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

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ɩɨɬɨɦɭ ɱɬɨ ɩɨɫɥɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɢɫɬɟɦɭ ɜɫɟɝɞɚ ɦɨɠɧɨ

ɡɚɩɢɫɚɬɶ ɥɨɤɚɥɶɧɨ ɜ «ɫɬɚɧɞɚɪɬɧɨɣ» ɮɨɪɦɟ (9.1.5).

ȿɫɥɢ ɜ (9.1.5) ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ r

(i = 1,..., m) ɨɱɟɧɶ ɦɚɥɵ, ɚ r

(> 0, s = m + 1,...,n)

ɤɨɧɟɱɧɵ, ɬɨ ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɩɪɢɧɰɢɩ «ɚɞɢɚɛɚɬɢɱɟɫɤɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ», ɩɨɥɨɠɢɜ

/dt = 0. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɵ ɞɨɥɠɧɵ ɫɥɟɞɢɬɶ, ɱɬɨɛɵ °x

° ɛɵɥɢ ɦɧɨɝɨ ɦɟɧɶɲɟ, ɱɟɦ

°x

°. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ ɩɨɫɥɟɞɧɢɯ (n — m) ɭɪɚɜɧɟɧɢɣ ɜ (9.1.5) ɦɵ ɦɨɠɟɦ

ɨɩɪɟɞɟɥɢɬɶ x

(s = m + 1,..., n) ɤɚɤ ɮɭɧɤɰɢɢ x

. ɉɨɜɟɞɟɧɢɟ ɫɢɫɬɟɦɵ ɚɩɩɪɨɤɫɢɦɢɪɭɟɬɫɹ

ɭɪɚɜɧɟɧɢɹɦɢ

Ɋɟɲɟɧɢɟ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ ɨɩɪɟɞɟɥɹɟɬ, ɛɭɞɟɬ ɥɢ ɜɨɡɦɨɠɧɨ ɧɟɧɭɥɟɜɨɟ ɩɨɜɟɞɟɧɢɟ

ɫɢɫɬɟɦɵ. ɉɨ ɚɧɚɥɨɝɢɢ ɦɨɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ ɦɨɞɵ x

(i = 1, ..., m) ɢɝɪɚɸɬ ɪɨɥɶ

ɩɚɪɚɦɟɬɪɨɜ ɩɨɪɹɞɤɚ, ɨɬ ɤɨɬɨɪɵɯ ɡɚɜɢɫɹɬ ɞɪɭɝɢɟ ɦɨɞɵ.

Ɍɚɤ ɤɚɤ ɜɨ ɦɧɨɝɢɯ ɩɪɢɥɨɠɟɧɢɹɯ ɜɫɬɪɟɱɚɸɬɫɹ ɨɞɧɚ ɢɥɢ ɧɟɫɤɨɥɶɤɨ ɧɟɭɫɬɨɣɱɢɜɵɯ

ɦɨɞ, ɦɵ ɦɨɠɟɦ ɡɧɚɱɢɬɟɥɶɧɨ ɩɨɧɢɡɢɬɶ ɪɚɡɦɟɪɧɨɫɬɶ ɡɚɞɚɱɢ. ɉɨɫɤɨɥɶɤɭ ɜɫɟ

ɡɚɬɭɯɚɸɳɢɟ ɦɨɞɵ ɤɨɩɢɪɭɸɬ ɩɨɜɟɞɟɧɢɟ ɩɚɪɚɦɟɬɪɨɜ ɩɨɪɹɞɤɚ, ɩɨɜɟɞɟɧɢɟ ɜɫɟɣ

ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɜɟɞɟɧɢɟɦ ɨɱɟɧɶ ɦɚɥɨɝɨ ɱɢɫɥɚ ɩɟɪɟɦɟɧɧɵɯ (ɩɚɪɚɦɟɬɪɨɜ

ɩɨɪɹɞɤɚ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɚɠɟ ɨɱɟɧɶ ɫɥɨɠɧɵɟ ɫɢɫɬɟɦɵ ɦɨɝɭɬ ɛɵɬɶ ɫɜɟɞɟɧɵ ɤ ɨɱɟɧɶ

ɩɪɨɫɬɵɦ. Ȼɨɥɟɟ ɬɨɝɨ, ɟɫɥɢ ɜ ɭɪɚɜɧɟɧɢɹɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɚɪɚɦɟɬɪɨɜ ɩɨɪɹɞɤɚ

ɜɨɡɧɢɤɚɸɬ ɛɢɮɭɪɤɚɰɢɢ, ɫɥɨɠɧɨɫɬɶ ɛɢɮɭɪɤɚɰɢɨɧɧɨɝɨ ɚɧɚɥɢɡɚ ɡɧɚɱɢɬɟɥɶɧɨ

ɫɧɢɠɚɟɬɫɹ.

ɇɚ ɩɪɚɤɬɢɤɟ ɦɵ ɱɚɫɬɨ ɢɦɟɟɦ ɞɟɥɨ ɫ ɢɟɪɚɪɯɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɨɣ, ɜ ɤɨɬɨɪɨɣ

ɤɨɧɫɬɚɧɬɵ ɭɛɵɜɚɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɫɝɪɭɩɩɢɪɨɜɚɧɵ ɬɚɤ, ɱɬɨ

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɩɪɨɰɟɞɭɪɭ ɫɧɚɱɚɥɚ ɤ ɩɟɪɟɦɟɧɧɵɦ ɝɪɭɩɩɵ r

(1)

, ɨɫɬɚɜɥɹɹ ɜ

ɫɢɫɬɟɦɟ ɞɪɭɝɢɟ ɩɟɪɟɦɟɧɧɵɟ, ɡɚɬɟɦ ɤ ɩɟɪɟɦɟɧɧɵɦ ɝɪɭɩɩɵ r

(2)

, ɢ ɬ. ɞ.

ɇɚ ɨɫɧɨɜɚɧɢɢ ɜɵɲɟɫɤɚɡɚɧɧɨɝɨ ɜɢɞɧɨ, ɱɬɨ ɩɪɢɧɰɢɩ ɩɨɞɱɢɧɟɧɢɹ ɩɨɡɜɨɥɹɟɬ

ɡɧɚɱɢɬɟɥɶɧɨ ɭɦɟɧɶɲɢɬɶ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢɡɭɱɚɟɦɨɣ ɫɢɫɬɟɦɵ. Ʉɪɨɦɟ ɬɨɝɨ, ɜ

ɤɧɢɝɟ ɏɚɤɟɧɚ (1977, ɪɚɡɞ. 12.4) ɦɨɠɧɨ ɧɚɣɬɢ ɢɧɬɟɪɟɫɧɵɣ ɩɪɢɦɟɪ ɫɜɹɡɢ ɦɟɠɞɭ

ɯɚɨɫɨɦ ɢ ɩɪɢɧɰɢɩɨɦ ɩɨɞɱɢɧɟɧɢɹ. ȼɡɹɜ ɜ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɭɪɚɜɧɟɧɢɹ Ʌɨɪɟɧɰɚ,

ɏɚɤɟɧ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɯɚɨɬɢɱɟɫɤɨɟ ɞɜɢɠɟɧɢɟ ɦɨɠɟɬ ɜɨɡɧɢɤɧɭɬɶ, ɤɨɝɞɚ ɩɪɢɧɰɢɩ

ɩɨɞɱɢɧɟɧɢɹ ɩɟɪɟɫɬɚɟɬ ɪɚɛɨɬɚɬɶ, ɢ ɭɫɬɨɣɱɢɜɚɹ ɩɪɟɠɞɟ ɦɨɞɚ, ɤɨɬɨɪɚɹ ɧɟ ɨɫɬɚɟɬɫɹ

ɞɨɥɟɟ ɩɨɞɱɢɧɟɧɧɨɣ, ɞɟɫɬɚɛɢɥɢɡɢɪɭɟɬɫɹ. ɉɪɢɧɰɢɩ ɩɨɞɱɢɧɟɧɢɹ ɩɪɢɦɟɧɢɦ ɬɚɤɠɟ ɤ

ɫɬɨɯɚɫɬɢɱɟɫɤɢɦ ɢ ɞɢɫɤɪɟɬɧɵɦ ɫɢɫɬɟɦɚɦ. ȼ ɩɪɢɥɨɠɟɧɢɢ ɤ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɩɪɢɜɟɞɟɦ

ɩɪɢɦɟɪ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɩɪɢɧɰɢɩɚ ɤ ɫɬɨɯɚɫɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɹɦ.

ɋ ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɩɪɢɧɰɢɩ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɵ ɦɨɠɟɦ ɧɚɣɬɢ

ɧɟɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ (ɫɨɜɨɤɭɩɧɵɯ ɢɥɢ ɩɪɟɨɛɪɚɡɨɜɚɧɧɵɯ)

ɩɟɪɟɦɟɧɧɵɯ, ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬ ɞɢɧɚɦɢɤɭ ɜɫɟɣ ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ

ɨɤɪɟɫɬɧɨɫɬɢ ɨɫɨɛɨɣ ɬɨɱɤɢ, ɬ. ɟ. ɨɫɬɚɥɶɧɵɟ ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɩɟɪɟɦɟɧɧɵɟ ɛɭɞɭɬ

ɡɚɜɢɫɢɦɵ ɨɬ ɧɢɯ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɫɧɨɜɧɨɟ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ

ɚɧɚɥɢɡɟ — ɷɬɨ ɨɩɪɚɜɞɚɧɢɟ ɡɚɤɨɧɧɨɫɬɢ ɪɟɞɭɤɰɢɢ ɡɚɞɚɱ, ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɯ ɜ

ɮɚɡɨɜɵɯ ɩɪɨɫɬɪɚɧɫɬɜɚɯ ɜɵɫɨɤɨɣ ɢ ɛɟɫɤɨɧɟɱɧɨɣ ɪɚɡɦɟɪɧɨɫɬɢ, ɤ ɡɚɞɚɱɚɦ ɪɚɡɦɟɪɧɨɫɬɢ

ɨɞɢɧ ɢɥɢ ɞɜɚ. Ⱥɧɚɥɢɡ ɬɚɤɢɯ ɧɢɡɤɨɪɚɡɦɟɪɧɵɯ ɡɚɞɚɱ ɱɚɫɬɨ ɫɜɹɡɚɧ ɫ ɛɢɮɭɪɤɚɰɢɹɦɢ ɞɥɹ

ɩɪɨɫɬɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ, ɢ ɦɟɬɨɞ ɛɢɮɭɪɤɚɰɢɣ, ɩɪɢɦɟɧɹɜɲɢɣɫɹ ɜ

ɩɪɟɞɵɞɭɳɢɯ ɝɥɚɜɚɯ, ɫɬɚɧɨɜɢɬɫɹ ɩɪɢɥɨɠɢɦ ɤ ɡɚɞɚɱɚɦ ɜɵɫɨɤɨɣ ɪɚɡɦɟɪɧɨɫɬɢ.

ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɢɦɟɸɬɫɹ ɢ ɧɟɤɨɬɨɪɵɟ ɞɪɭɝɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ

ɩɨɧɢɠɟɧɢɹ ɪɚɡɦɟɪɧɨɫɬɢ ɡɚɞɚɱ. ɇɚɩɪɢɦɟɪ, ɦɟɬɨɞ Ʌɹɩɭɧɨɜɚ-ɒɦɢɞɬɚ ɩɪɢɦɟɧɹɟɬɫɹ

ɞɥɹ ɞɟɤɨɦɩɨɡɢɰɢɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɪɟɲɟɧɢɣ ɢ ɭɪɚɜɧɟɧɢɣ ɧɚ ɤɨɧɟɱɧɨɦɟɪɧɭɸ ɢ

ɛɟɫɤɨɧɟɱɧɨɦɟɪɧɭɸ ɱɚɫɬɢ. ɉɪɢ ɷɬɨɦ ɭɪɚɜɧɟɧɢɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ

ɛɟɫɤɨɧɟɱɧɨɦɟɪɧɨɣ ɱɚɫɬɢ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɪɟɲɟɧɨ, ɚ ɨɫɬɚɜɲɚɹɫɹ ɤɨɧɟɱɧɨɦɟɪɧɚɹ

ɡɚɞɚɱɚ ɫɨɯɪɚɧɹɟɬ ɜɫɸ ɢɧɮɨɪɦɚɰɢɸ ɨ ɛɢɮɭɪɤɚɰɢɹɯ (ɫɦ. ɑɭ ɢ ɏɟɣɥ, 1982)

. Ƀɨɫɫ ɢ

Ⱦɠɨɡɟɮ (1980) ɩɪɢɦɟɧɢɥɢ ɬɟɨɪɟɦɭ ɨ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ ɞɥɹ ɨɛɨɫɧɨɜɚɧɢɹ ɩɪɹɦɨɝɨ

ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɜɵɱɢɫɥɟɧɢɹ ɪɟɲɟɧɢɣ ɜ ɜɢɞɟ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ ɪɚɡɥɨɠɟɧɢɹ ɩɨ

ɚɦɩɥɢɬɭɞɚɦ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɚɥɶɬɟɪɧɚɬɢɜɵ Ɏɪɟɞɝɨɥɶɦɚ. ɗɬɨɬ ɦɟɬɨɞ ɩɪɟɞɫɬɚɜɥɹɟɬ

ɫɨɛɨɣ ɷɤɨɧɨɦɢɱɧɵɣ ɫɩɨɫɨɛ ɨɩɪɟɞɟɥɟɧɢɹ ɤɚɱɟɫɬɜɟɧɧɵɯ ɫɜɨɣɫɬɜ ɛɢɮɭɪɤɚɰɢɨɧɧɵɯ

ɪɟɲɟɧɢɣ ɢ ɢɯ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɝɨ ɜɵɱɢɫɥɟɧɢɹ. Ɇɵ ɩɨɥɶɡɨɜɚɥɢɫɶ ɛɢɮɭɪɤɚɰɢɨɧɧɵɦ

ɦɟɬɨɞɨɦ Ƀɨɫɫɚ ɢ Ⱦɠɨɡɟɮɚ ɜ ɝɥ. 5. ɑɬɨɛɵ ɩɨɧɢɡɢɬɶ ɪɚɡɦɟɪɧɨɫɬɶ ɡɚɞɚɱɢ, ɦɵ ɦɨɠɟɦ

ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɬɚɤɠɟ ɬɟɨɪɟɦɨɣ ɨ ɰɟɧɬɪɚɥɶɧɨɦ ɦɧɨɝɨɨɛɪɚɡɢɢ (ɫɦ. ɧɢɠɟ). ɗɬɨɬ

ɦɟɬɨɞ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɬɨɦ ɮɚɤɬɟ, ɱɬɨ ɪɟɲɟɧɢɹ ɩɪɢɬɹɝɢɜɚɸɬɫɹ ɤ ɰɟɧɬɪɚɥɶɧɨɦɭ

ɦɧɨɝɨɨɛɪɚɡɢɸ, ɤɨɬɨɪɨɟ ɫɭɬɶ ɤɨɧɟɱɧɨɦɟɪɧɨ. ɋɥɟɞɭɟɬ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɜɵɛɨɪ ɦɟɬɨɞɚ

ɡɚɜɢɫɢɬ ɨɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɞɚɧɧɨɣ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɢ.

9.2 Ɍɟɨɪɟɦɚ ɨ ɰɟɧɬɪɚɥɶɧɨɦ

ɦɧɨɝɨɨɛɪɚɡɢɢ

ɉɪɢɧɰɢɩ ɩɨɞɱɢɧɟɧɢɹ ɏɚɤɟɧɚ ɞɚɟɬ ɷɮɮɟɤɬɢɜɧɵɣ ɦɟɬɨɞ ɩɨɧɢɠɟɧɢɹ ɪɚɡɦɟɪɧɨɫɬɢ

ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ. Ɂɞɟɫɶ ɦɵ ɨɡɧɚɤɨɦɢɦɫɹ; ɤɚɤ ɞɥɹ ɷɬɨɣ ɰɟɥɢ ɦɨɠɟɬ ɛɵɬɶ

ɢɫɩɨɥɶɡɨɜɚɧɚ ɬɟɨɪɟɦɚ ɨ ɰɟɧɬɪɚɥɶɧɨɦ ɦɧɨɝɨɨɛɪɚɡɢɢ. Ⱦɚɧɧɚɹ ɬɟɨɪɟɦɚ ɨɤɚɡɵɜɚɟɬɫɹ

ɩɨɥɟɡɧɨɣ ɜ ɰɟɥɨɦ ɪɹɞɟ ɩɪɢɤɥɚɞɧɵɯ ɡɚɞɚɱ.

Ʉɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜ ɝɥ. 3, ɥɢɧɟɣɧɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ

ɋɦ. ɬɚɤɠɟ ȼɚɣɧɛɟɪɝ Ɇ. Ɇ., Ɍɪɟɧɨɝɢɣ ȼ. Ⱥ. Ɍɟɨɪɢɹ ɜɟɬɜɥɟɧɢɣ

ɪɟɲɟɧɢɣ ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ, Ɇ.: ɇɚɭɤɚ, 1969, 528 ɫ., ɇɢɪɟɧɛɟɪɝ Ʌ.,

Ʌɟɤɰɢɢ ɩɨ ɧɟɥɢɧɟɣɧɨɦɭ ɮɭɧɤɰɢɨɧɚɥɶɧɨɦɭ ɚɧɚɥɢɡɭ, Ɇ.: Ɇɢɪ, 1977, 232

ɫ. — ɉɪɢɦ. ɪɟɞ.

ɥɨɤɚɥɶɧɨ ɨɩɪɟɞɟɥɹɟɬ ɤɚɱɟɫɬɜɟɧɧɨɟ ɩɨɜɟɞɟɧɢɟ ɬɪɚɟɤɬɨɪɢɣ ɫɢɫɬɟɦɵ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ

ɜɫɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɚɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɜ ɬɨɱɤɟ x = 0 ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ.

Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɟɫɥɢ ɥɢɧɟɣɧɚɹ ɱɚɫɬɶ ɧɟ ɢɦɟɟɬ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ, ɥɟɠɚɳɢɯ ɧɚ

ɦɧɢɦɨɣ ɨɫɢ, ɬɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɚ ɜɟɞɟɬ ɫɟɛɹ ɬɚɤ ɠɟ, ɤɚɤ

ɥɢɧɟɣɧɚɹ. ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɧɚɜɨɞɢɬ ɧɚ ɦɵɫɥɶ, ɱɬɨ ɥɸɛɵɟ ɫɭɳɟɫɬɜɟɧɧɨ ɧɟɥɢɧɟɣɧɵɟ

ɹɜɥɟɧɢɹ ɜ ɫɢɫɬɟɦɟ (ɫɥɨɠɧɵɟ ɪɚɜɧɨɜɟɫɢɹ, ɩɪɟɞɟɥɶɧɵɟ ɰɢɤɥɵ, ɷɮɮɟɤɬɵ ɝɢɫɬɟɪɟɡɢɫɚ)

ɜ ɛɨɥɶɲɨɣ ɫɬɟɩɟɧɢ ɫɜɹɡɚɧɵ ɫ ɱɢɫɬɨ ɦɧɢɦɵɦɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ. ɗɬɨ

ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨɩɪɚɜɞɚɧɨ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɜ ɬɨɦ, ɱɬɨ ɤɚɫɚɟɬɫɹ ɭɫɥɨɜɢɣ ɭɫɬɨɣɱɢ-

ɜɨɫɬɢ.

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

ɝɞɟ x ∈ R

, ɭ ∈ R

, ɚ Ⱥ ɢ ȼ — ɩɨɫɬɨɹɧɧɵɟ ɤɜɚɞɪɚɬɧɵɟ ɦɚɬɪɢɰɵ. Ɇɵ ɩɪɟɞɩɨɥɚɝɚɟɦ,

ɱɬɨ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɚɬɪɢɰɵ Ⱥ ɜɫɟɝɞɚ ɢɦɟɸɬ ɧɭɥɟɜɭɸ ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɱɚɫɬɶ,

ɚ ɦɚɬɪɢɰɵ ȼ — ɨɬɪɢɰɚɬɟɥɶɧɭɸ, ɮɭɧɤɰɢɢ f ɢ J ɩɪɢɧɚɞɥɟɠɚɬ ɤɥɚɫɫɭ ɋ

, k > 0 ɢ ɜ

ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ 0 ɨɛɪɚɳɚɸɬɫɹ ɜ ɧɭɥɶ ɜɦɟɫɬɟ ɫɨ ɫɜɨɢɦɢ ɩɟɪɜɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ.

Ɉɩɪɟɞɟɥɟɧɢɟ 9.2.1. (ɂɧɜɚɪɢɚɧɬɧɨɟ ɦɧɨɝɨɨɛɪɚɡɢɟ.) Ɇɧɨɠɟɫɬɜɨ S ∈ R

n+m

ɹɜɥɹɟɬɫɹ

ɢɧɜɚɪɢɚɧɬɧɵɦ ɦɧɨɝɨɨɛɪɚɡɢɟɦ ɞɥɹ ɫɢɫɬɟɦɵ (9.2.1), ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ ɪɟɲɟɧɢɹ

(x(t), y(t)), ɧɚɱɚɥɶɧɵɟ ɞɚɧɧɵɟ (x(0), y(0)) ɤɨɬɨɪɨɝɨ ɩɪɢɧɚɞɥɟɠɚɬ S, ɫɭɳɟɫɬɜɭɟɬ

ɧɟɤɨɬɨɪɨɟ Ɍ > 0, ɬɚɤɨɟ, ɱɬɨ x(t), y(t)) ∈ S ɞɥɹ ɜɫɟɯ t ∈ [0, ȉ].

Ɉɩɪɟɞɟɥɟɧɢɟ 9.2.2. (ɐɟɧɬɪɚɥɶɧɨɟ ɦɧɨɝɨɨɛɪɚɡɢɟ.) ɂɧɜɚɪɢɚɧɬɧɨɟ ɦɧɨɝɨɨɛɪɚɡɢɟ

S = {(ɯ, h(x))°°x° < δ} ɞɥɹ ɫɢɫɬɟɦɵ (9.2.1) ɹɜɥɹɟɬɫɹ ɰɟɧɬɪɚɥɶɧɵɦ, ɟɫɥɢ h(0) = 0,

Dh(0) = 0, ɝɞɟ D — ɨɩɟɪɚɬɨɪ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ.

Ɍɟɨɪɟɦɚ 9.2.1. Ⱦɥɹ ɫɢɫɬɟɦɵ (9.2.1) ɫɭɳɟɫɬɜɭɟɬ ɰɟɧɬɪɚɥɶɧɨɟ ɦɧɨɝɨɨɛɪɚɡɢɟ ɤɥɚɫɫɚ

. ɉɨɬɨɤ ɧɚ ɷɬɨɦ ɦɧɨɝɨɨɛɪɚɡɢɢ ɡɚɞɚɟɬɫɹ ɫɢɫɬɟɦɨɣ ɪɚɡɦɟɪɧɨɫɬɢ n

Ɍɟɨɪɟɦɚ 9.2.2.

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (9.2.2) ɭɫɬɨɣɱɢɜɨ. Ɍɨɝɞɚ, ɟɫɥɢ

(x(t), y(t)) ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ (9.2.1)

ɫ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɦɢ ɧɚɱɚɥɶɧɵɦɢ (x(0), ɭ(0)), ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ

(9.2.2), ɬɚɤɨɟ, ɱɬɨ ɞɥɹ t > 0

ɝɞɟ r

ɢ r

ɩɪɢ t ĺ෱ ɫɨ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ. ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ

ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ (9.2.2) ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɭɫɬɨɣɱɢɜɨ, ɬɨ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ

ɭɫɬɨɣɱɢɜɵɦ ɛɭɞɟɬ ɢ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ (9.2.1).

ii)

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (9.2.2) ɧɟɭɫɬɨɣɱɢɜɨ. Ɍɨɝɞɚ ɛɭɞɟɬ

ɧɟɭɫɬɨɣɱɢɜɵɦ ɢ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (9.2.1).

ȿɫɥɢ ɦɵ ɩɨɞɫɬɚɜɢɦ y(t) = h(x(t)) ɜɨ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ (9.2.1) ɢ ɡɚɬɟɦ, ɢɫɩɨɥɶɡɭɹ

ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ, ɢɫɤɥɸɱɢɦ dx/dt, ɬɨ ɩɨɥɭɱɢɦ

ɗɬɨ ɭɪɚɜɧɟɧɢɟ, ɫɨɜɦɟɫɬɧɨ ɫ h(0) = 0 ɢ Dh(0) = 0, ɨɛɪɚɡɭɟɬ ɫɢɫɬɟɦɭ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ

ɰɟɧɬɪɚɥɶɧɨɝɨ ɦɧɨɝɨɨɛɪɚɡɢɹ. ɏɨɬɹ ɦɵ ɦɨɠɟɦ ɢ ɧɟ ɫɭɦɟɬɶ ɬɨɱɧɨ ɧɚɣɬɢ h ɢɡ ɷɬɢɯ

ɭɪɚɜɧɟɧɢɣ, ɦɵ ɦɨɠɟɦ ɧɚɣɬɢ ɟɝɨ ɩɪɢɛɥɢɠɟɧɧɨ. ɉɨɥɨɠɢɦ

ɞɥɹ ɮɭɧɤɰɢɣ j : R

ĺ R

ɢɡ ɤɥɚɫɫɚ ɋ

ɜ ɨɤɪɟɫɬɧɨɫɬɢ 0 ∈ R

. ɂɡ (9.2.3) ɫɥɟɞɭɟɬ

N(h) = 0.

Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ ɩɪɢɦɟɧɟɧɢɹ ɷɬɨɣ ɬɟɨɪɟɦɵ ɪɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

ɋɦ. ɬɚɤɠɟ Ɇɚɪcɞɟɧ, Ɇɚɤ-Ʉɪɚɤɟɧ (1984),Ƚɥ. 2; Ⱥɪɧɨɥɶɞ (1978),

(

ɫɫɵɥɤɚ ɜ ɪɚɡɞ. 3.2).—ɉɪɢɦ. ɪɟɞ.

ɝɞɟ Ⱥ, ȼ, f ɢ J ɨɛɨɡɧɚɱɚɸɬ ɬɨ ɠɟ, ɱɬɨ ɢ ɜ (9.2.1). ɉɭɫɬɶ (x, ɭ) = 0 — ɨɫɨɛɚɹ ɬɨɱɤɚ ɩɪɢ

ɦɚɥɵɯ r. ɋɬɨɢɬ ɭɩɨɦɹɧɭɬɶ, ɱɬɨ ɧɚɲɢ ɪɚɫɫɭɠɞɟɧɢɹ ɩɪɢɦɟɧɢɦɵ ɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ

dz/dt = f(z,r), ɝɞɟ z — ɩɟɪɟɦɟɧɧɚɹ ɪɚɡɦɟɪɧɨɫɬɢ (m+n), ɚ r— ɩɚɪɚɦɟɬɪ ɪɚɡɦɟɪɧɨɫɬɢ q.

Ɇɵ ɦɨɠɟɦ ɩɨɬɪɟɛɨɜɚɬɶ, ɱɬɨɛɵ z = 0 ɛɵɥɚ ɛɵ ɨɫɨɛɨɣ ɬɨɱɤɨɣ ɞɥɹ ɦɚɥɵɯ r, ɬɚɤɨɣ, ɱɬɨ

ɨɩɟɪɚɬɨɪ D

f(0,0) ɢɦɟɥ ɛɵ n ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ, ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɚɫɬɢ ɤɨɬɨɪɵɯ

ɪɚɜɧɵ ɧɭɥɸ, ɢ ɬ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɫ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɞɟɣɫɬɜɢɬɟɥɶɧɵɦɢ

ɱɚɫɬɹɦɢ. ɗɬɨɬ ɨɛɳɢɣ ɫɥɭɱɚɣ ɜɫɟɝɞɚ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɤ (9.2.5). Ⱦɥɹ ɦɚɥɵɯ ɜɨɡɦɭ-

ɳɟɧɢɣ r (ɨɬ ɧɭɥɹ) ɬɚɤɢɟ ɫɢɫɬɟɦɵ «ɛɥɢɡɤɢ ɤ ɤɪɢɬɢɱɟɫɤɢɦ». ɉɪɢɡɧɚɧɨ, ɱɬɨ ɜɚɠɧɵɦ

ɲɚɝɨɦ ɜ ɚɧɚɥɢɡɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɝɨ ɩɨɜɟɞɟɧɢɹ ɦɚɥɵɯ ɪɟɲɟɧɢɣ ɜɛɥɢɡɢ ɨɫɨɛɨɣ ɬɨɱɤɢ

ɹɜɥɹɟɬɫɹ ɪɟɞɭɤɰɢɹ ɫɢɫɬɟɦɵ ɪɚɡɦɟɪɧɨɫɬɢ (ɩ + ɬ) ɤ ɫɢɫɬɟɦɟ ɪɚɡɦɟɪɧɨɫɬɢ n ɩɭɬɟɦ

ɢɫɤɥɸɱɟɧɢɹ ɱɚɫɬɢ ɩɟɪɟɦɟɧɧɵɯ, ɩɪɟɞɩɨɥɨɠɢɬɟɥɶɧɨ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɭɛɵɜɚɸɳɢɯ ɫɨ

ɜɪɟɦɟɧɟɦ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɬ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɫ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ

ɞɟɣɫɬɜɢɬɟɥɶɧɵɦɢ ɱɚɫɬɹɦɢ. Ɇɟɬɨɞɵ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ, ɨɫɧɨɜɚɧɧɵɟ ɧɚ ɜɵɛɨɪɟ

ɲɤɚɥɵ ɜɪɟɦɟɧɢ, ɨɛɟɫɩɟɱɢɜɚɸɬ ɨɞɢɧ ɢɡ ɫɩɨɫɨɛɨɜ ɪɟɞɭɤɰɢɢ ɬɚɤɨɝɨ ɪɨɞɚ. ɋɟɣɱɚɫ ɦɵ

ɩɨɤɚɠɟɦ, ɱɬɨ ɬɟɨɪɟɦɚ ɨ ɰɟɧɬɪɚɥɶɧɨɦ ɦɧɨɝɨɨɛɪɚɡɢɢ ɞɚɟɬ ɬɨɱɧɵɣ ɢ ɭɧɢɜɟɪɫɚɥɶɧɵɣ

ɦɟɬɨɞ ɞɥɹ ɪɟɞɭɤɰɢɢ ɫɢɫɬɟɦɵ, «ɛɥɢɡɤɨɣ ɤ ɤɪɢɬɢɱɟɫɤɨɣ».

ɉɨ ɬɟɨɪɟɦɟ 9.2.1 ɫɢɫɬɟɦɚ (9.2.5) ɨɛɥɚɞɚɟɬ ɰɟɧɬɪɚɥɶɧɵɦ ɦɧɨɝɨɨɛɪɚɡɢɟɦ ɭ = h(x, r),

°x° < r

, |r| < r

ɤɥɚɫɫɚ ɋ

. ɋɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ 9.2.2, ɩɨɜɟɞɟɧɢɟ ɦɚɥɵɯ ɪɟɲɟɧɢɣ

ɫɢɫɬɟɦɵ (9.2.5) ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɟɞɭɰɢɪɨɜɚɧɧɨɣ ɫɢɫɬɟɦɨɣ

ɇɚɤɨɧɟɰ, ɢɡ (9.2.4) ɫɥɟɞɭɟɬ, ɱɬɨ ɫɢɫɬɟɦɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ h ɢɦɟɟɬ ɜɢɞ

ɢ ɦɵ ɦɨɠɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɟɨɪɟɦɭ 9.2.3 ɞɥɹ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɪɟɲɟɧɢɹ.

ɉɵɬɚɹɫɶ ɪɟɲɢɬɶ (9.2.6), ɦɵ ɦɨɠɟɦ ɨɩɭɫɬɢɬɶ ɭɪɚɜɧɟɧɢɟ dr/dt = 0 ɢ ɫɧɨɜɚ ɫɱɢɬɚɬɶ

r ɩɪɨɫɬɨ ɩɚɪɚɦɟɬɪɨɦ.·Ɉɞɧɚɤɨ, ɩɪɢɦɟɧɹɹ ɬɟɨɪɟɦɭ 9.2.2, ɦɵ ɞɨɥɠɧɵ ɩɪɨɜɟɪɹɬɶ

ɭɫɬɨɣɱɢɜɨɫɬɶ ɧɭɥɟɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɵ (9.2.6), ɩɨɬɨɦɭ ɱɬɨ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ

ɭɪɚɜɧɟɧɢɟ dr/dt = 0 ɦɨɠɟɬ ɨɤɚɡɵɜɚɬɶ ɧɚ ɭɫɬɨɣɱɢɜɨɫɬɶ ɫɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ.

ɇɚɩɪɢɦɟɪ ɪɟɲɟɧɢɟ u = 0 ɭɪɚɜɧɟɧɢɹ

ɧɟɭɫɬɨɣɱɢɜɨ ɞɥɹ ɥɸɛɨɝɨ r, ɧɟ ɪɚɜɧɨɝɨ ɧɭɥɸ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɪɟɲɟɧɢɟ (ɢ, r) = (0,0)

ɫɢɫɬɟɦɵ

ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɭɫɬɨɣɱɢɜɨ. ɉɪɢɦɟɪɵ ɩɪɢɦɟɧɟɧɢɹ ɬɟɨɪɟɦɵ ɨ ɰɟɧɬɪɚɥɶɧɨɦ

ɦɧɨɝɨɨɛɪɚɡɢɢ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɪɚɛɨɬɚɯ Ʉɚɪɪɚ (1981), Kappa ɢ Ɇɚɧɤɚɫɬɟɪɚ (1983) ɢ

Ʉɚɫɬɢ (1985).

9.3 ɋɢɧɝɭɥɹɪɧɵɟ ɜɨɡɦɭɳɟɧɢɹ

Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɫɨɞɟɪɠɢɬ ɨɩɪɟɞɟɥɟɧɧɨɟ ɱɢɫɥɨ ɩɚɪɚɦɟɬɪɨɜ, ɜɨɡɧɢɤɚɸɳɢɯ ɜ

ɭɪɚɜɧɟɧɢɹɯ ɤɚɤ ɹɜɧɨ, ɬɚɤ ɢ ɧɟɹɜɧɨ. ɇɟɤɨɬɨɪɵɟ ɢɡ ɬɚɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɦɨɝɭɬ ɛɵɬɶ

ɦɚɥɵɦɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜ ɭɪɚɜɧɟɧɢɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɬɟɦɢ ɱɥɟɧɚɦɢ, ɤɨɬɨɪɵɟ ɢɯ

ɫɨɞɟɪɠɚɬ, ɢ ɩɨɥɭɱɢɬɶ ɯɨɪɨɲɭɸ ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ. Ɋɹɞ ɩɪɢɦɟɪɨɜ

ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɢɧɨɝɞɚ ɢ ɷɬɨɬ ɩɨɞɯɨɞ ɝɨɞɢɬɫɹ. Ⱦɚɠɟ ɤɨɝɞɚ ɦɵ ɧɟ ɦɨɠɟɦ ɧɟ

ɭɱɢɬɵɜɚɬɶ ɱɥɟɧɵ ɫ ɦɚɥɵɦ ɩɚɪɚɦɟɬɪɨɦ, ɨɤɚɡɵɜɚɟɬɫɹ ɜɨɡɦɨɠɧɵɦ ɩɨɥɭɱɢɬɶ

ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ, ɢɫɩɨɥɶɡɭɹ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɩɚɪɚɦɟɬɪ ɦɚɥ. ɗɬɨɬ ɪɚɡɞɟɥ

ɩɨɫɜɹɳɟɧ ɧɟɤɨɬɨɪɵɦ (ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ) ɦɟɬɨɞɚɦ ɩɨɫɬɪɨɟɧɢɹ ɬɚɤɢɯ ɩɪɢɛɥɢɠɟɧɢɣ.

ɉɪɢɥɨɠɟɧɢɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɬɟɫɧɨ ɫɜɹɡɚɧɵ ɫ ɜɨɡɦɭɳɟɧɢɹɦɢ ɢɥɢ

ɦɚɥɵɦɢ ɢɡɦɟɧɟɧɢɹɦɢ ɭɫɥɨɜɢɣ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɡɚɞɚɱ. ɗɬɨ ɦɨɠɟɬ ɛɵɬɶ ɞɨɛɚɜɥɟɧɢɟ

ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɱɥɟɧɨɜ ɜ ɭɪɚɜɧɟɧɢɹɯ ɢɥɢ ɢɡɦɟɧɟɧɢɟ ɨɞɧɨɝɨ ɢɡ ɩɚɪɚɦɟɬɪɨɜ ɡɚɞɚɱɢ.

ɉɭɫɬɶ İ ɛɭɞɟɬ ɦɟɪɨɣ ɜɟɥɢɱɢɧɵ ɜɨɡɦɭɳɟɧɢɹ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɦ, ɛɭɞɟɬ ɪɟɲɟɧɢɟ

ɜɨɡɦɭɳɟɧɧɨɣ ɡɚɞɚɱɢ ɛɥɢɡɤɨ ɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦɭ ɪɟɲɟɧɢɸ ɧɟɜɨɡɦɭɳɟɧɧɨɣ ɢɥɢ ɧɟɬ,

ɨɩɪɟɞɟɥɹɸɬ ɩɨɧɹɬɢɹ ɪɟɝɭɥɹɪɧɨɝɨ ɢ ɫɢɧɝɭɥɹɪɧɨɝɨ ɜɨɡɦɭɳɟɧɢɣ.

Ɉɩɪɟɞɟɥɟɧɢɟ 9.3.1 ɉɭɫɬɶ ɯ (İ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɜɨɡɦɭɳɟɧɧɨɣ ɡɚɞɚɱɟ ȇ (İ) (ɝɞɟ ȇ (İ)

ɛɭɞɟɬ ɨɛɵɱɧɨ ɫɨɫɬɨɹɬɶ ɢɡ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ

ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɨɣ ɨɛɥɚɫɬɢ Q ɢ ɧɟɤɨɬɨɪɵɯ ɧɚɱɚɥɶɧɵɯ ɢ ɝɪɚɧɢɱɧɵɯ

ɭɫɥɨɜɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɯ). Ɍɨɝɞɚ ɜɨɡɦɭɳɟɧɢɟ ɪɟɝɭɥɹɪɧɨ, ɟɫɥɢ ɯ — ɧɟɩɪɟɪɵɜɧɚɹ

ɮɭɧɤɰɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ İ ɜ ɬɨɱɤɟ İ = 0, ɬ. ɟ. ɟɫɥɢ ||u(İ) — u(0)|| ĺ 0 ɩɪɢ İ ĺ0, ɝɞɟ u(İ)

— ɷɬɨ ɪɟɲɟɧɢɟ, ɚ || · || — ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɧɨɪɦɚ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɜɨɡɦɭɳɟɧɢɟ

ɫɢɧɝɭɥɹɪɧɨ.

Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɨɡɦɭɳɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɪɟɝɭɥɹɪɧɨ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɞɧɨɣ ɧɨɪɦɟ ɢ

ɫɢɧɝɭɥɹɪɧɨ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɞɪɭɝɨɣ. Ɋɟɲɟɧɢɹ ɪɟɝɭɥɹɪɧɨ ɜɨɡɦɭɳɟɧɧɨɣ ɡɚɞɚɱɢ

ɡɚɱɚɫɬɭɸ ɩɪɟɞɫɬɚɜɢɦɨ ɤɚɤ ɫɬɟɩɟɧɧɨɣ

ɪɹɞ ɩɨ İ. Ⱥɫɢɦɩɬɨɬɢɱɟɫɤɨɟ ɪɚɡɥɨɠɟɧɢɟ x(İ) ɩɪɢ İ ĺ0 ɦɨɠɧɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ

ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ {İ

ɗɬɨɬ ɪɚɡɞɟɥ ɩɨɫɜɹɳɟɧ ɪɚɫɫɦɨɬɪɟɧɢɸ ɬɨɥɶɤɨ ɫɢɧɝɭɥɹɪɧɵɯ ɜɨɡɦɭɳɟɧɢɣ. Ɍɚɤɢɟ

ɡɚɞɚɱɢ ɜɨɡɧɢɤɚɸɬ, ɤɨɝɞɚ ɨɞɧɚ ɢɡ ɩɟɪɟɦɟɧɧɵɯ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɝɨɪɚɡɞɨ ɛɵɫɬɪɟɟ

ɨɫɬɚɥɶɧɵɯ. ȼ ɷɬɨɦ ɫɦɵɫɥɟ ɚɧɚɥɢɡ ɜɨɡɦɭɳɟɧɢɣ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɟɫɶɦɚ

ɫɭɳɟɫɬɜɟɧɧɵɦ ɞɥɹ ɷɤɨɧɨɦɢɤɢ, ɬɚɤ ɤɚɤ ɫɤɨɪɨɫɬɢ ɭɫɬɚɧɨɜɥɟɧɢɹ ɷɤɨɧɨɦɢɱɟɫɤɢɯ

ɩɟɪɟɦɟɧɧɵɯ ɱɚɫɬɨ ɫɢɥɶɧɨ ɪɚɡɥɢɱɚɸɬɫɹ.

Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ ɩɪɨɛɥɟɦɵ ɪɚɫɫɦɨɬɪɢɦ ɨɞɧɨɫɟɤɬɨɪɧɭɸ ɦɨɞɟɥɶ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ

ɪɨɫɬɚ

ɝɞɟ p — ɡɚɪɚɛɨɬɧɚɹ ɩɥɚɬɚ, k — ɜɟɥɢɱɢɧɚ ɤɚɩɢɬɚɥɚ ɧɚ ɞɭɲɭ ɧɚɫɟɥɟɧɢɹ, ɚ f ɢ J —

ɜɟɥɢɱɢɧɵ ɩɨɪɹɞɤɚ O(1), ɬ. ɟ. ɨɧɢ ɧɟ ɦɚɥɵ, ɧɨ ɢ ɧɟ ɜɟɥɢɤɢ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨ ɜɡɹɬɵɯ

ɡɧɚɱɟɧɢɣ p ɢ k, İ — ɦɚɥɵɣ ɩɚɪɚɦɟɬɪ. ɍɫɬɚɧɨɜɥɟɧɢɟ p ɩɪɨɢɫɯɨɞɢɬ ɡɧɚɱɢɬɟɥɶɧɨ

ɛɵɫɬɪɟɟ, ɱɟɦ k. ɂɦɟɧɧɨ ɷɬɨ ɢ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɜ ɧɟɨɤɥɚɫɫɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɪɨɫɬɚ, ɝɞɟ,

ɤɚɤ ɬɨɥɶɤɨ ɡɚɞɚɟɬɫɹ ɤɚɩɢɬɚɥ ɢ (ɩɨɥɧɚɹ) ɡɚɧɹɬɨɫɬɶ, ɜɫɟɝɞɚ ɨɩɪɟɞɟɥɟɧɵ ɡɚɪɚɛɨɬɧɵɟ

ɩɥɚɬɵ ɢ ɰɟɧɵ.

Ʌɟɝɤɨ ɡɚɦɟɬɢɬɶ ɫɢɧɝɭɥɹɪɧɭɸ ɩɪɢɪɨɞɭ ɩɨɞɨɛɧɨɣ ɡɚɞɚɱɢ. ɂɡ ɬɟɨɪɟɦɵ ɉɢɤɚɪɚ ɧɚɦ

ɢɡɜɟɫɬɧɨ: ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɪɟɲɟɧɢɟ ɜ ɨɤɪɟɫɬɧɨɫɬɢ t = 0, ɟɫɥɢ f ɢ J ɭɞɨɜɥɟɬɜɨɪɹɸɬ

ɭɫɥɨɜɢɸ Ʌɢɩɲɢɰɚ ɩɨ ɪ ɢ k. Ɉɞɧɚɤɨ ɟɫɥɢ ɧɚɱɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ

ɭɫɥɨɜɢɸ f(p

, k

) = 0, ɧɟɜɨɡɦɭɳɟɧɧɚɹ ɫɢɫɬɟɦɚ (İ = 0), ɨɱɟɜɢɞɧɨ, ɧɟ ɢɦɟɟɬ ɪɟɲɟɧɢɹ.

Ȼɨɥɟɟ ɬɨɝɨ, ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɢ ɩɨɞɯɨɞɹɳɢɯ ɭɫɥɨɜɢɹɯ ɛɵɫɬɪɨɣ ɩɟɪɟɦɟɧɧɨɣ

ɜɫɟɝɞɚ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɤɨɪɨɬɤɨɝɨ ɧɚɱɚɥɶɧɨɝɨ ɩɟɪɢɨɞɚ ɜɪɟɦɟɧɢ.

ɋɥɟɞɭɟɬ ɫɤɚɡɚɬɶ, ɱɬɨ ɜ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɟ ɩɨɞ ɨɞɧɢɦ ɧɚɡɜɚɧɢɟɦ ɦɟɬɨɞɚ

ɜɨɡɦɭɳɟɧɢɣ ɢɡɜɟɫɬɧɨ ɧɟɫɤɨɥɶɤɨ ɦɟɬɨɞɨɜ (ɫɦ., ɧɚɩɪɢɦɟɪ, Ʉɟɜɨɪɤɹɧ ɢ Ʉɨɭɥ, 1981,

Ȼɪɢɬɬɨɧ, 1986). ɉɪɢɜɟɞɟɦ ɩɪɢɦɟɪ ɩɪɢɦɟɧɟɧɢɹ ɨɞɧɨɝɨ ɢɡ ɧɢɯ — ɦɟɬɨɞɚ ɭɫɪɟɞɧɟɧɢɹ

—

ɤ ɭɪɚɜɧɟɧɢɸ ȼɚɧ ɞɟɪ ɉɨɥɹ. ɉɪɢɦɟɪ ɜɡɹɬ ɢɡ ɪɚɛɨɬɵ Ȼɪɢɬɬɨɧɚ (1986).

Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ ȼɚɧ ɞɟɪ ɉɨɥɹ

ɝɞɟ İ — ɦɚɥɵɣ ɩɚɪɚɦɟɬɪ, ɚ ɯ — ɫɤɚɥɹɪɧɚɹ ɩɟɪɟɦɟɧɧɚɹ. ɗɬɚ ɦɨɞɟɥɶ ɱɚɫɬɨ

ɷɤɫɩɥɭɚɬɢɪɭɟɬɫɹ ɜ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɫɢɧɝɭɥɹɪɧɨɝɨ ɜɨɡɦɭɳɟɧɢɹ, ɬɚɤ ɤɚɤ ɷɬɨ ɨɞɧɨ ɢɡ

ɩɪɨɫɬɟɣɲɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɨɛɥɚɞɚɸɳɟɟ

ɬɢɩɢɱɧɵɦ ɩɨɜɟɞɟɧɢɟɦ — ɨɫɰɢɥɥɹɰɢɹɦɢ ɢ, ɱɬɨ ɨɫɨɛɟɧɧɨ ɢɧɬɟɪɟɫɧɨ ɞɥɹ ɧɚɫ,

ɩɪɟɞɟɥɶɧɵɦɢ ɰɢɤɥɚɦɢ.

Ʉɨɝɞɚ İ = 0, ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɪɟɲɟɧɢɟ

ɬɚɤ ɱɬɨ

ɝɞɟ ɚ — ɤɨɦɩɥɟɤɫɧɚɹ ɤɨɧɫɬɚɧɬɚ. Ɋɟɡɭɥɶɬɚɬɨɦ ɩɪɢɫɭɬɫɬɜɢɹ ɧɟɥɢɧɟɣɧɨɝɨ ɱɥɟɧɚ ɜ

ɭɪɚɜɧɟɧɢɢ ȼɚɧ ɞɟɪ ɉɨɥɹ ɛɭɞɟɬ ɢɡɦɟɧɟɧɢɟ ɚ ɫɨ ɜɪɟɦɟɧɟɦ. Ɇɵ ɨɩɪɟɞɟɥɢɦ ɡɚɜɢɫɹɳɭɸ

ɨɬ ɜɪɟɦɟɧɢ ɮɭɧɤɰɢɸ a (t) ɱɟɪɟɡ ɮɭɧɤɰɢɸ x(t) ɩɪɢ ɩɨɦɨɳɢ ɭɪɚɜɧɟɧɢɣ

ɝɞɟ ɯ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ ȼɚɧ ɞɟɪ ɉɨɥɹ ɢ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ. ɉɨɬɪɟɛɭɟɦ,

ɱɬɨɛɵ ɩɪɚɜɚɹ ɱɚɫɬɶ (9.3.5) ɛɵɥɚ ɩɪɨɢɡɜɨɞɧɨɣ ɩɪɚɜɨɣ ɱɚɫɬɢ (9.3.4). ɗɬɨ ɬɨ ɠɟ ɫɚɦɨɟ,

ɱɬɨ ɩɨɬɪɟɛɨɜɚɬɶ

Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ (9.3.5), ɩɨɥɭɱɢɦ

Ɍɪɟɛɨɜɚɧɢɟ ɤ ɯ ɭɞɨɜɥɟɬɜɨɪɢɬɶ (9.3.2) ɩɪɢɜɨɞɢɬ ɤ

Ɂɞɟɫɶ ɦɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɫɨɨɬɧɨɲɟɧɢɟ (9.3.6). ɗɬɨ ɹɜɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ

ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɮɭɧɤɰɢɢ ɚ, ɨɩɪɟɞɟɥɟɧɧɨɣ ɜ (9.3.3). Ɉɬɫɸɞɚ ɜɢɞɧɨ, ɱɬɨ ɚ

ɦɟɞɥɟɧɧɨ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ, ɬ. ɟ. da/dt ɢɦɟɟɬ ɩɨɪɹɞɨɤ ɦɚɥɨɫɬɢ Ɉ(İ).

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɚ ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ ɨɫɬɚɸɬɫɹ ɩɨɫɬɨɹɧɧɵɦɢ

ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɡɚ ɩɟɪɢɨɞ ʌ. ȼɡɹɜ ɫɪɟɞɧɟɟ ɨɬ (9.3.7) ɡɚ ɷɬɨɬ ɩɟɪɢɨɞ,

ɦɵ ɩɨɥɭɱɢɦ

ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ ɚ ɢ da/dt ɨɫɬɚɸɬɫɹ ɩɪɢɦɟɪɧɨ ɩɨɫɬɨɹɧɧɵɦɢ ɜ ɬɟɱɟɧɢɟ

ɩɟɪɢɨɞɚ, ɩɪɢɜɨɞɢɬ ɤ ɫɨɨɬɧɨɲɟɧɢɸ

ɤɨɬɨɪɨɟ ɜɵɩɨɥɧɹɟɬɫɹ ɩɪɢɛɥɢɠɟɧɧɨ. ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɤɨɪɪɟɤɬɟɧ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ

ɱɥɟɧɨɜ ɩɨɪɹɞɤɚ Ɉ(İ). ɉɨɥɚɝɚɹ ɭ = |ɚ|

, ɛɭɞɟɦ ɢɦɟɬɶ

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɭ = 1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɟɲɟɧɢɸ, ɹɜɥɹɸɳɟɦɭɫɹ ɩɪɟɞɟɥɶɧɵɦ ɰɢɤɥɨɦ.

ȿɫɥɢ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɭ(0) ɧɟ ɪɚɜɧɨ 1, ɦɵ ɦɨɠɟɦ ɩɪɟɞɫɬɚɜɢɬɶ ɭ ɜ ɮɨɪɦɟ

ȼɢɞɧɨ, ɱɬɨ ɩɪɢ t ĺ෱ ɢɦɟɟɦ ɭ ĺ 1. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɩɪɟɞɟɥɶɧɵɣ ɰɢɤɥ ɭɫɬɨɣɱɢɜ.

ɇɚ ɷɬɨɦ ɩɪɢɦɟɪɟ ɫɬɚɧɨɜɢɬɫɹ ɩɨɧɹɬɧɨ, ɱɬɨ ɞɥɹ ɚɧɚɥɢɡɚ ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

ɦɟɬɨɞ ɭɫɪɟɞɧɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɨɱɟɧɶ ɷɮɮɟɤɬɢɜɟɧ. ɗɬɨɬ ɦɟɬɨɞ ɬɚɤɠɟ ɩɪɢɦɟɧɢɦ ɞɥɹ

ɫɢɫɬɟɦɵ ɫ ɨɫɰɢɥɥɢɪɭɸɳɟɣ ɪɟɚɤɰɢɟɣ

ɝɞɟ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɭɫɥɨɜɢɹɯ ɜɨɡɧɢɤɚɟɬ ɛɢɮɭɪɤɚɰɢɹ ɏɨɩɮɚ. ɍɪɚɜɧɟɧɢɟ ȼɚɧ ɞɟɪ

ɉɨɥɹ — ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɬɚɤɨɣ ɫɢɫɬɟɦɵ. Ȼɨɥɟɟ ɬɨɝɨ, ɧɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ ɷɬɨɣ

ɮɨɪɦɟ ɦɨɝɭɬ ɛɵɬɶ ɡɚɩɢɫɚɧɵ ɢ ɧɟɤɨɬɨɪɵɟ ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɦɨɞɟɥɢ, ɩɪɢɜɟɞɟɧɧɵɟ ɜ ɝɥ.

5. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɢɦɟɟɦ ɟɳɟ ɨɞɢɧ ɦɟɬɨɞ ɚɧɚɥɢɡɚ ɪɟɝɭɥɹɪɧɵɯ ɷɤɨɧɨɦɢɱɟɫɤɢɯ

ɨɫɰɢɥɥɹɰɢɢ.

Ⱦɪɭɝɢɦ ɤɥɚɫɫɨɦ ɦɟɬɨɞɨɜ ɜɨɡɦɭɳɟɧɢɹ, ɩɨɥɟɡɧɵɯ ɩɪɢ ɚɧɚɥɢɡɟ ɨɫɰɢɥɥɹɬɨɪɧɵɯ

ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢ ɭɪɚɜɧɟɧɢɣ ɪɟɚɤɰɢɢ-ɞɢɮɮɭɡɢɢ,

ɹɜɥɹɟɬɫɹ ɦɟɬɨɞ ɞɜɭɯ ɜɪɟɦɟɧɧɵɯ ɦɚɫɲɬɚɛɨɜ. Ɉɧ ɫɯɨɠ ɫ ɦɟɬɨɞɨɦ ɭɫɪɟɞɧɟɧɢɹ ɜ ɬɨɦ

ɨɬɧɨɲɟɧɢɢ, ɱɬɨ ɚɦɩɥɢɬɭɞɚ ɢ ɮɚɡɚ ɩɟɪɟɦɟɧɧɨɣ ɫɱɢɬɚɸɬɫɹ ɦɟɞɥɟɧɧɨ ɦɟɧɹɸɳɢɦɢɫɹ

ɮɭɧɤɰɢɹɦɢ ɜɪɟɦɟɧɢ, ɧɨ ɩɨɞɯɨɞ, ɪɚɡɜɢɜɚɟɦɵɣ ɜ ɧɟɦ, ɫɨɜɟɪɲɟɧɧɨ ɢɧɨɣ.

Ⱦɥɹ ɨɛɴɹɫɧɟɧɢɹ ɫɭɬɢ ɷɬɨɝɨ ɦɟɬɨɞɚ ɪɚɫɫɦɨɬɪɢɦ ɩɪɨɫɬɭɸ ɞɢɧɚɦɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ

ɝɞɟ İ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨ. Ⱦɥɹ ɡɧɚɱɟɧɢɹ İ = 0 ɫɭɳɟɫɬɜɭɟɬ ɨɫɰɢɥɥɢɪɭɸɳɟɟ ɪɟɲɟɧɢɟ.

ȼɫɩɨɦɧɢɜ ɬɟɨɪɟɦɭ ɏɨɩɮɚ ɨ ɛɢɮɭɪɤɚɰɢɹɯ, ɦɵ

ɩɨɧɢɦɚɟɦ, ɱɬɨ ɩɪɢ ɦɚɥɵɯ İ ɫɢɫɬɟɦɚ ɦɨɠɟɬ ɢɦɟɬɶ ɡɚɦɤɧɭɬɵɟ ɨɪɛɢɬɵ. Ɂɚɞɚɱɚ ɫɨɫɬɨɢɬ

ɜ ɬɨɦ, ɱɬɨɛɵ ɢɫɫɥɟɞɨɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɫɢɫɬɟɦɵ, ɢɫɩɨɥɶɡɭɹ ɦɟɬɨɞɵ ɜɨɡɦɭɳɟɧɢɣ, ɚ ɧɟ

ɬɟɨɪɢɸ ɛɢɮɭɪɤɚɰɢɣ. ɉɟɪɟɩɢɲɟɦ ɫɧɚɱɚɥɚ (9.3.10) ɜ ɜɢɞɟ

Ɋɟɝɭɥɹɪɧɨɟ» ɪɚɡɥɨɠɟɧɢɟ ɜɨɡɦɭɳɟɧɢɹ ɜ ɮɨɪɦɟ

ɞɚɟɬ

ɤɨɬɨɪɨɟ ɧɚ ɩɟɪɜɵɣ ɜɡɝɥɹɞ ɤɚɠɟɬɫɹ ɧɟɨɝɪɚɧɢɱɟɧɧɵɦ ɩɪɢ t ĺ෱ . Ɉɞɧɚɤɨ ɬɨɱɧɨɟ

ɪɟɲɟɧɢɟ ɡɚɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ

ɝɞɟ J(t, İ) = t(1 — İ

)

1/2

. ȼ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɦɚɥɨɫɬɢ İ t ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɫɨɝɥɚɫɭɟɬɫɹ ɫ

ɪɟɝɭɥɹɪɧɵɦ ɪɚɡɥɨɠɟɧɢɟɦ, ɱɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɜɪɟɦɹ ɧɟ ɞɨɥɠɧɨ ɛɵɬɶ

ɨɱɟɧɶ ɛɨɥɶɲɢɦ. Ɉɬɦɟɬɢɦ, ɱɬɨ x ɹɜɥɹɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɜ ɦɟɞɥɟɧɧɨɦ

ɦɚɫɲɬɚɛɟ J(t, İ), ɧɨ ɦɨɞɭɥɢɪɨɜɚɧɚ ɧɚ İ t.

ɗɬɨ ɧɚɜɨɞɢɬ ɧɚ ɦɵɫɥɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɞɜɟ ɲɤɚɥɵ ɜɪɟɦɟɧɢ —

ɛɵɫɬɪɭɸ ɢ ɦɟɞɥɟɧɧɭɸ. Ɉɩɪɟɞɟɥɢɦ

Ⱦɥɹ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɜɵɲɟ ɩɪɢɦɟɪɚ T

ɜɜɟɞɟɧɨ ɞɥɹ ɩɟɪɟɦɟɧɧɨɣ İ t, ɚ Ɍ

— ɞɥɹ J (t, İ).

Ⱦɚɥɟɟ ɩɨɥɨɠɢɦ x(t; İ) = ɯ·(Ɍ

, T

;İ) ɢ ɛɭɞɟɦ ɢɫɤɚɬɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɟ ɪɚɡɥɨɠɟɧɢɟ

ɪɟɲɟɧɢɹ ɜ ɮɨɪɦɟ

ɗɬɨɬ ɦɟɬɨɞ ɢɡɜɟɫɬɟɧ ɤɚɤ ɦɟɬɨɞ ɞɜɭɯ ɜɪɟɦɟɧɧɵɯ ɦɚɫɲɬɚɛɨɜ ɢɥɢ ɦɟɬɨɞ ɞɜɭɯ ɜɪɟɦɟɧ.

ȼ ɪɚɛɨɬɟ Ȼɪɢɬɬɨɧɚ (1986) ɷɬɨɬ ɦɟɬɨɞ ɩɪɢɦɟɧɹɥɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ȼɚɧ ɞɟɪ ɉɨɥɹ ɢ

ɩɪɢɜɟɥ ɤ ɬɟɦ ɠɟ ɪɟɡɭɥɶɬɚɬɚɦ, ɱɬɨ ɢ ɦɟɬɨɞ ɭɫɪɟɞɧɟɧɢɹ.

9.4 ɋɜɹɡɶ ɛɵɫɬɪɵɯ ɢ ɦɟɞɥɟɧɧɵɯ

ɩɟɪɟɦɟɧɧɵɯ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ

ɚɧɚɥɢɡɟ

Ɉɞɧɢɦ ɢɡ ɩɪɟɞɦɟɬɨɜ ɫɟɪɶɟɡɧɨɝɨ ɨɛɫɭɠɞɟɧɢɹ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɞɢɧɚɦɢɤɟ ɹɜɥɹɟɬɫɹ

ɜɨɩɪɨɫ ɨ ɫɤɨɪɨɫɬɹɯ ɭɫɬɚɧɨɜɥɟɧɢɹ, ɤɨɬɨɪɵɟ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɜɪɟɦɟɧɚ ɞɨɫɬɢɠɟɧɢɹ

ɩɟɪɟɦɟɧɧɵɦɢ ɪɚɜɧɨɜɟɫɢɹ. Ɏɚɤɬɢɱɟɫɤɢ ɩɪɢɦɟɧɟɧɢɟ ɷɬɨɝɨ ɩɨɧɹɬɢɹ ɫɜɹɡɚɧɨ ɫɨ

ɲɤɚɥɨɣ ɜɪɟɦɟɧɢ, ɤɨɬɨɪɭɸ ɦɵ ɞɨɥɠɧɵ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɢɡɭɱɟɧɢɹ ɹɜɥɟɧɢɹ.