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

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Ɉɧɢ ɹɜɥɹɸɬɫɹ ɨɞɧɨɪɨɞɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɩɨ ɪ ɢ ɩɨɞɱɢɧɹɸɬɫɹ ɡɚɤɨɧɭ ȼɚɥɶɪɚɫɚ

ɉɭɫɬɶ {D = ɪ⏐ɪ > 0,

}

p , ɢ ɩɭɫɬɶ G — ɝɪɚɧɢɰɚ ɨɛɥɚɫɬɢ D.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.4.1. (Ɋɚɜɧɨɜɟɫɢɟ.) Ɍɨɱɤɚ ɪ* ∈ D\G ɹɜɥɹɟɬɫɹ ɪɚɜɧɨɜɟɫɢɟɦ, ɟɫɥɢ

ɞɥɹ ɤɚɠɞɨɝɨ i (I) ɮɭɧɤɰɢɹ s

(ɪ*, w*) ɧɟɨɬɪɢɰɚɬɟɥɶɧɚ ɢ (II)

(ɪ*,w*) = 0.

ɉɪɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɭɫɥɨɜɢɹɯ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ

ɝɚɪɚɧɬɢɪɭɟɬɫɹ. Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɣ ɫɥɭɱɚɣ ɞɢɧɚɦɢɤɢ ɰɟɧ:

Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɪɟɲɟɧɢɟ (3.4.4) ɨɝɪɚɧɢɱɟɧɨ. Ɉɩɪɟɞɟɥɢɦ

ɉɨ ɡɚɤɨɧɭ ȼɚɥɶɪɚɫɚ ɢɦɟɟɦ dV/dt = 2p*s(p). Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ ɜɫɟ ɬɨɜɚɪɵ

ɨɛɥɚɞɚɸɬ ɫɜɨɣɫɬɜɨɦ ɜɚɥɨɜɨɣ ɡɚɦɟɧɢɦɨɫɬɢ, ɬɨ p*s(p) < 0, ɩɪɢ ɪ ɧɟ ɪɚɜɧɨɦ kp* ɞɥɹ

k > 0. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɟɧ ɫɥɟɞɭɸɳɢɣ ɪɟɡɭɥɶɬɚɬ:

Ɍɟɨɪɟɦɚ 3.4.4. ȿɫɥɢ ɜɫɟ ɬɨɜɚɪɵ ɨɛɥɚɞɚɸɬ ɫɜɨɣɫɬɜɨɦ ɜɚɥɨɜɨɣ ɡɚɦɟɧɢɦɨɫɬɢ, ɬɨ

ɩɪɢ ɭɫɥɨɜɢɢ (3.4.5) ɟɞɢɧɫɬɜɟɧɧɨɟ ɩɨɥɨɠɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɵ (3.4.4) ɹɜɥɹɟɬɫɹ

ɝɥɨɛɚɥɶɧɨ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɵɦ

Ɉɩɢɫɚɧɧɵɣ ɜɵɲɟ ɩɪɨɰɟɫɫ ɭɫɬɚɧɨɜɥɟɧɢɹ ɦɨɠɧɨ ɨɛɨɛɳɢɬɶ (ɏɚɧ, 1982). Ʉɪɨɦɟ

ɬɨɝɨ, ɜ ɥɢɬɟɪɚɬɭɪɟ ɦɨɠɧɨ ɧɚɣɬɢ ɦɧɨɠɟɫɬɜɨ ɞɪɭɝɢɯ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɩɪɢɦɟɪɨɜ

ɚɧɚɥɨɝɢɱɧɵɯ ɩɪɨɰɟɫɫɨɜ ɭɫɬɚɧɨɜɥɟɧɢɹ (ɫɦ. ɗɪɪɨɭ ɢ ɏɚɧ, 1971).

ɋɦ. ɩɨɞɪɨɛɧɟɟ Ʌɚɧɤɚɫɬɟɪ Ʉ., Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɷɤɨɧɨɦɢɤɚ, Ɇ.: ɋɨɜɟɬɫɤɨɟ Ɋɚɞɢɨ, 1972. — ɉɪɢɦ. ɪɟɞ.

3.5 ɋɬɪɭɤɬɭɪɧɚɹ ɭɫɬɨɣɱɢɜɨɫɬɶ

Ɋɚɡɜɢɬɚɹ ɜɵɲɟ ɤɨɧɰɟɩɰɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɨɬɜɟɱɚɟɬ ɧɚ ɜɨɩɪɨɫ, ɤɚɤɢɦ ɨɛɪɚɡɨɦ

ɞɢɧɚɦɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɪɟɚɝɢɪɭɟɬ ɧɚ ɜɨɡɦɭɳɟɧɢɟ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ. Ɉɞɧɚɤɨ ɢɧɨɝɞɚ

ɧɚɫ ɢɧɬɟɪɟɫɭɸɬ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɮɨɪɦɵ ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ.

ɇɚɩɪɢɦɟɪ, ɛɵɜɚɟɬ ɜɚɠɧɚ ɢɧɮɨɪɦɚɰɢɹ ɨɛ ɭɫɬɨɣɱɢɜɨɫɬɢ ɫɚɦɨɣ ɮɭɧɤɰɢɢ ɩɪɟɞɥɨɠɟɧɢɹ

ɞɟɧɟɝ. Ʉɚɱɟɫɬɜɟɧɧɵɟ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɣ ɧɚɯɨɞɹɬ ɨɬɪɚɠɟɧɢɟ ɜ ɤɨɧɰɟɩɰɢɢ

ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ.

Ɇɵ ɫɬɪɨɢɦ ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɦɨɞɟɥɢ, ɱɬɨɛɵ ɨɛɴɹɫɧɢɬɶ ɪɟɚɥɶɧɵɟ ɹɜɥɟɧɢɹ. Ɉɞɧɚɤɨ

ɪɟɡɭɥɶɬɚɬ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɜɟɫɶɦɚ ɱɭɜɫɬɜɢɬɟɥɶɧɵɦ ɤ ɦɚɥɟɣɲɢɦ ɢɡɦɟɧɟɧɢɹɦ

ɦɨɞɟɥɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɨɢɡɜɨɥɶɧɨ ɦɚɥɨɟ ɢɡɦɟɧɟɧɢɟ ɦɨɞɟɥɢ ɩɪɢɜɨɞɢɬ ɧɚɫ ɤ ɞɪɭɝɨɣ

ɦɨɞɟɥɢ ɫ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ. ɑɬɨɛɵ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ

ɤɨɧɰɟɩɰɢɸ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ, ɨɛɪɚɬɢɦɫɹ ɤ ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɨɣ ɛɢɨɥɨɝɢ-

ɱɟɫɤɨɣ ɦɨɞɟɥɢ «ɯɢɳɧɢɤ-ɠɟɪɬɜɚ», ɤɨɬɨɪɭɸ ɪɹɞ ɚɜɬɨɪɨɜ ɪɚɫɫɦɚɬɪɢɜɚɥ ɜ ɩɪɢɥɨɠɟɧɢɢ

ɤ ɷɤɨɧɨɦɢɤɟ. ɋɢɫɬɟɦɚ «ɯɢɳɧɢɤ-ɠɟɪɬɜɚ» ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ

ɭɪɚɜɧɟɧɢɣ ɫɥɟɞɭɸɳɟɝɨ ɜɢɞɚ:

ȼ ɥɢɬɟɪɚɬɭɪɟ ɩɨ ɞɢɧɚɦɢɤɟ ɝɨɪɨɞɨɜ ɷɬɚ ɦɨɞɟɥɶ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɨɩɢɫɚɧɢɹ ɞɢɧɚɦɢɤɢ

ɦɚɥɵɯ ɝɨɪɨɞɫɤɢɯ ɚɪɟɚɥɨɜ (Ⱦɟɧɞɪɢɧɨɫ ɢ Ɇɭɥɥɚɥɥɢ, 1983, Ɂɚɧɝ, 1988ɫ). ɉɟɪɟɦɟɧɧɚɹ ɯ

ɜ (3.5.1) ɨɡɧɚɱɚɟɬ ɩɥɨɬɧɨɫɬɶ ɡɟɦɥɟɩɨɥɶɡɨɜɚɧɢɹ, ɭ — ɡɟɦɟɥɶɧɭɸ ɪɟɧɬɭ,

, y

—

ɧɟɤɨɬɨɪɵɟ ɩɚɪɚɦɟɬɪɵ. ɋɢɫɬɟɦɚ ɨɩɢɫɵɜɚɟɬ ɩɪɨɫɬɭɸ ɦɨɞɟɥɶ ɫɩɪɨɫɚ-ɩɪɟɞɥɨɠɟɧɢɹ

ɫɩɟɤɭɥɹɬɢɜɧɨɣ ɡɟɦɟɥɶɧɨɣ ɪɟɧɬɵ ɫ ɭɱɟɬɨɦ ɛɭɞɭɳɢɯ ɩɪɨɰɟɧɬɨɜ ɩɪɢ ɱɚɫɬɢɱɧɨ

ɫɨɜɩɚɞɚɸɳɢɯ ɨɠɢɞɚɧɢɹɯ ɫɨ ɫɬɨɪɨɧɵ ɡɟɦɥɟɩɨɥɶɡɨɜɚɬɟɥɟɣ ɢ ɡɟɦɥɟɜɥɚɞɟɥɶɰɟɜ.

Ⱦɪɭɝɨɟ ɩɪɢɥɨɠɟɧɢɟ ɦɨɞɟɥɢ ɷɬɨɝɨ ɬɢɩɚ ɤ ɷɤɨɧɨɦɢɤɟ ɞɚɧɨ Ƚɭɞɜɢɧɨɦ (ɫɦ. ɬɚɤɠɟ

Ƚɚɛɢɲ ɢ Ʌɨɪɟɧɰ, 1986). ɉɨɫɤɨɥɶɤɭ ɫɢɫɬɟɦɚ «ɯɢɳɧɢɤ-ɠɟɪɬɜɚ» ɛɭɞɟɬ ɧɟ ɪɚɡ

ɭɩɨɦɢɧɚɬɶɫɹ ɢ ɞɚɥɟɟ, ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɩɨɥɟɡɧɵɦ ɨɩɢɫɚɬɶ ɦɨɞɟɥɶ Ƚɭɞɜɢɧɚ ɛɨɥɟɟ

ɞɟɬɚɥɶɧɨ.

Ɇɨɞɟɥɶ Ƚɭɞɜɢɧɚ ɩɨɫɬɪɨɟɧɚ ɞɥɹ ɨɩɢɫɚɧɢɹ ɤɥɚɫɫɨɜɨɣ ɛɨɪɶɛɵ. Ɋɚɫɫɦɨɬɪɢɦ ɞɜɚ

ɬɢɩɚ ɝɪɚɠɞɚɧ: ɪɚɛɨɱɢɯ ɢ ɤɚɩɢɬɚɥɢɫɬɨɜ. Ɋɚɛɨɱɢɟ ɬɪɚɬɹɬ ɜɟɫɶ ɫɜɨɣ ɞɨɯɨɞ wL ɧɚ

ɩɨɬɪɟɛɥɟɧɢɟ, ɤɚɩɢɬɚɥɢɫɬɵ ɧɚɤɚɩɥɢɜɚɸɬ ɫɜɨɣ ɞɨɯɨɞ Y — wL, ɝɞɟ Y — ɩɪɨɞɭɤɰɢɹ

ɩɪɨɢɡɜɨɞɫɬɜɚ. ɐɟɧɚ ɩɨɬɪɟɛɢɬɟɥɶɫɤɢɯ ɬɨɜɚɪɨɜ ɨɬɧɨɪɦɢɪɨɜɚɧɚ ɤ ɟɞɢɧɢɰɟ. ɉɭɫɬɶ Ʉ

ɨɡɧɚɱɚɟɬ ɤɚɩɢɬɚɥ, ɚ = ɚ

ɟɯɪ(Jt) = Y/L — ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɬɪɭɞɚ, ɜɨɡɪɚɫɬɚɸɳɭɸ ɫ

ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ J, k = K/Y — ɤɨɷɮɮɢɰɢɟɧɬ ɤɚɩɢɬɚɥɨɟɦɤɨɫɬɢ ɩɪɨɞɭɤɰɢɢ, ɚ

N =N

exp(nt) —ɩɪɟɞɥɨɠɟɧɢɟ ɧɚ ɪɵɧɤɟ ɪɚɛɨɱɟɣ ɫɢɥɵ, ɤɨɬɨɪɨɟ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ

ɬɟɦɩɨɦ ɪɨɫɬɚ n. Ⱦɨɥɹ ɡɚɬɪɚɬ ɧɚ ɨɩɥɚɬɭ ɬɪɭɞɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɚɰɢɨɧɚɥɶɧɨɦɭ

ɞɨɯɨɞɭ ɫɨɫɬɚɜɥɹɟɬ wL/Y = w/a. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɨɥɹ ɩɪɢɛɵɥɢ ɤɚɩɢɬɚɥɢɫɬɨɜ

ɫɨɫɬɚɜɥɹɟɬ 1 - w/a. ɉɨɫɤɨɥɶɤɭ ɫɛɟɪɟɠɟɧɢɹ ɨɩɪɟɞɟɥɟɧɵ ɤɚɤ S == Y —wL = (1 — w/a)Y,

ɞɨɥɹ ɢɧɜɟɫɬɢɰɢɣ ɫɨɫɬɚɜɥɹɟɬ dK/dt = S = (1 - w/a)Y ɢɥɢ (dK/dt)/K = (1 - w/a)/k, ɩɪɢɱɟɦ

ɜɵɛɵɬɢɟɦ ɤɚɩɢɬɚɥɚ ɦɵ ɩɪɟɧɟɛɪɟɝɥɢ. ɉɪɢ ɩɨɫɬɨɹɧɧɨɦ ɡɧɚɱɟɧɢɢ ɤɚɩɢɬɚɥɨɟɦɤɨɫɬɢ k

ɩɨɥɭɱɚɟɦ, ɱɬɨ YdK/dt = KdY/dt. ɂɬɚɤ, ɜ ɫɢɥɭ

ɢ (dK/dt)/K = (1 -w/a)/k ɩɨɥɭɱɢɦ (dL/dt)/L = (1 - w/a)/k - J. ȼɜɨɞɹ ɧɨɜɵɟ ɩɟɪɟɦɟɧɧɵɟ

—

ɞɨɥɸ ɡɚɬɪɚɬ ɧɚ ɨɩɥɚɬɭ ɬɪɭɞɚ, ɭ = w/a, ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɧɹɬɨɫɬɢ ɯ = L/N, ɦɨɠɧɨ

ɩɨɤɚɡɚɬɶ, ɱɬɨ

ɝɞɟ (da/dt)/a = J, Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɫɬɚɜɤɭ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɵ ɛɵɫɬɪɨɣ ɩɟɪɟɦɟɧɧɨɣ,

ɤɨɬɨɪɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɪɢɜɨɣ Ɏɢɥɥɢɩɫɚ, ɬ.ɟ.

Ʌɢɧɟɣɧɚɹ ɚɩɩɪɨɤɫɢɦɚɰɢɹ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ (dw/dt)/w = —r + bɯ ɩɪɢɜɨɞɢɬ ɧɚɫ ɤ

(

dy/dt)/y = -r +bɯ - J. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɦɢ ɩɨɥɭɱɟɧɚ ɦɨɞɟɥɶ Ƚɭɞɜɢɧɚ ɜ ɫɥɟɞɭɸɳɟɦ

ɜɢɞɟ:

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɨɞɟɥɶ Ƚɭɞɜɢɧɚ ɫɨɜɩɚɞɚɟɬ ɫ ɦɨɞɟɥɶɸ (3.5.1). ȼɫɟ ɜɵɜɨɞɵ ɨɛɳɟɝɨ

ɯɚɪɚɤɬɟɪɚ, ɫɩɪɚɜɟɞɥɢɜɵɟ ɞɥɹ ɫɢɫɬɟɦɵ (3.5.1), ɫɩɪɚɜɟɞɥɢɜɵ ɢ ɞɥɹ ɦɨɞɟɥɢ Ƚɭɞɜɢɧɚ.

Ɏɨɪɦɚɥɶɧɨɟ ɫɯɨɞɫɬɜɨ ɦɨɞɟɥɢ Ƚɭɞɜɢɧɚ ɫ ɦɨɞɟɥɶɸ Ʌɨɬɤɢ-ȼɨɥɶɬɟɪɪɚ «ɯɢɳɧɢɤ-

ɠɟɪɬɜɚ» ɩɨɡɜɨɥɹɟɬ ɭɫɬɚɧɨɜɢɬɶ ɚɧɚɥɨɝɢɸ ɦɟɠɞɭ ɹɜɥɟɧɢɹɦɢ ɤɥɚɫɫɨɜɨɣ ɛɨɪɶɛɵ ɢ

ɛɨɪɶɛɨɣ ɛɢɨɥɨɝɢɱɟɫɤɢɯ ɫɨɨɛɳɟɫɬɜ

. Ɇɨɞɟɥɶ Ƚɭɞɜɢɧɚ, ɭɱɢɬɵɜɚɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ

Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɛɢɨɥɨɝɢɱɟɫɤɨɣ ɤɨɧɤɭɪɟɧɰɢɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɪɭɝɢɟ ɦɨɞɟɥɢ ɧɟɠɟɥɢ ɦɨɞɟɥɶ «ɯɢɳɧɢɤ-

ɠɟɪɬɜɚ». Ʉ ɧɢɦ, ɜ ɱɚɫɬɧɨɫɬɢ, ɨɬɧɨɫɢɬɫɹ ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɦɨɞɟɥɶ ɤɨɧɤɭɪɟɧɰɢɢ ɡɚ ɪɟɫɭɪɫ (ɫɦ., ɧɚɩɪɢɦɟɪ,

ɋɜɢɪɟɠɟɜ ɘ.Ɇ., Ʌɨɝɨɮɟɬ Ⱦ.Ɉ., ɍɫɬɨɣɱɢɜɨɫɬɶ ɛɢɨɥɨɝɢɱɟɫɤɢɯ ɫɨɨɛɳɟɫɬɜ,- Ɇ: ɇɚɭɤɚ. 1978). Ɍɚɤɚɹ ɦɨɞɟɥɶ

ɞɥɹ ɞɜɭɯ ɤɨɧɤɭɪɢɪɭɸɳɢɯ ɜɢɞɨɜ ɨɬɥɢɱɚɟɬɫɹ ɨɬ (3.5.1) ɡɧɚɤɨɦ ɥɟɜɨɣ ɱɚɫɬɢ ɜɨ ɜɬɨɪɨɦ ɭɪɚɜɧɟɧɢɢ. -

ɉɪɢɦ.ɪɟɞ

ɦɟɠɞɭ ɭɪɨɜɧɟɦ ɡɚɧɹɬɨɫɬɢ ɢ ɡɚɤɨɧɨɞɚɬɟɥɶɧɨ ɭɫɬɚɧɨɜɥɟɧɧɨɣ ɞɨɥɟɣ ɨɬɱɢɫɥɟɧɢɣ ɧɚ

ɨɩɥɚɬɭ ɬɪɭɞɚ, ɜɟɫɶɦɚ ɧɚɩɨɦɢɧɚɟɬ ɤɥɚɫɫɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɩɨɥɢɬɢɱɟɫɤɨɣ ɷɤɨɧɨɦɢɢ.

ɇɚɡɵɜɚɟɦɚɹ ɢɧɨɝɞɚ ɧɟɨɦɚɪɤɫɢɫɬɫɤɨɣ ɦɨɞɟɥɶɸ, ɨɧɚ ɜɧɨɜɶ ɩɪɢɜɥɟɤɥɚ ɜɧɢɦɚɧɢɟ ɤ

ɬɪɭɞɚɦ ɷɤɨɧɨɦɢɫɬɨɜ-ɤɥɚɫɫɢɤɨɜ, ɬɚɤɢɯ, ɤɚɤ Ɋɢɤɚɪɞɨ, ɋɦɢɬ ɢ Ɇɚɪɤɫ. ɋɨɜɪɟɦɟɧɧɵɟ

ɨɛɨɛɳɟɧɢɹ ɞɚɧɧɨɣ ɦɨɞɟɥɢ ɩɪɢɧɚɞɥɟɠɚɬ, ɧɚɩɪɢɦɟɪ, Ⱦɟɫɚɢ (1973), ȼɟɥɭɩɢɥɥɚɢ

(1978),

ɒɚɯɭ ɢ Ⱦɟɫɚɢ (1981), ȼɚɧ ɞɟɪ ɉɥɸɝɭ (1983, 1987), Ɏɥɚɲɟɥɸ (1984) ɢ Ɂɚɧɝɭ

(1988

ɚ).

Ɉɛɫɭɠɞɚɟɦɚɹ ɦɨɞɟɥɶ ɨɱɟɧɶ ɩɪɨɫɬɚ ɢ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɤ ɨɫɰɢɥɥɹɬɨɪɧɵɦ

ɷɮɮɟɤɬɚɦ. Ɉɞɧɚɤɨ ɟɟ ɩɪɢɥɨɠɟɧɢɹ ɨɝɪɚɧɢɱɟɧɵ ɜ ɫɢɥɭ ɟɟ ɫɬɪɭɤɬɭɪɧɨɣ

ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ. Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɞɚɠɟ ɦɚɥɵɟ ɢɡɦɟɧɟɧɢɹ

ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɮɨɪɦɵ ɛɭɞɭɬ ɜɥɢɹɬɶ ɧɚ ɤɚɱɟɫɬɜɟɧɧɵɟ ɫɜɨɣɫɬɜɚ ɫɢɫɬɟɦɵ. Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ, ɩɪɟɞɥɨɠɟɧɧɚɹ ɦɨɞɟɥɶ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɩɟɪɟɧɟɫɟɧɚ ɧɚ ɪɟɚɥɶɧɵɟ ɢɡɭɱɚɟɦɵɟ

ɹɜɥɟɧɢɹ, ɩɨɬɨɦɭ ɱɬɨ, ɫɬɪɨɹ ɦɨɞɟɥɶ ɪɟɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ, ɦɵ ɢɞɟɚɥɢɡɢɪɭɟɦ ɢ ɭɩɪɨɳɚ-

ɟɦ ɟɟ, ɬɚɤ ɤɚɤ ɜɫɟ ɜɯɨɞɹɳɢɟ ɩɚɪɚɦɟɬɪɵ ɢɡɜɟɫɬɧɵ ɧɚɦ ɥɢɲɶ ɩɪɢɛɥɢɠɟɧɧɨ.

ȿɫɬɟɫɬɜɟɧɧɨ, ɞɚɥɟɟ ɜɨɡɧɢɤɚɟɬ ɜɨɩɪɨɫ, ɤɚɤɢɦ ɨɛɪɚɡɨɦ ɜɵɞɟɥɢɬɶ ɬɟ ɫɜɨɣɫɬɜɚ ɦɨɞɟɥɢ

ɢɡɭɱɚɟɦɨɝɨ ɹɜɥɟɧɢɹ, ɤɨɬɨɪɵɟ ɧɟ ɛɭɞɭɬ ɫɥɢɲɤɨɦ ɱɭɜɫɬɜɢɬɟɥɶɧɵ ɤ ɦɚɥɵɦ

ɢɡɦɟɧɟɧɢɹɦ ɦɨɞɟɥɢ, ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɝɭɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɫɜɨɣɫɬɜɚ

ɪɟɚɥɶɧɨɝɨ ɩɪɨɰɟɫɫɚ.

ɇɚ ɬɚɤɢɟ ɫɜɨɣɫɬɜɚ ɭɤɚɡɵɜɚɟɬ ɧɚɦ ɩɨɧɹɬɢɟ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ.

ɏɨɬɹ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɟ ɢɞɟɢ ɤɨɧɰɟɩɰɢɢ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɪɢɧɚɞɥɟɠɚɬ

ɉɭɚɧɤɚɪɟ, ɫɨɜɪɟɦɟɧɧɵɣ ɜɢɞ ɨɧɢ ɩɪɢɨɛɪɟɥɢ ɜ ɪɚɛɨɬɚɯ Ⱥɧɞɪɨɧɨɜɚ ɢ ɉɨɧɬɪɹɝɢɧɚ

1937

ɝɨɞɚ. Ɂɧɚɱɢɬɟɥɶɧɨɝɨ ɩɪɨɝɪɟɫɫɚ ɜ ɬɟɨɪɢɢ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɞɥɹ

ɮɚɡɨɜɵɯ ɩɪɨɫɬɪɚɧɫɬɜ ɦɚɥɨɣ ɪɚɡɦɟɪɧɨɫɬɢ ɞɨɛɢɥɫɹ ɋɦɟɣɥ (1967). Ɉɧ ɩɨɤɚɡɚɥ, ɱɬɨ ɞɥɹ

ɮɚɡɨɜɵɯ ɩɪɨɫɬɪɚɧɫɬɜ ɛɨɥɶɲɢɯ ɪɚɡɦɟɪɧɨɫɬɟɣ ɫɭɳɟɫɬɜɭɸɬ ɫɢɫɬɟɦɵ, ɜ ɨɤɪɟɫɬɧɨɫɬɢ

ɤɨɬɨɪɵɯ ɧɟɬ ɫɬɪɭɤɬɭɪɧɨ ɭɫɬɨɣɱɢɜɵɯ ɫɢɫɬɟɦ. ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɨɛɥɟɦɚ

ɩɨɥɧɨɣ ɬɨɩɨɥɨɝɢɱɟɫɤɨɣ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɦɧɨɝɨ-

ɦɟɪɧɨɦ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɛɟɡɧɚɞɟɠɧɚ, ɞɚɠɟ ɟɫɥɢ ɨɝɪɚɧɢɱɢɬɶɫɹ ɬɨɥɶɤɨ

ɬɢɩɨɜɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ ɢ ɬɨɥɶɤɨ ɧɟɜɵɪɨɠɞɟɧɧɵɦɢ ɫɥɭɱɚɹɦɢ.

ɑɬɨɛɵ ɩɨɹɫɧɢɬɶ ɢɞɟɸ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ, ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɧɟɤɨɬɨɪɨɦ

ɦɧɨɝɨɨɛɪɚɡɢɢ Ɇ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ dx/dt = f(x), ɝɞɟ f ɩɪɢɧɚɞɥɟɠɢɬ

ɡɚɞɚɧɧɨɦɭ ɜɟɤɬɨɪɧɨɦɭ ɩɨɥɸ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.5.1. Ƚɨɜɨɪɹɬ, ɱɬɨ ɞɜɟ ɫɢɫɬɟɦɵ ɬɨɩɨɥɨɝɢɱɟɫɤɢ ɨɪɛɢɬɚɥɶɧɨ

ɷɤɜɢɜɚɥɟɧɬɧɵ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɝɨɦɟɨɦɨɪɮɢɡɦ ɮɚɡɨɜɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɩɟɪɜɨɣ

ɫɢɫɬɟɦɵ ɧɚ ɮɚɡɨɜɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɜɬɨɪɨɣ, ɩɟɪɟɜɨɞɹɳɢɣ ɨɪɢɟɧɬɢɪɨɜɚɧɧɵɟ ɮɚɡɨɜɵɟ

ɤɪɢɜɵɟ ɩɟɪɜɨɣ ɫɢɫɬɟɦɵ ɧɚ ɨɪɢɟɧɬɢɪɨɜɚɧɧɵɟ ɮɚɡɨɜɵɟ ɤɪɢɜɵɟ ɜɬɨɪɨɣ ɫɢɫɬɟɦɵ. ɉɪɢ

ɷɬɨɦ ɧɟ ɬɪɟɛɭɟɬɫɹ ɧɢɤɚɤɨɣ ɤɨɨɪɞɢɧɚɰɢɢ ɞɜɢɠɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɮɚɡɨɜɵɯ

ɤɪɢɜɵɯ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.5.2. ɉɭɫɬɶ Ɇ — ɤɨɦɩɚɤɬɧɨɟ ɦɧɨɝɨɨɛɪɚɡɢɟ (ɤɥɚɫɫɚ C

k-1

, k > 0).

ɉɭɫɬɶ f — ɜɟɤɬɨɪɧɨɟ ɩɨɥɟ ɤɥɚɫɫɚ k (ɟɫɥɢ Ɇ ɨɛɥɚɞɚɟɬ ɝɪɚɧɢɰɟɣ, ɬɨ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ,

ɱɬɨ f ɧɟ ɤɚɫɚɬɟɥɶɧɨ ɤ ɧɟɣ). ɋɢɫɬɟɦɚ (Ɇ, f) ɧɚɡɵɜɚɟɬɫɹ ɫɬɪɭɤɬɭɪɧɨ ɭɫɬɨɣɱɢɜɨɣ, ɟɫɥɢ ɜ

ɩɪɨɫɬɪɚɧɫɬɜɟ ɋ

ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɚɹ ɨɤɪɟɫɬɧɨɫɬɶ f, ɱɬɨ ɤɚɠɞɨɟ ɜɟɤɬɨɪɧɨɟ ɩɨɥɟ,

ɥɟɠɚɳɟɟ ɜ ɷɬɨɣ ɨɤɪɟɫɬɧɨɫɬɢ, ɨɩɪɟɞɟɥɹɟɬ ɫɢɫɬɟɦɭ, ɤɨɬɨɪɚɹ ɬɨɩɨɥɨɝɢɱɟɫɤɢ

ɨɪɛɢɬɚɥɶɧɨ ɷɤɜɢɜɚɥɟɧɬɧɚ ɢɫɯɨɞɧɨɣ, ɢ ɝɨɦɟɨɦɨɪɮɢɡɦ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɛɥɢɡɨɤ

ɬɨɠɞɟɫɬɜɟɧɧɨɦɭ ɝɨɦɟɨɦɨɪɮɢɡɦɭ.

ɋɢɫɬɟɦɚ, ɤɨɬɨɪɚɹ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ,

ɧɚɡɵɜɚɟɬɫɹ ɫɬɪɭɤɬɭɪɧɨ ɧɟɭɫɬɨɣɱɢɜɨɣ. ȼ ɷɬɨɦ ɫɦɵɫɥɟ ɫɢɫɬɟɦɚ (3.5.1) ɹɜɥɹɟɬɫɹ

ɧɟɭɫɬɨɣɱɢɜɨɣ. ɉɪɢɜɟɞɟɦ ɟɳɟ ɨɞɢɧ ɩɪɢɦɟɪ ɫɬɪɭɤɬɭɪɧɨ ɧɟɭɫɬɨɣɱɢɜɨɣ ɫɢɫɬɟɦɵ.

Ⱦɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɫ ɬɪɟɧɢɟɦ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ d

x/dt

= dy/dt = —ɯ — rɭ.

ȿɫɥɢ r = 0, ɬɨ ɜɫɟ ɮɚɡɨɜɵɟ ɤɪɢɜɵɟ ɡɚɦɤɧɭɬɵ. ȿɫɥɢ r > 0, ɨɧɢ ɧɚɦɚɬɵɜɚɸɬɫɹ ɩɨ

ɫɩɢɪɚɥɢ ɧɚ ɬɨɱɤɭ ɯ = ɭ = 0, ɤɨɬɨɪɚɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɢɡɨɥɢɪɨɜɚɧɧɭɸ ɨɫɨɛɭɸ

ɬɨɱɤɭ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɛɵɥ ɪɚɜɟɧ ɧɭɥɸ, ɟɝɨ ɦɚɥɨɟ

ɢɡɦɟɧɟɧɢɟ ɤɚɱɟɫɬɜɟɧɧɨ ɢɡɦɟɧɹɟɬ ɯɚɪɚɤɬɟɪ ɩɨɜɟɞɟɧɢɹ ɮɚɡɨɜɵɯ ɤɪɢɜɵɯ. ȿɫɥɢ

ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɛɵɥ ɨɬɥɢɱɟɧ ɨɬ ɧɭɥɹ ɢ ɩɨɥɨɠɢɬɟɥɟɧ, ɤɚɱɟɫɬɜɟɧɧɨɝɨ ɢɡɦɟɧɟɧɢɹ

ɨɛɳɟɣ ɤɚɪɬɢɧɵ ɧɟ ɧɚɛɥɸɞɚɟɬɫɹ.

Ɇɵ ɦɨɠɟɦ ɞɚɬɶ ɟɳɟ ɛɨɥɟɟ ɩɪɨɫɬɨɟ ɩɨɹɫɧɟɧɢɟ ɷɬɨɝɨ ɩɨɧɹɬɢɹ, ɤɨɬɨɪɨɝɨ, ɜɩɪɨɱɟɦ,

ɞɥɹ ɧɚɫ ɛɭɞɟɬ ɜɩɨɥɧɟ ɞɨɫɬɚɬɨɱɧɨ. ɋɢɫɬɟɦɚ dx/dt = f(x) ɹɜɥɹɟɬɫɹ ɫɬɪɭɤɬɭɪɧɨ

ɭɫɬɨɣɱɢɜɨɣ, ɟɫɥɢ ɞɥɹ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɯ ɜɨɡɦɭɳɟɧɢɣ ɪ(ɯ) ɫɭɳɟɫɬɜɭɟɬ

ɝɨɦɟɨɦɨɪɮɢɡɦ, ɩɟɪɟɜɨɞɹɳɢɣ ɬɪɚɟɤɬɨɪɢɢ dx/dt = f(x) ɜ ɬɪɚɟɤɬɨɪɢɢ dx/dt = f(x) +ɪ(ɯ).

Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Ɇ ɜɧɭɬɪɟɧɧɸɸ ɨɛɥɚɫɬɶ ɡɚɦɤɧɭɬɨɣ ɤɪɢɜɨɣ, ɧɟ ɢɦɟɸɳɟɣ

ɤɚɫɚɧɢɣ ɫ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ ɜɟɤɬɨɪɧɵɦɢ ɩɨɥɹɦɢ, ɢ ɩɭɫɬɶ G — ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ

ɷɬɢɯ ɜɟɤɬɨɪɧɵɯ ɩɨɥɟɣ C

. ɋɩɪɚɜɟɞɥɢɜɚ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ:

Ɍɟɨɪɟɦɚ 3.5.1. Ɏɭɧɤɰɢɹ f(x) ɜ G ɫɬɪɭɤɬɭɪɧɨ ɭɫɬɨɣɱɢɜɚ ɜ ɬɨɦ ɢ ɬɨɥɶɤɨ ɬɨɦ

ɫɥɭɱɚɟ, ɟɫɥɢ ɤɚɠɞɚɹ ɬɨɱɤɚ ɪɚɜɧɨɜɟɫɢɹ ɢ ɤɚɠɞɚɹ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɨɪɛɢɬɚ ɢɦɟɸɬ

ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɣ ɬɢɩ, ɢ ɦɟɠɞɭ ɫɟɞɥɨɜɵɦɢ ɬɨɱɤɚɦɢ ɧɟɬ ɫɨɟɞɢɧɟɧɢɣ. Ɇɧɨɠɟɫɬɜɨ

ɫɬɪɭɤɬɭɪɧɨ ɭɫɬɨɣɱɢɜɵɯ ɫɢɫɬɟɦ ɹɜɥɹɟɬɫɹ ɨɬɤɪɵɬɵɦ ɢ ɩɥɨɬɧɵɦ ɜ G.

ɗɬɢ ɢɞɟɢ, ɤɚɤ ɢ ɬɟɨɪɟɦɚ 3.5.1, ɩɪɢɧɚɞɥɟɠɚɬ ɑɭ ɢ ɏɟɣɥɭ (1982)

. Ɍɟɨɪɟɦɚ

ɮɨɪɦɭɥɢɪɭɟɬ ɧɟɨɛɯɨɞɢɦɨɟ ɢ ɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ

ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ. Ɉɞɧɚɤɨ ɩɪɢɦɟɧɢɬɶ ɷɬɢ ɪɟɡɭɥɶɬɚɬɵ ɜ ɪɟɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ ɧɟ

ɬɚɤ-ɬɨ ɩɪɨɫɬɨ, ɩɨɫɤɨɥɶɤɭ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɣ ɬɟɨɪɟɦɵ ɬɪɭɞɧɨ ɩɪɨɜɟɪɢɬɶ.

ɋɦ. ɬɚɤɠɟ Ⱥɧɞɪɨɧɨɜ, ɉɨɧɬɪɹɝɢɧ (1937) ɢ Ⱥɪɧɨɥɶɞ (ɫɧɨɫɤɚ ɜ ɪɚɡɞ. 3.2). — ɉɪɢɦ. ɪɟɞ.

3.6 Ʉɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ

ɷɬɨɦ ɪɚɡɞɟɥɟ ɦɵ ɨɩɪɟɞɟɥɢɦ ɩɨɧɹɬɢɟ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɫɬɟɦɵ, ɢɡɭɱɢɦ ɫɜɨɣɫɬɜɚ

ɬɚɤɢɯ ɫɢɫɬɟɦ ɢ ɩɨɤɚɠɟɦ, ɤɚɤɨɜɚ ɫɜɹɡɶ ɦɟɠɞɭ ɤɨɧɫɟɪɜɚɬɢɜɧɨɫɬɶɸ ɫɢɫɬɟɦɵ ɢ ɟɟ

ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɶɸ.

Ɋɚɫɫɦɨɬɪɢɦ ɞɢɧɚɦɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ dx/dt = f(ɯ). Ɏɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɫɜɨɣɫɬɜɨɦ

ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɫɬɟɦɵ ɹɜɥɹɟɬɫɹ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɬɚɤɨɣ ɮɭɧɤɰɢɢ ɡɚɜɢɫɢɦɵɯ

ɩɟɪɟɦɟɧɧɵɯ ɫɢɫɬɟɦɵ, ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɤɨɧɫɬɚɧɬɨɣ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɢ ɢɝɪɚɟɬ

ɪɨɥɶ «ɷɧɟɪɝɢɢ». ɋɢɫɬɟɦɚ ɹɜɥɹɟɬɫɹ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɮɭɧɤɰɢɹ G(x),

ɧɚɡɵɜɚɟɦɚɹ ɩɟɪɜɵɦ ɢɧɬɟɝɪɚɥɨɦ ɢɥɢ ɩɪɨɫɬɨ ɢɧɬɟɝɪɚɥɨɦ ɫɢɫɬɟɦɵ, ɬɚɤɚɹ, ɱɬɨ

ɉɨɥɶɡɭɹɫɶ ɮɢɡɢɱɟɫɤɨɣ ɬɟɪɦɢɧɨɥɨɝɢɟɣ, ɦɨɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ

ɯɚɪɚɤɬɟɪɧɵ ɬɟɦ, ɱɬɨ ɜ ɩɪɨɰɟɫɫɟ ɷɜɨɥɸɰɢɢ ɷɥɟɦɟɧɬ «ɨɛɴɟɦɚ» ɜ ɮɚɡɨɜɨɦ

ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɦɟɧɹɟɬ ɬɨɥɶɤɨ ɮɨɪɦɭ, ɫɨɯɪɚɧɹɹ ɫɜɨɸ ɜɟɥɢɱɢɧɭ

, ɬɨɝɞɚ ɤɚɤ ɞɥɹ

ɞɢɫɫɢɩɚɬɢɜɧɵɯ ɫɢɫɬɟɦ ɨɛɴɟɦ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɭɦɟɧɶɲɚɟɬɫɹ. ɗɬɨ ɪɚɡɥɢɱɢɟ

ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɧɨ ɧɚ ɪɢɫ. 3.10. ȼ ɞɢɫɫɢɩɚɬɢɜɧɵɯ ɫɢɫɬɟɦɚɯ ɬɪɚɟɤɬɨɪɢɢ ɩɪɢɬɹ-

ɝɢɜɚɸɬɫɹ ɤ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɟ, ɢ ɮɚɡɨɜɵɣ ɨɛɴɟɦ ɫɠɢɦɚɟɬɫɹ, ɚ ɜ ɤɨɧɫɟɪɜɚɬɢɜɧɵɯ —

ɬɨɱɤɢ ɨɛɪɚɳɚɸɬɫɹ ɜɨɤɪɭɝ ɷɥɥɢɩɬɢɱɟɫɤɨɣ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɢ, ɫɨɯɪɚɧɹɹ ɮɚɡɨɜɵɣ

ɨɛɴɟɦ.

Ɋɢɫ. 3.10. (ɚ) Ⱦɢɫɫɢɩɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ, (b) ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ.

Ȼɭɞɟɦ ɧɚɡɵɜɚɬɶ ɫɢɫɬɟɦɭ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

dx/dt = f(x,t), x ∈R

ɞɢɫɫɢɩɚɬɢɜɧɨɣ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɸɬ ɬɚɤɢɟ ɱɢɫɥɚ R > 0 ɢ t

> 0, ɱɬɨ

ɞɥɹ ɜɫɟɯ ɪɟɲɟɧɢɣ ɯ(.) ɫɢɫɬɟɦɵ ɢɡ ɭɫɥɨɜɢɹ ⏐x(0)⏐≤ R ɜɫɟɝɞɚ ɫɥɟɞɭɟɬ ⏐x(t)⏐ < R ɜ

ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ≥ t

. Ⱦɥɹ ɞɢɫɫɢɩɚɬɢɜɧɵɯ ɫɢɫɬɟɦ ɦɵ ɢɦɟɟɦ ɫɥɟɞɭɸɳɭɸ

ɜɚɠɧɭɸ ɬɟɨɪɟɦɭ:

ɉɪɚɜɢɥɶɧɟɟ ɝɨɜɨɪɢɬɶ ɨ ɫɨɯɪɚɧɟɧɢɢ ɷɥɟɦɟɧɬɚ ɦɚɫɫɵ ɩɪɢ ɩɨɞɯɨɞɹɳɢɦ ɨɛɪɚɡɨɦ ɡɚɞɚɧɧɨɣ ɩɥɨɬɧɨɫɬɢ,

ɤɨɬɨɪɭɸ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɨɣ ɞɥɹ ɥɢɧɟɣɧɵɯ ɫɢɫɬɟɦ. ȼ ɨɤɪɟɫɬɧɨɫɬɢ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ (ɤɚɤ ɧɚ

ɪɢɫ. 3.10(b)) ɬɚɤɚɹ ɩɥɨɬɧɨɫɬɶ ɛɥɢɡɤɚ ɤ ɩɨɫɬɨɹɧɧɨɣ. — ɉɪɢɦ. ɪɟɞ

ɍɬɜɟɪɠɞɟɧɢɟ. Ⱦɢɫɫɢɩɚɬɢɜɧɚɹ ɫɢɫɬɟɦɚ dx/dt = f(x,t), x ∈ R

ɢɦɟɟɬ ɩɟɪɢɨɞɢɱɟɫɤɨɟ

ɪɟɲɟɧɢɟ ɩɟɪɢɨɞɚ ɪ > 0, ɟɫɥɢ (I) ɮɭɧɤɰɢɹ f ɪ-ɩɟɪɢɨɞɢɱɧɚ ɩɨ t, ɢ (II) ɞɥɹ ɤɚɠɞɨɝɨ

ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ x

∈ R

ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ x(.), ɬɚɤɨɟ, ɱɬɨ

ɯ(0) = x

, x(t) ɨɩɪɟɞɟɥɟɧɨ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ∈ [0,) ɢ ɧɟɩɪɟɪɵɜɧɨ ɩɨ x

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɑɬɨɛɵ ɞɨɤɚɡɚɬɶ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ, ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɥɟɞɭɸɳɟɣ

ɬɟɨɪɟɦɨɣ ɨ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɟ, ɩɪɢɧɚɞɥɟɠɚɳɟɣ Ȼɪɚɭɷɪɭ.

ɉɭɫɬɶ Ⱥ : ɏ ĺ ɏ — ɤɨɦɩɚɤɬɧɵɣ ɨɩɟɪɚɬɨɪ ɧɚ R

(ɝɞɟ ɏ ɨɞɧɨɫɜɹɡɧɨ).

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɞɥɹ ɧɟɤɨɬɨɪɨɝɨ ɡɚɞɚɧɧɨɝɨ ɧɚɬɭɪɚɥɶɧɨɝɨ ɬ ɦɧɨɠɟɫɬɜɨ Ⱥ

(X)

ɨɝɪɚɧɢɱɟɧɨ. Ɍɨɝɞɚ Ⱥ ɢɦɟɟɬ ɧɟɩɨɞɜɢɠɧɭɸ ɬɨɱɤɭ.

ɉɨɫɬɪɨɢɦ ɨɩɟɪɚɬɨɪ ɫɞɜɢɝɚ Ⱥ : R

ĺ R

ɜɢɞɚ Ⱥɯ

= ɯ(ɪ). Ɂɞɟɫɶ ɯ(.) — ɪɟɲɟɧɢɟ

ɫɢɫɬɟɦɵ. Ɍɨɝɞɚ A

= ɯ(ɬɪ). Ɇɧɨɠɟɫɬɜɨ G ɨɩɪɟɞɟɥɢɦ ɤɚɤ G = {x ∈ R

: |x| < R}.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, A

∈G ɩɪɢ ɜɫɟɯ x

, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɡɚɦɵɤɚɧɢɸ G, ɢ ɞɨɫɬɚɬɨɱɧɨ

ɛɨɥɶɲɢɯ ɬ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, Ⱥ ɢɦɟɟɬ ɧɟɩɨɞɜɢɠɧɭɸ ɬɨɱɤɭ, ɤɨɬɨɪɨɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ

ɢɫɤɨɦɨɟ ɩɟɪɢɨɞɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ.

Ʉɚɤ ɩɪɚɜɢɥɨ, ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ ɨɛɥɚɞɚɸɬ ɩɟɪɢɨɞɢɱɟɫɤɢɦɢ ɪɟɲɟɧɢɹɦɢ ɢ

ɛɥɚɝɨɞɚɪɹ ɷɬɨɦɭ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɹɜɥɟɧɢɣ, ɩɨɞɨɛɧɵɯ

ɨɫɰɢɥɥɹɰɢɹɦ ɩɨɩɭɥɹɰɢɣ ɯɢɳɧɢɤɨɜ ɢ ɠɟɪɬɜ, ɜɡɚɢɦɨɫɜɹɡɢ ɝɨɪɨɞɫɤɨɣ ɡɟɦɟɥɶɧɨɣ

ɪɟɧɬɵ ɢ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɡɟɦɥɟɩɨɥɶɡɨɜɚɧɢɹ, ɛɟɡɪɚɛɨɬɢɰɵ ɢ ɞɢɧɚɦɢɤɢ

ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɨɫɬɚ, ɢ ɬ. ɞ.

ɑɬɨɛɵ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɫɢɫɬɟɦɚ (3.5.1) ɤɨɧɫɟɪɜɚɬɢɜɧɚ, ɩɪɨɜɟɞɟɦ ɫɥɟɞɭɸɳɟɟ

ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ:

ɉɪɢ ɷɬɨɦ ɫɢɫɬɟɦɚ (3.5.1) ɩɟɪɟɩɢɲɟɬɫɹ ɤɚɤ

Ʌɟɝɤɨ ɧɚɣɬɢ ɩɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɷɬɨɣ ɫɢɫɬɟɦɵ:

ɝɞɟ Ⱥ — ɤɨɧɫɬɚɧɬɚ. ɉɨɫɤɨɥɶɤɭ ɩɪɢ ɞɜɢɠɟɧɢɢ ɜɞɨɥɶ ɬɪɚɟɤɬɨɪɢɢ, ɩɪɟɞɫɬɚɜɥɹɸɳɟɣ

ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ, G(u, v) ɧɟ ɦɟɧɹɟɬɫɹ, ɷɬɢ ɬɪɚɟɤɬɨɪɢɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɪɚɜɧɟɧɢɟɦ

G(u,v) = Ⱥ ɢ ɡɧɚɱɟɧɢɟɦ ɤɨɧɫɬɚɧɬɵ Ⱥ. ɂɡ ɷɬɨɝɨ ɧɟɦɟɞɥɟɧɧɨ ɫɥɟɞɭɟɬ, ɱɬɨ ɨɫɨɛɚɹ ɬɨɱɤɚ

(1,1)

ɧɟ ɦɨɠɟɬ

ɛɵɬɶ ɭɫɬɨɣɱɢɜɵɦ ɮɨɤɭɫɨɦ. ȿɫɥɢ ɛɵ ɷɬɨ ɛɵɥɨ ɧɟ ɬɚɤ, ɬɨ ɜɫɟ ɤɪɢɜɵɟ ɜ ɟɟ ɨɤɪɟɫɬɧɨɫɬɢ

ɭɫɬɪɟɦɥɹɥɢɫɶ ɛɵ ɤ ɷɬɨɣ ɬɨɱɤɟ, ɢ, ɤɚɤ ɫɥɟɞɫɬɜɢɟ, ɢɦɟɥɨ ɛɵ ɦɟɫɬɨ G(u, v) = G(1, 1) ɜ

ɫɢɥɭ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɮɭɧɤɰɢɢ J. ɇɨ ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ (1,1)

ɮɭɧɤɰɢɹ G ɩɨɫɬɨɹɧɧɚ, ɱɬɨ ɩɪɨɬɢɜɨɪɟɱɢɬ ɟɟ ɨɩɪɟɞɟɥɟɧɢɸ. ɂɡ ɬɟɯ ɠɟ ɫɨɨɛɪɚɠɟɧɢɣ

ɫɥɟɞɭɟɬ, ɱɬɨ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɧɢ ɭɫɬɨɣɱɢɜɵɯ, ɧɢ ɧɟɭɫɬɨɣɱɢɜɵɯ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ ɫ

ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ (1,1). ȼɫɟ ɬɪɚɟɤɬɨɪɢɢ, ɛɟɪɭɳɢɟ ɧɚɱɚɥɨ ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɤɜɚɞɪɚɧɬɟ,

ɨɝɪɚɧɢɱɟɧɵ, ɬɚɤ ɱɬɨ ɨɫɬɚɟɬɫɹ ɟɞɢɧɫɬɜɟɧɧɚɹ ɜɨɡɦɨɠɧɨɫɬɶ — ɤɨɝɞɚ ɮɚɡɨɜɚɹ

ɩɥɨɫɤɨɫɬɶ ɩɨɤɪɵɬɚ ɡɚɦɤɧɭɬɵɦɢ ɬɪɚɟɤɬɨɪɢɹɦɢ ɫ ɰɟɧɬɪɨɦ ɜ ɨɫɨɛɨɣ ɬɨɱɤɟ, ɩɪɢɱɟɦ

ɪɚɡɧɵɦ ɬɪɚɟɤɬɨɪɢɹɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɚɡɥɢɱɧɵɟ ɡɧɚɱɟɧɢɹ ɷɧɟɪɝɢɢ G(u,v).

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɞɟɥɶ ɨɪɛɢɬɚɥɶɧɨ ɭɫɬɨɣɱɢɜɚ, ɯɨɬɹ ɢ ɧɟɭɫɬɨɣɱɢɜɚ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ.

ɉɪɢ ɜɫɹɤɨɣ ɩɨɩɵɬɤɟ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɦɵ ɜɵɧɭɠɞɟɧɵ ɩɪɟɧɟɛɪɟɝɚɬɶ ɤɚɤɢɦɢ-ɬɨ

ɷɮɮɟɤɬɚɦɢ. Ɉɞɧɚɤɨ ɧɭɠɧɨ ɛɵɬɶ ɭɜɟɪɟɧɧɵɦ, ɱɬɨ ɬɚɤɨɟ: ɩɪɟɧɟɛɪɟɠɟɧɢɟ ɧɟ ɨɤɚɠɟɬ

ɫɤɨɥɶɤɨ-ɧɢɛɭɞɶ ɫɟɪɶɟɡɧɨɝɨ ɜɥɢɹɧɢɹ ɧɚ ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ. Ɍɨ ɟɫɬɶ ɱɬɨɛɵ

ɦɨɞɟɥɶ ɯɨɪɨɲɨ ɨɬɪɚɠɚɥɚ ɩɪɨɛɥɟɦɭ, ɧɟɨɛɯɨɞɢɦɨ ɩɨɬɪɟɛɨɜɚɬɶ ɟɟ ɫɬɪɭɤɬɭɪɧɭɸ

ɭɫɬɨɣɱɢɜɨɫɬɶ

ɑɬɨɛɵ ɩɨɤɚɡɚɬɶ, ɤɚɤ ɦɚɥɵɟ ɜɨɡɦɭɳɟɧɢɹ ɜɥɢɹɸɬ ɧɚ ɩɨɜɟɞɟɧɢɟ ɫɢɫɬɟɦɵ (3.6.1),

ɞɚɜɚɣɬɟ ɩɪɢɛɚɜɢɦ ɤ ɩɟɪɜɨɦɭ ɭɪɚɜɧɟɧɢɸ ɱɥɟɧ -rɢ

. ɉɨɥɭɱɢɦ

ȿɫɥɢ ɩɚɪɚɦɟɬɪ r ɛɟɫɤɨɧɟɱɧɨ ɦɚɥ, ɟɫɬɟɫɬɜɟɧɧɨ ɬɪɟɛɨɜɚɬɶ, ɱɬɨɛɵ ɨɧ ɧɟ ɨɤɚɡɚɥ

ɡɧɚɱɢɬɟɥɶɧɨɝɨ ɜɥɢɹɧɢɹ ɧɚ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ. Ɉɞɧɚɤɨ ɚɧɚɥɢɡ ɫɨɛɫɬɜɟɧɧɵɯ

ɡɧɚɱɟɧɢɣ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɪɚɜɧɨɜɟɫɢɟ ɜ ɬɨɱɤɟ (1,1 - r)

ɹɜɥɹɟɬɫɹ ɭɫɬɨɣɱɢɜɵɦ ɮɨɤɭɫɨɦ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɬɨɣɱɢɜɵɦ ɮɨɤɭɫɨɦ

ɪɚɫɲɢɪɟɧɧɨɣ ɧɟɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ, ɤɚɤɢɦ ɛɵ ɦɚɥɵɦ ɧɢ ɛɵɥɨ r. Ɏɚɤɬɢɱɟɫɤɢ

ɮɭɧɤɰɢɹ V(u,v) = σ(u — logu) + v — (1 — r) log v ɹɜɥɹɟɬɫɹ ɝɥɨɛɚɥɶɧɨɣ ɮɭɧɤɰɢɟɣ

Ʌɹɩɭɧɨɜɚ ɫɢɫɬɟɦɵ (ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɤɜɚɞɪɚɧɬɟ). Ɋɟɲɟɧɢɟ ɧɨɜɨɣ ɫɢɫɬɟɦɵ ɩɨ

ɫɩɢɪɚɥɢ ɧɚɦɚɬɵɜɚɸɬɫɹ ɧɚ ɪɚɜɧɨɜɟɫɢɟ, ɢ ɬɚɤɚɹ ɫɢɫɬɟɦɚ ɛɨɥɶɲɟ ɧɟ ɦɨɠɟɬ

ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɨɫɰɢɥɥɹɰɢɢ ɝɨɪɨɞɫɤɨɣ ɞɢɧɚɦɢɤɢ.

Ɉɛɳɚɹ ɬɟɨɪɢɹ ɜɨɡɦɭɳɟɧɢɣ ɜ ɩɪɢɥɨɠɟɧɢɢ ɤ ɫɢɫɬɟɦɟ (3.6.1)

ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɩɟɪɜɵɯ ɢɧɬɟɝɪɚɥɨɜ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɭɫɦɨɬɪɟɧɨ ɜ ɢɫɯɨɞɧɨɣ ɩɨɫɬɚɧɨɜɤɟ (ɤɚɤ,

ɧɚɩɪɢɦɟɪ, ɜ ɡɚɞɚɱɚɯ ɧɟɛɟɫɧɨɣ ɦɟɯɚɧɢɤɢ). — ɉɪɢɦ. ɪɟɞ.

ɝɞɟ f

(i = 1,2) — ɮɭɧɤɰɢɢ ɜɨɡɦɭɳɟɧɢɹ, ɢɡɭɱɟɧɧɵɟ Ɏɪɢɞɦɚɧɨɦ ɢ ȼɚɥɶɬɦɚɧɨɦ (1975).

ɂɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ ɨ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ, ɨɧɢ ɞɨɤɚɡɚɥɢ, ɱɬɨ ɞɚɠɟ ɟɫɥɢ ɟ ɩɪɨɢɡɜɨɥɶɧɨ

ɦɚɥɨ, ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɮɭɧɤɰɢɣ f ɜɨɡɦɨɠɧɨ ɩɨɹɜɥɟɧɢɟ ɭɫɬɨɣɱɢɜɵɯ (ɢ ɧɟɭɫɬɨɣɱɢɜɵɯ)

ɰɢɤɥɨɜ, ɤɨɬɨɪɵɟ ɤɚɱɟɫɬɜɟɧɧɨ ɨɬɥɢɱɧɵ ɨɬ ɢɫɯɨɞɧɨɝɨ ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ

Ʉɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ ɯɨɪɨɲɢ ɬɟɦ, ɱɬɨ, ɤɚɤ ɩɪɚɜɢɥɨ, ɩɨɞɞɚɸɬɫɹ ɚɧɚɥɢɡɭ, ɧɨ ɜ

ɤɚɱɟɫɬɜɟ ɦɨɞɟɥɟɣ ɪɟɚɥɶɧɵɯ ɫɢɫɬɟɦ ɢɦɟɸɬ ɪɹɞ ɛɨɥɶɲɢɯ ɧɟɞɨɫɬɚɬɤɨɜ — ɩɨɫɤɨɥɶɤɭ

ɜɫɟ ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ ɫɬɪɭɤɬɭɪɧɨ ɧɟɭɫɬɨɣɱɢɜɵ (Ȼɪɢɬɬɨɧ, 1986), ɢɯ ɧɭɠɧɨ

ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɪɚɣɧɟ ɨɫɬɨɪɨɠɧɨ.

ȼ ɡɚɤɥɸɱɟɧɢɟ ɩɪɢɜɟɞɟɦ ɩɪɢɦɟɪ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɦɨɞɟɥɢ,

ɩɪɟɞɥɨɠɟɧɧɨɣ Ⱥɧɞɟɪɫɫɨɧɨɦ ɢ Ɂɚɧɝɨɦ (1988ɚ). Ɇɨɞɟɥɶ ɫɬɪɨɢɬɫɹ ɜ ɪɚɦɤɚɯ ɫɢɫɬɟɦɵ

Ʌɟɨɧɬɶɟɜɚ «ɡɚɬɪɚɬɵ-ɜɵɩɭɫɤ». ɗɤɨɧɨɦɢɤɚ ɫɨɫɬɨɢɬ ɢɡ n ɫɟɤɬɨɪɨɜ, ɧɟ ɢɦɟɸɳɢɯ

ɨɛɴɟɞɢɧɟɧɧɵɯ ɩɪɨɢɡɜɨɞɫɬɜ. ɋɤɨɪɨɫɬɶ ɪɨɫɬɚ ɷɤɨɧɨɦɢɤɢ ɪɟɝɭɥɢɪɭɟɬɫɹ

ɩɪɚɜɢɬɟɥɶɫɬɜɨɦ. Ɇɨɞɟɥɶ ɨɩɢɫɵɜɚɟɬ ɞɢɧɚɦɢɤɭ ɰɟɧ ɢ ɩɪɨɢɡɜɨɞɫɬɜɚ ɫɢɫɬɟɦɨɣ

ɭɪɚɜɧɟɧɢɣ

ɝɞɟ ɪ=(ɪ

,.. . ,p

)

- «ɧɨɪɦɢɪɨɜɚɧɧɵɣ» ɜɟɤɬɨɪ ɰɟɧ, x=(x

,..., x

) -«ɧɨɪɦɢɪɨɜɚɧɧɵɣ»

ɜɵɩɭɫɤ ɩɪɨɞɭɤɰɢɢ, A = (ɚ

) ɢ ȼ = (b

) — ɦɚɬɪɢɰɵ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ

ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɧɜɟɫɬɢɰɢɣ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɏɭɧɤɰɢɹ J(ɪ,ɯ) ɜ (3.6.3) ɨɩɢɫɵɜɚɟɬ

ɫɤɨɪɨɫɬɶ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɨɫɬɚ ɫɢɫɬɟɦɵ. ɋɤɨɪɨɫɬɶ ɪɨɫɬɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɚ-

ɜɢɬɟɥɶɫɬɜɨɦ, ɤɨɬɨɪɨɟ ɦɚɤɫɢɦɢɡɢɪɭɟɬ ɮɭɧɤɰɢɸ «ɫɨɰɢɚɥɶɧɨɣ» ɩɨɥɟɡɧɨɫɬɢ,

ɜɨɡɞɟɣɫɬɜɭɹ ɧɚ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɦɨɳɧɨɫɬɢ.

ɍɪɚɜɧɟɧɢɟ ɰɟɧ ɜ (3.6.3) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɟɫɥɢ ɫɩɪɨɫ ɧɚ i-ɵɣ ɬɨɜɚɪ ɩɪɟɜɵɲɚɟɬ

ɩɪɟɞɥɨɠɟɧɢɟ ɷɬɨɝɨ ɬɨɜɚɪɚ, ɬɨ ɰɟɧɚ i-ɨɝɨ ɬɨɜɚɪɚ ɞɨɥɠɧɚ ɜɨɡɪɚɫɬɢ, ɢ ɧɚɨɛɨɪɨɬ.

ɍɪɚɜɧɟɧɢɟ ɭɱɢɬɵɜɚɟɬ ɩɨɜɟɞɟɧɢɟ ɩɨɬɪɟɛɢɬɟɥɟɣ. ȼ ɤɨɥɢɱɟɫɬɜɟɧɧɨɦ ɜɵɪɚɠɟɧɢɢ ɷɬɨ

ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɨɦɭ, ɱɬɨ ɟɫɥɢ ɨɛɳɚɹ ɫɟɛɟɫɬɨɢɦɨɫɬɶ ɟɞɢɧɢɰɵ i-ɨɣ ɩɪɨɞɭɤɰɢɢ

ɩɪɟɜɵɲɚɟɬ ɰɟɧɭ ɩɪɨɞɚɠɢ ɷɬɨɝɨ ɬɨɜɚɪɚ, ɬɨ ɜ i-ɨɦ ɫɟɤɬɨɪɟ ɨɛɴɟɦ ɩɪɨɢɡɜɨɞɫɬɜɚ

ɞɨɥɠɟɧ ɛɵɬɶ ɭɦɟɧɶɲɟɧ, ɱɬɨɛɵ ɭɦɟɧɶɲɢɬɶ ɩɨɬɟɪɢ.

Ɍɟɨɪɟɦɚ 3.6.1. Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ (3.6.3) ɹɜɥɹɟɬɫɹ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ.

ȼ ɝɪɭɛɨɦ ɫɥɭɱɚɟ ɭɫɬɨɣɱɢɜɨɫɬɶ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɬɪɚɟɤɬɨɪɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɤɨɦ ɩɨɤɚɡɚɬɟɥɹ,

ɜɵɱɢɫɥɹɟɦɨɝɨ ɤɚɤ ɢɧɬɟɝɪɚɥ ɩɨ ɬɪɚɟɤɬɨɪɢɢ ɨɬ ɫɥɟɞɚ ɥɢɧɟɚɪɢɡɚɰɢɢ ɫɢɫɬɟɦɵ (ɫɦ. Ȼɚɭɬɢɧ ɇ. ɇ., Ʌɟɨɧɬɨɜɢɱ

ȿ. Ⱥ., Ɇɟɬɨɞɵ ɢ ɩɪɢɟɦɵ ɤɚɱɟɫɬɜɟɧɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɧɚ ɩɥɨɫɤɨɫɬɢ, Ɇ: ɇɚɭɤɚ,

1990, ɝɥ. 5). — ɉɪɢɦ. ɪɟɞ.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ʌɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɮɭɧɤɰɢɹ

ɢɝɪɚɟɬ ɪɨɥɶ ɩɟɪɜɨɝɨ ɢɧɬɟɝɪɚɥɚ ɫɢɫɬɟɦɵ.

Ɂɚɦɟɬɢɦ, ɱɬɨ Ⱥɧɞɟɪɫɫɨɧɨɦ ɢ Ɂɚɧɝɨɦ (1988ɚ) ɩɪɟɞɥɨɠɟɧɵ ɢ ɛɨɥɟɟ ɨɛɳɢɟ

ɞɢɧɚɦɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɭɫɬɚɧɨɜɥɟɧɢɹ, ɧɨ ɨɫɬɚɧɚɜɥɢɜɚɬɶɫɹ ɡɞɟɫɶ ɧɚ ɧɢɯ ɦɵ ɧɟ

ɛɭɞɟɦ.

ȼ ɡɚɤɥɸɱɟɧɢɟ ɨɬɦɟɬɢɦ ɬɚɤɠɟ, ɱɬɨ ɯɨɬɹ ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɫɬɟɦɵ ɧɟ ɨɛɥɚɞɚɸɬ

ɚɬɬɪɚɤɬɨɪɚɦɢ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬ.ɟ. ɧɟ ɢɦɟɸɬ ɧɢ ɩɪɢɬɹɝɢɜɚɸɳɢɯ

ɧɟɩɨɞɜɢɠɧɵɯ ɬɨɱɟɤ, ɧɢ ɩɪɢɬɹɝɢɜɚɸɳɢɯ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ, ɧɢ «ɫɬɪɚɧɧɵɯ»

ɚɬɬɪɚɤɬɨɪɨɜ, ɜɫɟ ɠɟ ɢ ɡɞɟɫɶ ɦɵ ɨɛɧɚɪɭɠɢɜɚɟɦ «ɯɚɨɫ» ɫ ɩɨɥɨɠɢɬɟɥɶɧɨɣ Ʉ-

ɷɧɬɪɨɩɢɟɣ, ɬ.ɟ. ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɩɪɢɫɭɬɫɬɜɭɸɬ «ɫɬɪɚɧɧɵɟ», «ɯɚɨɬɢɱɟɫɤɢɟ»

ɨɛɥɚɫɬɢ, ɧɨ ɨɧɢ ɧɟ ɹɜɥɹɸɬɫɹ ɨɛɥɚɫɬɹɦɢ ɩɪɢɬɹɠɟɧɢɹ ɢ ɦɨɝɭɬ ɬɟɫɧɨ ɩɟɪɟɩɥɟɬɚɬɶɫɹ ɫ

ɨɛɥɚɫɬɹɦɢ ɪɟɝɭɥɹɪɧɨɫɬɢ ɬɚɤɢɯ ɫɢɫɬɟɦ (ɫɦ. ɒɭɫɬɟɪ, 1988). Ɉɞɧɚɤɨ ɜ ɧɚɲɟɣ ɤɧɢɝɟ

ɦɵ ɧɟ ɛɭɞɟɦ ɡɚɬɪɚɝɢɜɚɬɶ ɷɬɢɯ ɜɟɫɶɦɚ ɫɥɨɠɧɵɯ ɩɪɨɛɥɟɦ.

3.7 Ɍɟɨɪɢɹ ɛɢɮɭɪɤɚɰɢɣ

Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɬɤɪɵɬɢɹ, ɦɚɥɵɟ ɢ ɛɨɥɶɲɢɟ, ...

ɧɢɤɨɝɞɚ ɧɟ ɪɨɠɞɚɸɬɫɹ ɫɩɨɧɬɚɧɧɨ. ɂɯ ɩɨɹɜɥɟɧɢɟ ɜɫɟɝɞɚ

ɩɪɟɞɩɨɥɚɝɚɟɬ, ɱɬɨ ɩɨɱɜɚ ɛɵɥɚ ɨɛɢɥɶɧɨ ɡɚɫɟɹɧɚ

ɫɟɦɟɧɚɦɢ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɯ ɡɧɚɧɢɣ ɢ ɯɨɪɨɲɨ

ɩɨɞɝɨɬɨɜɥɟɧɚ ɩɪɨɰɟɫɫɨɦ ɤɚɤ ɫɨɡɧɚɬɟɥɶɧɨɣ, ɬɚɤ ɢ

ɩɨɞɫɨɡɧɚɬɟɥɶɧɨɣ ɪɚɛɨɬɵ.

Ⱥɧɪɢ ɉɭɚɧɤɚɪɟ

Ɉɫɧɨɜɧɨɟ ɧɚɡɧɚɱɟɧɢɟ ɷɬɨɝɨ ɪɚɡɞɟɥɚ — ɜɜɟɞɟɧɢɟ ɧɟɤɨɬɨɪɵɯ ɩɨɧɹɬɢɣ ɬɟɨɪɢɢ

ɛɢɮɭɪɤɚɰɢɣ.

Ɋɚɡɥɢɱɚɸɬ ɞɜɚ ɚɫɩɟɤɬɚ ɬɟɨɪɢɢ ɛɢɮɭɪɤɚɰɢɣ: ɫɬɚɬɢɱɟɫɤɢɣ ɢ ɞɢɧɚɦɢɱɟɫɤɢɣ.

ɋɬɚɬɢɱɟɫɤɚɹ ɬɟɨɪɢɹ ɛɢɮɭɪɤɚɰɢɣ ɢɦɟɟɬ ɞɟɥɨ ɫ ɢɡɦɟɧɟɧɢɹɦɢ, ɜɨɡɧɢɤɚɸɳɢɦɢ ɜ

ɫɬɪɭɤɬɭɪɟ ɦɧɨɠɟɫɬɜɚ ɧɭɥɟɣ ɮɭɧɤɰɢɣ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɩɚɪɚɦɟɬɪɨɜ, ɜɯɨɞɹɳɢɯ ɜ ɷɬɢ

ɮɭɧɤɰɢɢ. ȼ ɫɥɭɱɚɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɧɵɟ ɪɟɲɟɧɢɹ ɹɜɥɹɸɬɫɹ

ɧɭɥɹɦɢ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɤ ɧɢɦ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɪɢɦɟɧɢɦɵ

ɦɟɬɨɞɵ ɫɬɚɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɛɢɮɭɪɤɚɰɢɣ. Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɬɟɨɪɢɹ ɛɢɮɭɪɤɚɰɢɣ ɢɡɭɱɚɟɬ

ɢɡɦɟɧɟɧɢɹ, ɤɨɬɨɪɵɟ ɜɨɡɧɢɤɚɸɬ ɜ ɫɬɪɭɤɬɭɪɟ ɪɟɲɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɩɚɪɚɦɟɬɪɨɜ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ.

ɂɡɦɟɧɟɧɢɟ ɤɚɱɟɫɬɜɟɧɧɵɯ ɫɜɨɣɫɬɜ ɦɨɠɟɬ ɨɡɧɚɱɚɬɶ ɢ ɢɡɦɟɧɟɧɢɟ ɫɜɨɣɫɬɜɚ

ɭɫɬɨɣɱɢɜɨɫɬɢ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɫɬɟɦɚ ɞɨɥɠɧɚ

ɨɛɥɚɞɚɬɶ ɟɳɟ ɤɚɤɢɦ-ɬɨ ɫɨɫɬɨɹɧɢɟɦ, ɨɬɥɢɱɧɵɦ ɨɬ ɢɫɯɨɞɧɨɝɨ. ɇɟ ɞɚɜɚɹ ɫɬɪɨɝɢɯ

ɨɩɪɟɞɟɥɟɧɢɣ, ɫɤɚɠɟɦ ɥɢɲɶ,