɢɢ ɪ ɢ ɬ*, ɬ.ɟ. s = s(p,w*) ɢ

z = z(p,w*).