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Исследование устойчивости неподвижных точек систем с трансверсальным односторонним ограничением.

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4/2011
ВЕСТНИК
МГСУ
ɂɋɋɅȿȾɈȼȺɇɂȿ ɍɋɌɈɃɑɂȼɈɋɌɂ ɇȿɉɈȾȼɂɀɇɕɏ ɌɈɑȿɄ
ɋɂɋɌȿɆ ɋ ɌɊȺɇɋȼȿɊɋȺɅɖɇɕɆ ɈȾɇɈɋɌɈɊɈɇɇɂɆ
ɈȽɊȺɇɂɑȿɇɂȿɆ
FIXED POINTS STABILITY INVESTIGATION FOR SYSTEMS
WITH TRANSVERSAL ONE-SIDED RESTRICTIONS
ȿ.ɂ. Ɉɬɫɬɚɜɧɨɜ
E.I. Otstavnov
ȽɈɍ ȼɉɈ ɆȽɋɍ
ȼɜɨɞɢɬɫɹ ɩɨɧɹɬɢɟ ɫɢɫɬɟɦɵ ɈȾɍ ɫ ɬɪɚɧɫɜɟɪɫɚɥɶɧɵɦ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ. ɍɤɚɡɵɜɚɸɬɫɹ ɢɯ ɫɬɚɰɢɨɧɚɪɧɵɟ ɪɟɲɟɧɢɹ ɢ ɫɩɨɫɨɛ ɢɫɫɥɟɞɨɜɚɧɢɹ ɢɯ ɭɫɬɨɣɱɢɜɨɫɬɢ. ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɞɚɺɬɫɹ ɨɛɨɛɳɟɧɢɟ ɨɞɧɨɣ ɡɚɞɚɱɢ Ƚ.Ʉ. ɋɭɫɥɨɜɚ.
ODE systems with transversal one-sided restrictions are introduced. Fixed points and
their stability conditions are investigated. A generalization of one G.K. Suslov’s problem is
provided as an example.
ɉɪɟɞɜɚɪɢɬɟɥɶɧɵɟ ɫɨɨɛɪɚɠɟɧɢɹ. Ɉɩɪɟɞɟɥɟɧɢɟ ɫɢɫɬɟɦɵ ɈȾɍ ɫ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ.
ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɚɤɬɢɜɧɨ ɢɫɫɥɟɞɭɟɬɫɹ ɤɥɚɫɫ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ ɨɞɧɨɫɬɨɪɨɧɧɢɦɢ ɫɜɹɡɹɦɢ, ɜ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɧɢɯ ɫɬɪɨɹɬɫɹ ɦɟɬɨɞɢɤɢ ɢɫɫɥɟɞɨɜɚɧɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ.
Ʉɧɢɝɚ [1] ɪɚɫɤɪɵɜɚɟɬ ɜɨɩɪɨɫɵ ɨɛ ɢɦɩɭɥɶɫɢɜɧɨɦ ɞɜɢɠɟɧɢɢ, ɜɨɡɧɢɤɚɸɳɟɦ ɩɪɢ ɧɚɥɢɱɢɢ
ɧɟɭɞɟɪɠɢɜɚɸɳɢɯ ɫɜɹɡɟɣ, ɢ ɩɪɢɜɨɞɢɬ ɚɜɬɨɪɫɤɢɟ ɨɛɨɛɳɟɧɢɹ ɦɟɬɨɞɚ ɮɭɧɤɰɢɣ Ʌɹɩɭɧɨɜɚ
ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɨɥɨɠɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɢ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɞɜɢɠɟɧɢɣ ɜ
ɬɚɤɢɯ ɫɢɫɬɟɦɚɯ. ȼ [8] ɩɨɫɬɪɨɟɧɚ ɮɨɪɦɚɥɶɧɚɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɫɜɹɡɟɣ ɢ ɭɤɚɡɚɧɵ ɯɚɪɚɤɬɟɪɧɵɟ ɦɨɦɟɧɬɵ ɜ ɩɨɜɟɞɟɧɢɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ.
1.
ɋɢɫɬɟɦɨɣ ɈȾɍ ɫ ɬɪɚɧɫɜɟɪɫɚɥɶɧɵɦ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ (ɢɥɢ ɩɪɨɫɬɨ
ɫɢɫɬɟɦɨɣ) ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɬɪɨɣɤɭ ሺܺǡ ‫ݎ‬ǡ ܴሻ, ɫɨɫɬɨɹɳɭɸ ɢɡ:
1) ɋɢɫɬɟɦɵ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ‫ݔ‬ሶ ൌ ܺሺ‫ݔ‬ሻǡ ‫ܴ א ݔ‬௡ , ɧɚɡɵɜɚɟɦɨɣ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɨɣ.
2) ɋɤɚɥɹɪɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ ‫ݎ‬ሺ‫ݔ‬ሻ ൒ Ͳ ɧɚɡɵɜɚɟɦɨɝɨ ɫɜɹɡɶɸ,
3) ȼɟɤɬɨɪ-ɮɭɧɤɰɢɢ ܴሺ‫ݔ‬ሻ, ɧɚɡɵɜɚɟɦɨɣ ɪɟɚɤɰɢɟɣ ɫɜɹɡɢ, ɩɪɢɱɺɦ ɩɨɥɟ ܺ ɬɪɚɧɫɜɟɪɫɚɥɶɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ‫ ݎ‬ൌ Ͳ ɩɨɱɬɢ ɜɫɸɞɭ.
253
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МГСУ
ɋɤɚɱɨɤ ɪɟɲɟɧɢɹ ɤɚɤ ɮɭɧɤɰɢɢ ɜɪɟɦɟɧɢ ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ
‫ݎ‬ሺ‫ݔ‬ሻ ൌ Ͳ. Ɇɨɦɟɧɬ ɫɤɚɱɤɚ ɧɢɤɚɤ ɧɟ ɩɪɢɜɹɡɚɧ ɤ ɡɧɚɱɟɧɢɸ ɮɚɡɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ, ɱɬɨ ɧɟ
ɩɨɡɜɨɥɹɟɬ ɜɵɞɟɥɢɬɶ ɜ ɪɚɫɲɢɪɟɧɧɨɦ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɩɨɜɟɪɯɧɨɫɬɶ, ɝɞɟ ɩɪɨɢɫɯɨɞɢɬ ɫɤɚɱɨɤ ɫ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ ‫ ݐ‬ൌ ߬ሺ‫ݔ‬ሻ ɥɢɛɨ ɩɪɨɫɬɨ ɡɚɮɢɤɫɢɪɨɜɚɬɶ ɦɨɦɟɧɬɵ
ɫɤɚɱɤɨɜ, ɤɚɤ ɷɬɨ ɫɞɟɥɚɧɨ ɜ [2]-[4].
ȼɜɟɞɟɧɢɟ ɮɭɧɤɰɢɢ ܴሺ‫ݔ‬ሻ, ɢɫɩɨɥɶɡɭɸɳɟɣɫɹ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ ɫɢɫɬɟɦɵ ɩɪɢ ‫ݎ‬ሺ‫ݔ‬ሻ ൌ Ͳ, ɨɬɥɢɱɚɟɬ ɩɪɟɞɥɚɝɚɟɦɵɣ ɩɨɞɯɨɞ ɨɬ ɩɪɢɦɟɧɹɟɦɨɝɨ ɜ [5],[6] ɢ ɞɚɺɬ ɞɨɩɨɥɧɢɬɟɥɶɧɭɸ ɝɢɛɤɨɫɬɶ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɨɞɟɥɟɣ. ȼ ɦɟɯɚɧɢɤɟ ɫɭɳɟɫɬɜɭɸɬ ɫɬɚɧɞɚɪɬɧɵɟ ɫɩɨɫɨɛɵ ɜɵɱɢɫɥɟɧɢɹ ɪɟɚɤɰɢɣ ɞɥɹ ɪɚɡɧɵɯ ɬɢɩɨɜ ɫɜɹɡɟɣ, ɢɯ ɪɚɡɥɨɠɟɧɢɹ ɧɚ ɧɨɪɦɚɥɶɧɭɸ ɢ ɤɚɫɚɬɟɥɶɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɦɨɞɟɥɢ ɬɪɟɧɢɹ. Ⱦɥɹ ɨɛɳɟɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ, ɨɞɧɚɤɨ, ɭɞɨɛɧɨ ɨɩɭɫɬɢɬɶ ɬɪɚɞɢɰɢɨɧɧɵɟ ɞɨɩɭɳɟɧɢɹ ɨ
(ɧɟ)ɢɞɟɚɥɶɧɨɫɬɢ ɫɜɹɡɢ ɢ ɫɱɢɬɚɬɶ ‫ݎ‬ሺ‫ݔ‬ሻ ɢ ܴሺ‫ݔ‬ሻ ɧɢɤɚɤ ɧɟ ɫɜɹɡɚɧɧɵɦɢ ɜ ɷɬɨɦ ɫɦɵɫɥɟ ɨɛɴɟɤɬɚɦɢ.
Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɞɚɥɶɧɟɣɲɟɝɨ ɢɡɥɨɠɟɧɢɹ ɮɭɧɤɰɢɢ ܺሺ‫ݔ‬ሻ, ‫ ݎ‬ൌ ‫ݎ‬ሺ‫ݔ‬ሻ ɢ ܴሺ‫ݔ‬ሻ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɚɧɚɥɢɬɢɱɟɫɤɢɦɢ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ ‫ ݔ‬ൌ Ͳ. ȼɫɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɧɧɨɣ
‫ ݔ‬ɜɫɸɞɭ ɧɢɠɟ ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɥɟɠɚɳɢɦɢ ɜ ɷɬɨɣ ɨɤɪɟɫɬɧɨɫɬɢ, ɟɫɥɢ ɧɟ ɨɝɨɜɨɪɟɧɨ ɨɛɪɚɬɧɨɟ. Ɍɨɝɞɚ ɫɩɪɚɜɟɞɥɢɜɵ ɪɚɡɥɨɠɟɧɢɹ:
ଵ
‫ݔ‬ሶ ൌ ݂ ൅ ‫ ݔܣ‬൅ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ ൅ ‫ڮ‬
ଶ
(1.1)
ଵ
‫ ݎ‬ൌ ‫ݎ‬ሺ‫ݔ‬ሻ ൌ ሺܿǡ ‫ݔ‬ሻ ൅ ሺܷ‫ݔ‬ǡ ‫ݔ‬ሻ ൅ ‫ڮ‬
ଶ
ɝɞɟ ‫ݎ‬ሺͲሻ ൌ Ͳǡ ݂ ൌ ܺሺͲሻ ് Ͳǡ ‫ ܣ‬ൌ
ܿൌ
డ௥
డ௫
ሺͲሻ ് Ͳǡ ܷ ൌ
డమ ௥
డ௫ మ
డ௑
డ௫
ሺͲሻǡ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ ؔ
డమ ௑
డ௫ మ
௜ ௝ ௞
௜
௜
ሺͲሻ ൌ ൫ܾ௝௞
‫ ݔ ݔ‬൯ǡ ܾ௝௞
ൌ ܾ௞௝
ǡ
ሺͲሻ, ɬɪɨɟɬɨɱɢɟɦ ɨɛɨɡɧɚɱɟɧɵ ɫɥɚɝɚɟɦɵɟ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɩɨ-
ɪɹɞɤɨɜ ɩɨ ‫ݔ‬. Ʉɪɭɝɥɵɦɢ ɫɤɨɛɤɚɦɢ ɨɛɨɡɧɚɱɟɧɨ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ (ɤɪɨɦɟ
ɭɫɥɨɜɧɨɝɨ ɨɛɨɡɧɚɱɟɧɢɹ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ ɞɥɹ ɤɜɚɞɪɚɬɢɱɧɵɯ ɫɥɚɝɚɟɦɵɯ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɈȾɍ).
ɇɢɠɟ ɧɨɪɦɚ ɜɟɤɬɨɪɚ ɩɨɥɚɝɚɟɬɫɹ ɟɜɤɥɢɞɨɜɨɣ, ɚ ɧɨɪɦɚ ɥɢɧɟɣɧɨɝɨ ɨɩɟɪɚɬɨɪɚ ɟɣ ɩɨɞɱɢɧɟɧɚ.
ɉɨɥɨɠɢɦ, ɱɬɨ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɪɟɲɟɧɢɟ ɜɯɨɞɹɳɟɣ ɜ ɧɟɺ ɈȾɍ,
ɟɫɥɢ ‫ ݎ‬൐ Ͳ.
Ɋɚɫɫɦɨɬɪɢɦ ɢɡɦɟɧɟɧɢɟ ɮɭɧɤɰɢɢ ‫ ݎ‬ɧɚ ɪɟɲɟɧɢɢ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ
ɧɭɥɹ. Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ, ɩɨɥɭɱɢɦ:
‫ݎ‬ሶ ሺ‫ݔ‬ሻ ൌ ሺܿǡ ݂ሻ ൅ ‫ڮ‬
(1.2)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡɦɟɧɟɧɢɟ ɮɭɧɤɰɢɢ ‫ ݎ‬ɜ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɟɢɦɭɳɟɫɬɜɟɧɧɨ ɩɨɫɬɨɹɧɧɵɦ ɫɥɚɝɚɟɦɵɦ ɢɡ ɩɪɚɜɨɣ ɱɚɫɬɢ (1.2).
ȼɨɡɦɨɠɧɵ ɫɥɟɞɭɸɳɢɟ ɫɢɬɭɚɰɢɢ:
- ȿɫɥɢ ሺܿǡ ݂ሻ ൐ Ͳ, ɬɨ, ɩɨ ɩɨɥɨɠɟɧɢɸ, ɮɭɧɤɰɢɹ ‫ݎ‬ሺ‫ݔ‬ሻ ɛɭɞɟɬ ɜɨɡɪɚɫɬɚɬɶ ɜɞɨɥɶ ɪɟɲɟɧɢɣ ɈȾɍ, ɧɚɱɚɜɲɢɯɫɹ ɞɨɫɬɚɬɨɱɧɨ ɛɥɢɡɤɨ ɤ ɬɨɱɤɟ ‫ ݔ‬ൌ Ͳ ɩɪɢ ‫ݎ‬ሺ‫ݔ‬ሻ ൐ Ͳ. ɗɬɨɬ ɫɥɭɱɚɣ
ɧɟ ɛɭɞɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ.
- Ɉɫɬɚɸɬɫɹ ɞɜɚ ɫɥɭɱɚɹ: ሺܿǡ ݂ሻ ൏ Ͳ ɢ ሺܿǡ ݂ሻ ൌ Ͳ. Ʉɚɠɞɵɣ ɢɡ ɧɢɯ ɩɪɟɞɩɨɥɚɝɚɟɬ ɫɭɳɟɫɬɜɟɧɧɨɟ ɪɚɡɥɢɱɢɟ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɈȾɍ ɫ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ ɢ ɬɪɟɛɭɟɬ ɫɜɨɢɯ ɩɪɟɞɩɨɥɨɠɟɧɢɣ. ȼ ɫɬɚɬɶɟ ɪɚɡɛɢɪɚɟɬɫɹ ɩɟɪɜɵɣ ɢɡ ɧɢɯ, ɤɨɝɞɚ
ɜɟɤɬɨɪɧɨɟ ɩɨɥɟ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɬɪɚɧɫɜɟɪɫɚɥɶɧɨ ɝɪɚɧɢɰɟ ɫɜɹɡɢ.
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2. Ɍɪɚɧɫɜɟɪɫɚɥɶɧɵɣ ɫɥɭɱɚɣ ሺܿǡ ݂ሻ ൏ Ͳ.
2.1.Ɉɩɪɟɞɟɥɟɧɢɟ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɩɪɢ ሺܿǡ ݂ሻ ൏ Ͳ.
Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ, ɨɬɜɟɱɚɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ‫ݔ‬଴ ɨɩɪɟɞɟɥɢɦ ɤɨɧɫɬɪɭɤɬɢɜɧɨ:
a) ȿɫɥɢ (ɩɨɤɚ) ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ൯ ൐ Ͳ, ɬɨ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɩɨɤɚ ɧɟ ɜɵɩɨɥɧɟɧɨ b).
b) ȿɫɥɢ ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሻ൯ ൌ Ͳ, ‫ݎ‬ሶ ǣ ൌ
డ௥
డ௫
ܺሺ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሻሻ ൑ Ͳ ɢ ɫɭɳɟɫɬɜɭɟɬ İ ൐ Ͳ, ɱɬɨ ɞɥɹ ɜɫɟɯ
‫ א ݐ‬ሺ‫ כ ݐ‬ǡ ‫ כ ݐ‬൅ İሻ ɫɩɪɚɜɟɞɥɢɜɨ ‫ݎ‬ሶ ൏ Ͳ ɜ ɫɢɥɭ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ, ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɨɬ
a) ɤ ɫ), ɥɢɛɨ ɪɟɲɟɧɢɟ ɫɪɚɡɭ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ c), ɟɫɥɢ ‫ כ ݐ‬ൌ Ͳ. ɗɬɨɬ ɩɟɪɟɯɨɞ ɧɚɡɵɜɚɟɬɫɹ ɜɵɯɨɞɨɦ ɧɚ ɫɜɹɡɶ.
c) ȿɫɥɢ (ɩɨɤɚ) ‫ݎ‬ሺ‫ݔ‬ሻ ൌ Ͳ, ɬɨ ɪɟɲɟɧɢɟ ɫɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ
‫ݔ‬ሶ ൌ ܺሺ‫ݔ‬ሻ ൅ ܴሺ‫ݔ‬ሻ ɩɨɤɚ ɧɟ ɜɵɩɨɥɧɟɧɨ d). Ɏɭɧɤɰɢɹ ܴሺ‫ݔ‬ሻ ɜɵɛɢɪɚɟɬɫɹ ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɫɨɯɪɚɧɟɧɢɟ ɪɚɜɟɧɫɬɜɚ ‫ ݎ‬ൌ Ͳ ɞɥɹ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ.
d) ȿɫɥɢ ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ ככ ݐ‬ሻ൯ ൌ Ͳǡ ‫ݎ‬ሶ ൒ Ͳ ɢ ɫɭɳɟɫɬɜɭɟɬ İ ൐ Ͳ, ɱɬɨ ɞɥɹ ɜɫɟɯ ‫ א ݐ‬ሺ‫ ככ ݐ‬ǡ ‫ ככ ݐ‬൅ İሻ
ɫɩɪɚɜɟɞɥɢɜɨ ‫ݎ‬ሶ ൐ Ͳ ɜ ɫɢɥɭ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ, ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɨɬ c) ɤ a), ɥɢɛɨ
ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ a) ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ‫ ככ ݐ‬ൌ Ͳ. ɗɬɨɬ ɩɟɪɟɯɨɞ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɨɦ ɫɨ ɫɜɹɡɢ.
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɪɟɲɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ ɟɞɢɧɫɬɜɟɧɧɵ ɞɥɹ
ɩɨɥɭɱɚɟɦɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɢ ɩɪɨɞɨɥɠɢɦɵ ɜ ɛɭɞɭɳɟɟ ɧɚ ɧɟɨɛɯɨɞɢɦɵɣ ɞɥɹ ɪɚɫɫɦɨɬɪɟɧɢɹ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ. Ɍɨɝɞɚ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɟɧɨ ɟɞɢɧɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ ɢ ɩɪɨɞɨɥɠɢɦɨ ɜ ɛɭɞɭɳɟɟ.
2.2. ɂɫɫɥɟɞɨɜɚɧɢɟ ɭɫɬɨɣɱɢɜɨɫɬɢ ɪɟɲɟɧɢɹ ‫ݔ‬ሺͲǡ ‫ݐ‬ሻ ‫ Ͳ ؠ‬ɩɪɢ ሺܿǡ ݂ሻ ൏ Ͳ.
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ܴሺͲሻ ൌ െܺሺͲሻ ൌ ݂, ɬɨɝɞɚ ɜ ɫɢɥɭ ‫ݎ‬ሺͲሻ ൌ Ͳ ɢ ‫ݎ‬ሶ ሺͲሻ ൌ ሺܿǡ ݂ሻ ൏ Ͳ
‫ݔ‬ሺͲǡ ‫ݐ‬ሻ ‫ Ͳ ؠ‬ɹɜɥɹɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ ɢ ɜɫɟɣ ɫɢɫɬɟɦɵ ɜ ɰɟɥɨɦ. ɂɫɫɥɟɞɭɟɦ ɜɨɩɪɨɫ ɨ ɟɝɨ ɭɫɬɨɣɱɢɜɨɫɬɢ ɜ ɫɦɵɫɥɟ Ʌɹɩɭɧɨɜɚ.
Ʌɟɦɦɚ 1. ɋɭɳɟɫɬɜɭɸɬ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ İǡ ݃ǡ ݀ ɬɚɤɢɟ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ ‫ݔ‬଴ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɩɪɢ ԡ‫ݔ‬଴ ԡ ൏ İ ɪɟɲɟɧɢɟ ɜɵɣɞɟɬ ɧɚ ɫɜɹɡɶ ɡɚ
ɜɪɟɦɹ ܶሺ‫ݔ‬଴ ሻ, ɢ ɫɩɪɚɜɟɞɥɢɜɵ ɨɰɟɧɤɢ:
ԡ‫ݔ‬ሺܶሺ‫ݔ‬଴ ሻǡ ‫ݔ‬଴ ሻԡ ൑ ݃İǡ ܶሺ‫ݔ‬଴ ሻ ൑ ݀İ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɋɟɲɟɧɢɟ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ ɹɜɥɹɟɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ ɩɨ ɩɪɟɞɩɨɥɨɠɟɧɢɸ. Ɍɨɝɞɚ ɮɭɧɤɰɢɹ ‫ݎ‬ሺ‫ݐ‬ሻ ൌ ‫ݎ‬ሺ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻሻ ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ
ɚɧɚɥɢɬɢɱɟɫɤɨɣ: ‫ݎ‬ሺ‫ݐ‬ሻ ൌ ‫ݎ‬ሺ‫ݐ‬ሻȁ௧ୀ଴ ൅ ሺܿǡ ݂ሻ‫ ݐ‬൅ ‫ ڮ‬Ⱦɥɹ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɢɡ ɧɟɤɨɬɨɪɨɣ
ɦɚɥɨɣ (ɡɚɦɤɧɭɬɨɣ) ɲɚɪɨɜɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɪɚɞɢɭɫɚ ߝ:
ȁ‫ݎ‬ሺ‫ݔ‬ሻ െ ‫ݎ‬ሺ‫ݕ‬ሻȁ ൌ ȁሺܿǡ ‫ݔ‬ሻ െ ሺܿǡ ‫ݕ‬ሻ ൅ ‫ ڮ‬ȁ ൑ ȁሺܿǡ ‫ ݔ‬െ ‫ݕ‬ሻȁ ൅ ԡ‫ ݔ‬െ ‫ݕ‬ԡ ൑
൑ ԡܿԡ ȉ ԡ‫ ݔ‬െ ‫ݕ‬ԡ ൅ ԡ‫ ݔ‬െ ‫ݕ‬ԡ ൌ ሺԡܿԡ ൅ ͳሻ ȉ ԡ‫ ݔ‬െ ‫ݕ‬ԡ ‫ ؠ‬ɫଵ ȉ ԡ‫ ݔ‬െ ‫ݕ‬ԡ
ȼɬɨɪɨɣ ɩɟɪɟɯɨɞ ɜɨɡɦɨɠɟɧ ɜ ɫɢɥɭ ɝɥɚɞɤɨɫɬɢ ‫ݎ‬ሺ‫ݔ‬ሻ ɢ ɩɨɞɯɨɞɹɳɟɣ ɨɰɟɧɤɢ ɫɜɟɪɯɭ ɦɥɚɞɲɢɯ ɱɥɟɧɨɜ, ɚ ɬɪɟɬɢɣ ɩɪɨɢɡɜɟɞɺɧ ɜ ɫɢɥɭ ɧɟɪɚɜɟɧɫɬɜɚ Ʉɨɲɢ-Ȼɭɧɹɤɨɜɫɤɨɝɨ. ɑɥɟɧɵ ɜɬɨɪɨɝɨ ɢ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɩɨɪɹɞɤɨɜ ɪɚɡɥɨɠɟɧɢɹ ‫ݎ‬ሺ‫ݐ‬ሻ ɞɨɩɭɫɤɚɸɬ ɨɰɟɧɤɭ ɫɜɟɪɯɭ ɜɟɥɢɱɢɧɨɣ
ܿଶ ԡ‫ݔ‬଴ ԡǡ ܿଶ ൐ Ͳɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ ‫ݐ‬. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ:
‫ݎ‬ሺ‫ݐ‬ሻ ൌ ‫ݎ‬ሺ‫ݐ‬ሻȁ௧ୀ଴ ൅ ሺܿǡ ݂ሻ‫ ݐ‬൅ ‫ ڮ‬൑ ܿଵ ԡ‫ݔ‬଴ ԡ ൅ ሺܿǡ ݂ሻ‫ ݐ‬൅ ܿଶ ԡ‫ݔ‬଴ ԡ
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ВЕСТНИК
МГСУ
4/2011
Ɍɚɤ ɤɚɤ ሺܿǡ ݂ሻ ൏ Ͳ, ɬɨ ɩɪɢ ‫ ݐ‬൐
௖భ ା௖మ
ିሺ௖ǡ௙ሻ
ɞɚɧɧɨɝɨ ‫ݔ‬଴ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ܶሺ‫ݔ‬଴ ሻ ,
ԡ‫ݔ‬଴ ԡ ൐ Ͳ ɜɵɩɨɥɧɹɟɬɫɹ ‫ݎ‬ሺ‫ݐ‬ሻ ൏ Ͳ, ɡɧɚɱɢɬ ɞɥɹ
ܶሺ‫ݔ‬଴ ሻ ൑
௖భ ା௖మ
ିሺ௖ǡ௙ሻ
ԡ‫ݔ‬଴ ԡ ൏
௖భ ା௖మ
ିሺ௖ǡ௙ሻ
İ , ɩɪɢ ɤɨɬɨɪɨɦ
‫ݎ‬൫ܶሺ‫ݔ‬଴ ሻ൯ ൌ Ͳ. ȿɫɥɢ ߝ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨ, ɬɨ ɧɚ ɩɪɨɦɟɠɭɬɤɟ ɜɪɟɦɟɧɢ ቂͲǡ
௖భ ା௖మ
ିሺ௖ǡ௙ሻ
İቃ ɪɚɡɥɨɠɟ-
ɧɢɟ ɪɟɲɟɧɢɹ ɜ ɪɹɞ ɩɨ ɜɪɟɦɟɧɢ ɨɫɬɚɺɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦ ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜɵɩɨɥɧɟɧɵ
ɨɰɟɧɤɢ ɜɵɲɟ. ɉɨɥɨɠɢɦ ݀ ൌ
௖భ ା௖మ
ିሺ௖ǡ௙ሻ
. Ɋɚɡɥɨɠɢɦ ɪɟɲɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɜ ɪɹɞ ɩɨ
ɜɪɟɦɟɧɢ ɢ ɨɰɟɧɢɦ ɫɥɚɝɚɟɦɵɟ ɩɪɢ ‫ א ݐ‬ሾͲǡ ܶሿ, ԡ‫ݔ‬଴ ԡ ൑ İ. ɐɟɩɨɱɤɚ ɧɟɪɚɜɟɧɫɬɜ
ԡ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻԡ ൌ ԡ‫ݔ‬଴ ൅ ܺሺ‫ݔ‬଴ ሻ‫ ݐ‬൅ ‫ ڮ‬ԡ ൑ ԡ‫ݔ‬଴ ԡ ൅ ԡܺሺ‫ݔ‬଴ ሻԡ‫ ݐ‬൅ ԡ‫ݔ‬଴ ԡ ൑
൑ ʹԡ‫ݔ‬଴ ԡ ൅ ԡ݂ ൅ ‫ݔܣ‬଴ ൅ ‫ ڮ‬ԡ‫ ݐ‬൑
൑ ʹİ ൅ ԡ݂ԡ‫ ݐ‬൅ ሺԡ‫ܣ‬ԡ ൅ ͳሻԡ‫ݔ‬଴ ԡ ൑ ሺ͵ ൅ ԡ‫ܣ‬ԡሻİ ൅ ݀ԡ݂ԡİ ൌ ሺ͵ ൅ ԡ‫ܣ‬ԡ ൅ ݀ԡ݂ԡሻİ,
ɡɚɜɟɪɲɚɟɬ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫ ݃ ൌ ͵ ൅ ԡ‫ܣ‬ԡ ൅ ݀ԡ݂ԡ.
Ɍɟɨɪɟɦɚ 1. Ⱦɥɹ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ ɜ ɫɥɭɱɚɟ ሺܿǡ ݂ሻ ൏ Ͳ ɩɪɢ
ܴሺͲሻ ൌ െܺሺͲሻ ɭɫɬɨɣɱɢɜɨɫɬɶ ɪɟɲɟɧɢɹ ‫ݔ‬ሺͲǡ ‫ݐ‬ሻ ‫ Ͳ ؠ‬ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɢɫɫɥɟɞɨɜɚɧɢɟɦ
ɟɝɨ ɭɫɬɨɣɱɢɜɨɫɬɢ ɤɚɤ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɋɚɫɫɦɨɬɪɢɦ ɦɚɥɨɟ ɜɨɡɦɭɳɟɧɢɟ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ‫ݔ‬଴ . ȿɫɥɢ
‫ݎ‬ሺ‫ݔ‬଴ ሻ ൌ Ͳ, ɬɨ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɫɢɫɬɟɦɨɣ ɧɚ ɫɜɹɡɢ ɜ ɫɢɥɭ ɨɩɪɟɞɟɥɟɧɢɹ ɢ
ɜɵɩɨɥɟɧɢɹ ‫ݎ‬ሶ ൌ ሺܿǡ ݂ሻ ൅ ‫ ڮ‬൏ Ͳ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. ɍɫɬɨɣɱɢɜɨɫɬɶ ɩɨɞɨɛɧɵɯ ɪɟɲɟɧɢɣ ɫɢɫɬɟɦɵ ɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦ ɭɫɥɨɜɢɟɦ ɭɫɬɨɣɱɢɜɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɵɯ (ɦɚɥɵɯ) ɜɨɡɦɭɳɟɧɢɣ. Ɉɬɫɸɞɚ ɠɟ ɫɥɟɞɭɟɬ ɧɟɭɫɬɨɣɱɢɜɨɫɬɶ
ɜɫɟɣ ɫɢɫɬɟɦɵ ɩɪɢ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ. ɇɢɠɟ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ 0
– ɭɫɬɨɣɱɢɜɚɹ ɩɨ Ʌɹɩɭɧɨɜɭ ɫɬɚɰɢɨɧɚɪɧɚɹ ɬɨɱɤɚ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɦɚɥɨɟ ɩɨ ɧɨɪɦɟ ‫ݔ‬଴ , ɬɚɤɨɟ, ɱɬɨ ‫ݎ‬ሺ‫ݔ‬଴ ሻ ൐ Ͳ. ɋɨɝɥɚɫɧɨ ɥɟɦɦɟ ɪɟɲɟɧɢɟ
ɜɵɣɞɟɬ ɧɚ ɫɜɹɡɶ ɡɚ ɦɚɥɨɟ ɜɪɟɦɹ ɛɥɢɡɤɨ ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɨɧɨ ɛɭɞɟɬ ɩɪɨɞɨɥɠɚɬɶɫɹ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ ɜ ɫɢɥɭ ‫ݎ‬ሶ ൏ Ͳ, ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɩɨɤɢɧɟɬ ɧɟɤɨɬɨɪɭɸ ɤɨɧɟɱɧɭɸ ɨɤɪɟɫɬɧɨɫɬɶ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɭɫɬɨɣɱɢɜɨɫɬɢ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ İ ൐ Ͳ ɧɚɣɞɺɬɫɹ ɬɚɤɨɟ į ൌ įሺİሻ ൐ Ͳ, ɱɬɨ ɜɫɟ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɧɚ
ɫɜɹɡɢ, ɧɚɱɚɜɲɢɟɫɹ ɜ ɲɚɪɟ ɪɚɞɢɭɫɚ į, ɜɟɱɧɨ ɛɭɞɭɬ ɨɫɬɚɜɚɬɶɫɹ ɜ ɲɚɪɟ ɪɚɞɢɭɫɚ İ (ɨɫɬɚɜɚɹɫɶ ɩɪɢ ɷɬɨɦ ɧɚ ɫɜɹɡɢ ɜ ɫɢɥɭ ɫɨɯɪɚɧɟɧɢɹ ‫ݎ‬ሶ ൏ Ͳ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɦ İ). ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜɨɡɦɭɳɟɧɢɹ, ɧɟ ɥɟɠɚɳɢɟ ɧɚ ɫɜɹɡɢ, ɬɨ ɢɯ ɜɵɛɨɪ ɢɡ
į
೎భ శ೎మ
ԡ௙ԡ
ଷାԡ஺ԡି ሺ೎ǡ೑ሻ
- ɨɤɪɟɫɬ-
ɧɨɫɬɢ ɬɨɱɤɢ ‫ ݔ‬ൌ Ͳ ɝɚɪɚɧɬɢɪɭɟɬ (ɫɦ. ɥɟɦɦɭ), ɱɬɨ ɨɧɢ ɨɤɚɠɭɬɫɹ ɧɚ ɫɜɹɡɢ ɜɧɭɬɪɢ į – ɨɤɪɟɫɬɧɨɫɬɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɢ ɜɟɱɧɨ ɛɭɞɭɬ ɨɫɬɚɜɚɬɶɫɹ ɜ İ - ɨɤɪɟɫɬɧɨɫɬɢ, ɫɨɝɥɚɫɧɨ
ɩɪɟɞɩɨɥɨɠɟɧɢɸ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɂɚɦɟɱɚɧɢɟ 1. Ɋɟɡɭɥɶɬɚɬɵ ɨɫɬɚɸɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦɢ, ɟɫɥɢ ɩɪɚɜɵɟ ɱɚɫɬɢ (1.1) ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɤɥɚɫɫɚ ‫ ܥ‬ଵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ, ɬ.ɤ. ɢɫɩɨɥɶɡɨɜɚɥɚɫɶ ɥɢɲɶ
ɦɚɥɨɫɬɶ ɨɬɛɪɚɫɵɜɚɟɦɵɯ ɫɥɚɝɚɟɦɵɯ ɜ ɪɚɡɥɨɠɟɧɢɹɯ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɡɚɦɟɧɢɬɶ ɪɹɞɵ Ɍɟɣɥɨɪɚ ɞɥɹ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɪɚɡɥɨɠɟɧɢɹɦɢ Ɇɚɤɥɨɪɟɧɚ.
Ɂɚɦɟɱɚɧɢɟ 2. ɋɢɫɬɟɦɚ ɧɚ ɫɜɹɡɢ ɹɜɥɹɟɬɫɹ ɨɛɵɱɧɨɣ ɫɢɫɬɟɦɨɣ ɈȾɍ, ɢ ɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɤɥɚɫɫɢɱɟɫɤɢɣ ɚɩɩɚɪɚɬ ɬɟɨɪɢɢ ɭɫɬɨɣɱɢɜɨɫɬɢ [7], ɜ ɱɚɫɬɧɨɫɬɢ, ɬɟɨɪɟɦɭ Ʌɹɩɭɧɨɜɚ ɨɛ
ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɛɥɢɠɟɧɢɸ.
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Ɂɚɦɟɱɚɧɢɟ 3. ȼ ɤɧɢɝɟ [1] ɞɨɤɚɡɵɜɚɟɬɫɹ ɩɨɯɨɠɟɟ ɭɬɜɟɪɠɞɟɧɢɟ ɞɥɹ ɦɟɯɚɧɢɱɟɫɤɢɯ
ɫɢɫɬɟɦ ɫ ɧɟɭɩɪɭɝɢɦ ɜɵɯɨɞɨɦ ɧɚ ɝɪɚɧɢɰɭ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɝɨɥɨɧɨɦɧɨɣ ɫɜɹɡɢ. ȼɵɜɨɞ ɨɛ
ɭɫɬɨɣɱɢɜɨɫɬɢ ɬɚɤɠɟ ɞɟɥɚɟɬɫɹ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ ɫɢɫɬɟɦɭ ɧɚ ɫɜɹɡɢ. ɗɬɨ ɫɯɨɞɫɬɜɨ ɜɧɟɲɧɟɟ. ȼɨ ɜɜɟɞɺɧɧɨɦ ɜɵɲɟ ɨɩɪɟɞɟɥɟɧɢɢ ɪɟɲɟɧɢɹ ɧɟ ɩɪɨɢɫɯɨɞɢɬ ɫɤɚɱɤɨɜ ɪɟɲɟɧɢɹ ɩɪɢ
ɜɵɯɨɞɟ ɧɚ ɝɪɚɧɢɰɭ ɫɜɹɡɢ (ɧɢɤɚɤɢɟ ɫɨɫɬɚɜɥɹɸɳɢɟ “ɢɦɩɭɥɶɫɚ” ɧɟ ɬɟɪɹɸɬɫɹ). Ɍɚɤɚɹ ɫɢɬɭɚɰɢɹ ɬɢɩɢɱɧɚ ɩɪɢ ɜɵɯɨɞɟ ɧɚ ɝɪɚɧɢɰɭ ɧɟɭɞɟɪɠɢɜɚɸɳɟɣ ɧɟɝɨɥɨɧɨɦɧɨɣ ɫɜɹɡɢ [8].
3. Ɂɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɫ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɨɣ, ɧɚ ɤɨɬɨɪɨɟ ɧɚɥɨɠɟɧɚ ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ ɫɜɹɡɶ.
Ɋɚɫɫɦɨɬɪɢɦ ɚɛɫɨɥɸɬɧɨ ɬɜɺɪɞɨɟ ɬɟɥɨ, ɬɨɱɤɚ O ɤɨɬɨɪɨɝɨ ɡɚɮɢɤɫɢɪɨɜɚɧɚ ɜ ɚɛɫɨɥɸɬɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. ɉɪɢɦɟɦ, ɱɬɨ ɜɧɟɲɧɢɟ ɫɢɥɵ ɨɬɫɭɬɫɬɜɭɸɬ, ɥɢɛɨ ɧɟ ɫɨɡɞɚɸɬ ɦɨɦɟɧɬɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ O. ɉɭɫɬɶ ૑ - ɜɟɤɬɨɪ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ, ɚ ‫ ܍‬- ɨɪɬ, ɜɦɨɪɨɠɟɧɧɵɣ ɜ ɬɟɥɨ. ɇɚ ɬɟɥɨ ɧɚɥɨɠɟɧɚ ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ ɫɜɹɡɶ (ɢɞɟɚɥɶɧɚɹ ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɧɟɝɨɥɨɧɨɦɧɚɹ, ɫɨɝɥɚɫɧɨ ɬɟɪɦɢɧɨɥɨɝɢɢ [8], ɟɫɥɢ ɧɟ ɡɚɛɵɜɚɬɶ ɩɪɨ ɭɝɥɨɜɨɟ
ɩɨɥɨɠɟɧɢɟ ɬɟɥɚ).
ሺ૑ǡ ‫܍‬ሻ ൒ Ͳ
(3.1)
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ɡɚɞɚɱɢ ɋɭɫɥɨɜɚ[11] ɩɭɬɺɦ ɡɚɦɟɧɵ ɪɚɜɟɧɫɬɜɚ ɧɚ ɧɟɪɚɜɟɧɫɬɜɨ ɜ ɭɪɚɜɧɟɧɢɢ ɫɜɹɡɢ.
Ⱦɥɹ ɟɺ ɜɨɩɥɨɳɟɧɢɹ ɜ ɪɟɚɥɶɧɨɫɬɢ ɩɨɞɯɨɞɢɬ ɫɥɟɞɭɸɳɚɹ ɦɨɞɟɥɶ: ɪɚɫɫɦɨɬɪɢɦ ɫɯɟɦɭ
ȼɚɝɧɟɪɚ [10] ɫ ɡɚɦɟɧɨɣ ɤɨɥɺɫɢɤɨɜ ɫ ɨɫɬɪɨɣ ɤɪɨɦɤɨɣ (ɷɤɜɢɜɚɥɟɧɬɧɵɯ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
ɤɨɧɶɤɚɦ ɑɚɩɥɵɝɢɧɚ) ɧɚ ɚɧɚɥɨɝɢɱɧɨ ɡɚɤɪɟɩɥɺɧɧɵɟ ɨɞɧɨɫɬɨɪɨɧɧɢɟ ɤɨɧɶɤɢ [8], ɥɟɠɚɳɢɟ
ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɜɦɨɪɨɠɟɧɧɨɣ ɜ ɬɟɥɨ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɟɩɨɞɜɢɠɧɭɸ ɬɨɱɤɭ. ɂɯ
ɞɨɩɭɫɬɢɦɵɟ ɞɥɹ ɫɤɨɥɶɠɟɧɢɹ ɫɬɨɪɨɧɵ ɪɚɫɩɨɥɨɠɢɦ ɰɟɧɬɪɚɥɶɧɨ-ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɫɥɟɞɧɟɣ. Ɍɟɦ ɫɚɦɵɦ ɩɨɥɭɱɢɦ ɬɪɟɛɭɟɦɭɸ ɜɨɡɦɨɠɧɨɫɬɶ ɜɪɚɳɟɧɢɹ ɬɨɥɶɤɨ ɜ
ɨɞɧɭ ɫɬɨɪɨɧɭ. ɗɬɨ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɟɣɲɢɦ ɨɛɨɛɳɟɧɢɟɦ ɪɟɚɥɢɡɚɰɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ
ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɫɜɹɡɢ ɢɡ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜ [9].
ɋɜɹɠɟɦ ɫ ɬɜɺɪɞɵɦ ɬɟɥɨɦ ɩɪɚɜɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ȟȘȗ, ɩɨɫɬɪɨɟɧɧɭɸ ɧɚ ɝɥɚɜɧɵɯ ɨɫɹɯ ɢɧɟɪɰɢɢ ɞɥɹ ɬɨɱɤɢ O, ɤɨɬɨɪɵɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ǡ ǡ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɍɜɺɪɞɨɟ ɬɟɥɨ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ ɞɢɧɚɦɢɱɟɫɤɢ ɧɟɜɵɪɨɠɞɟɧɧɵɦ – ൐
Ͳ.Ⱦɚɥɶɧɟɣɲɟɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɛɭɞɟɦ ɩɪɨɜɨɞɢɬɶ “ɜ ɫɤɨɪɨɫɬɹɯ” ૑. ɉɪɢ ɷɬɨɦ ɧɟ ɢɦɟɸɬ
ɡɧɚɱɟɧɢɹ ɭɫɥɨɜɢɹ ɢɧɬɟɝɪɢɪɭɟɦɨɫɬɢ ɫɜɹɡɢ (3.1), ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɢ ɫɚɦɨ ɩɨɧɹɬɢɟ ɧɟɝɨɥɨɧɨɦɧɨɫɬɢ ɬɟɪɹɟɬ ɫɦɵɫɥ. ɗɬɢɦ ɨɛɭɫɥɚɜɥɢɜɚɟɬɫɹ ɩɪɢɦɟɧɟɧɢɟ ɬɟɪɦɢɧɚ “ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ”.
ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɢɦɟɸɬ ɜɢɞ:
Ȝሺ૑ሻ ൒ Ͳǡ ɩɪɢሺ૑ǡ ‫܍‬ሻ ൌ Ͳǡ ሺ૑ሶǡ ‫܍‬ሻ ൑ Ͳ
૑ሶ ൅ ૑ ൈ ૑ ൌ Ȝ‫܍‬, Ȝ ൌ ൜
(3.2)
Ͳǡ ɜɨɫɬɚɥɶɧɵɯɫɥɭɱɚɹɯǤ
Ɉɧɢ ɩɨɥɭɱɚɸɬɫɹ ɤɚɤ ɭɪɚɜɧɟɧɢɹ Ⱥɩɩɟɥɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɤɨɦɩɨɧɟɧɬɨɜ ૑ ɜ ɤɚɱɟɫɬɜɟ ɤɜɚɡɢɫɤɨɪɨɫɬɟɣ, Ȝ – ɧɟɨɩɪɟɞɟɥɺɧɧɵɣ ɦɧɨɠɢɬɟɥɶ Ʌɚɝɪɚɧɠɚ, ɭɫɥɨɜɢɹ ɞɥɹ ɧɟɝɨ ɩɪɢɜɟɞɟɧɵ ɫ ɭɱɺɬɨɦ ɬɨɝɨ, ɱɬɨ ɫɜɹɡɶ ɧɟɭɞɟɪɠɢɜɚɸɳɚɹ.
ȼ ɥɟɜɨɣ ɱɚɫɬɢ (3.2) ɧɚɩɢɫɚɧɵ ɤɥɚɫɫɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɗɣɥɟɪɚ, ɚ ɬɨɱɤɨɣ ɨɛɨɡɧɚɱɟɧɚ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɜɨ ɜɦɨɪɨɠɟɧɧɵɯ ɨɫɹɯ. ɋɩɪɚɜɚ ɫɬɨɢɬ ɪɟɚɤɬɢɜɧɵɣ ɦɨɦɟɧɬ,
ɜɨɡɧɢɤɚɸɳɢɣ ɩɪɢ ɜɵɯɨɞɟ ɧɚ ɝɪɚɧɢɰɭ ɫɜɹɡɢ ɢ ɩɪɟɩɹɬɫɬɜɭɸɳɢɣ ɜɵɯɨɞɭ ɫɢɫɬɟɦɵ ɜ ɡɚɩɪɟɳɺɧɧɭɸ ɨɛɥɚɫɬɶ ሺ૑ǡ ‫܍‬ሻ ൏ Ͳ.
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Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɫɢɫɬɟɦɚ ɫɨɜɟɪɲɚɟɬ ɞɜɢɠɟɧɢɟ ɩɨ ɫɜɹɡɢ. ɉɨ ɩɪɟɞɩɨɥɨɠɟɧɢɸ ɬɟɧɡɨɪ
ɢɧɟɪɰɢɢ ɧɟɜɵɪɨɠɞɟɧ, ɩɨɷɬɨɦɭ ɭɦɧɨɠɢɦ ɭɪɚɜɧɟɧɢɟ (3.2) ɫɥɟɜɚ ɧɚ ିଵ ɢ ɡɚɬɟɦ ɫɤɚɥɹɪɧɨ ɧɚ ‫܍‬. Ɍɚɤ ɤɚɤ ɨɩɟɪɚɬɨɪ ɫɢɦɦɟɬɪɢɱɟɧ ɢ ‫܍‬ଶ ൌ ͳ, ɩɨɥɭɱɢɦ
ሺ૑ሶǡ ‫܍‬ሻ ൅ ሺିଵ ‫܍‬ǡ ૑ ൈ ૑ሻ ൌ Ȝሺ ିଵ ‫܍‬ǡ ‫܍‬ሻ,
(3.3)
Ɍɚɤ ɤɚɤ ɩɪɢ ɞɜɢɠɟɧɢɢ ɧɚ ɫɜɹɡɢ ሺ૑ሺ–ሻǡ ‫܍‬ሻ ‫Ͳ ؠ‬, ɬɨ ሺ૑ሶǡ ‫܍‬ሻ ൌ Ͳ ɜ (3.3). ȼɵɪɚɠɚɹ Ȝ,
ɩɨɥɭɱɢɦ:
ଵ
Ȝ ൌ Ȝሺ૑ሻ ൌ ሺ୎షభ
‫܍‬ǡ‫܍‬ሻ
ሺ ିଵ ‫܍‬ǡ ૑ ൈ ૑ሻ.
(3.4)
ɉɨɫɥɟɞɧɟɟ ɜɵɪɚɠɟɧɢɟ ɪɚɜɧɹɟɬɫɹ ɜɡɹɬɨɣ ɫ ɨɛɪɚɬɧɵɦ ɡɧɚɤɨɦ ɩɪɨɟɤɰɢɢ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ‫ ܍‬ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ ɝɪɚɧɢɰɵ
ɫɜɹɡɢ. Ɍɨ ɟɫɬɶ ɭɫɥɨɜɢɟ Ȝ ൐ Ͳ ɜ ɨɛɳɢɯ ɨɛɨɡɧɚɱɟɧɢɹɯ ɞɚɺɬ ܿ ൌ ‫܍‬ǡ ݂ ൌ െ‫ିܬ‬ଵ ሺ૑ ൈ ‫ܬ‬૑ሻ ,
ሺܿǡ ݂ሻ ൏ Ͳ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɫɢɫɬɟɦɵ ɜ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɞɚɧɧɨɣ ɬɨɱɤɢ ૑.
Ʉɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜ [8], ɜɵɯɨɞ ɧɚ ɝɪɚɧɢɰɭ ɧɟɭɞɟɪɠɢɜɚɸɳɟɣ ɝɨɥɨɧɨɦɧɨɣ ɫɜɹɡɢ
ɩɪɨɢɫɯɨɞɢɬ ɛɟɡ ɭɞɚɪɚ-ɫɤɚɱɤɚ (ɤɜɚɡɢ) ɫɤɨɪɨɫɬɢ, ɩɪɨɫɬɨ ɜɨɡɧɢɤɚɟɬ ɞɨɩɨɥɧɢɬɟɥɶɧɚɹ ɫɢɥɚ
ɢ ɥɢɲɶ ɭɫɤɨɪɟɧɢɟ ɩɪɟɬɟɪɩɟɜɚɟɬ ɪɚɡɪɵɜ. ɗɬɨ ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɞɬɜɟɪɠɞɚɟɬ ɫɨɨɬɜɟɬɫɬɜɢɟ
ɩɨɜɟɞɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɟɧɢɸ, ɜɜɟɞɺɧɧɨɦɭ ɜɵɲɟ.
ɋɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ 1 ɭɫɬɨɣɱɢɜɨɫɬɶ ɧɟɩɨɞɜɢɠɧɵɯ ɬɨɱɟɤ (ɩɟɪɦɚɧɟɧɬɧɵɯ ɜɪɚɳɟɧɢɣ
ɬɜɺɪɞɨɝɨ ɬɟɥɚ) ɩɨ ɩɟɪɟɦɟɧɧɵɦ ૑ ɛɭɞɟɬ ɭɧɚɫɥɟɞɨɜɚɧɚ ɨɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɪɟɲɟɧɢɣ
ɡɚɞɚɱɢ ɋɭɫɥɨɜɚ, ɫɪɟɞɢ ɤɨɬɨɪɵɯ ɛɭɞɭɬ ɜɵɛɢɪɚɬɶɫɹ ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨɦɭ
ɬɪɟɛɨɜɚɧɢɸ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɫɬɢ ɩɪɨɟɤɰɢɢ ɪɟɚɤɬɢɜɧɨɝɨ ɦɨɦɟɧɬɚ ɧɚ ɜɟɤɬɨɪ ‫܍‬.
Ⱥɜɬɨɪ ɜɵɪɚɠɚɟɬ ɛɥɚɝɨɞɚɪɧɨɫɬɶ ȼ.ȼ. Ȼɟɥɟɰɤɨɦɭ ɡɚ ɨɛɫɭɠɞɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ ɷɬɨɣ
ɫɬɚɬɶɢ. Ɋɚɛɨɬɵ ɜɵɩɨɥɧɟɧɚ ɩɪɢ ɩɨɞɞɟɪɠɤɟ ɝɪɚɧɬɚ ɊɎɎɂ ʋ 10-01-00406.
Ʌɢɬɟɪɚɬɭɪɚ
1. ɂɜɚɧɨɜ Ⱥ.ɉ. Ⱦɢɧɚɦɢɤɚ ɫɢɫɬɟɦ ɫ ɦɟɯɚɧɢɱɟɫɤɢɦɢ ɫɨɭɞɚɪɟɧɢɹɦɢ. Ɇ.: Ɇɟɠɞɭɧɚɪɨɞɧɚɹ ɩɪɨɝɪɚɦɦɚ ɨɛɪɚɡɨɜɚɧɢɹ, 1997.
2. Ɇɵɲɤɢɫ Ⱥ.Ⱦ., ɋɚɦɨɣɥɟɧɤɨ Ⱥ.Ɇ. ɋɢɫɬɟɦɚ ɫ ɬɨɥɱɤɚɦɢ ɜ ɡɚɞɚɧɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ.//
Ɇɚɬ. ɋɛɨɪɧɢɤ. Ɍ.74 ɜɵɩ 2., 1967, ɫɬɪ. 202-208.
3. Ɇɵɲɤɢɫ Ⱥ.Ⱦ. ɍɫɬɨɣɱɢɜɨɫɬɶ ɪɟɲɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɪɢ ɨɛɨɛɳɺɧɧɵɯ
ɢɦɩɭɥɶɫɧɵɯ ɜɨɡɦɭɳɟɧɢɹɯ.//Ⱥɜɬɨɦɚɬɢɤɚ ɢ ɬɟɥɟɦɟɯɚɧɢɤɚ. 2007, ʋ10, ɫɬɪ. 125-133.
4. ɉɟɪɟɫɬɸɤ ɇ.Ⱥ. ɋɚɦɨɣɥɟɧɤɨ Ⱥ.Ɇ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɢɦɩɭɥɶɫɧɵɦ ɜɨɡɞɟɣɫɬɜɢɟɦ. Ʉɢɟɜ, 1987.
5. Ƚɨɪɛɢɤɨɜ ɋ.ɉ. Ɉɫɨɛɟɧɧɨɫɬɢ ɫɬɪɨɟɧɢɹ ɮɚɡɨɜɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ
ɭɞɚɪɧɵɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ.// ɂɡɜ. Ⱥɇ ɋɋɋɊ ɆɌɌ. 1987, ʋ3, ɫɬɪ. 23-26.
6. Ƚɨɪɛɢɤɨɜ ɋ.ɉ. Ʌɨɤɚɥɶɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ ɭɞɚɪɧɵɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ. // Ɇɚɬ. Ɂɚɦɟɬɤɢ ɬ.64 ɜɵɩ.4. 1998, ɫɬɪ. 531-542.
7. ɑɟɬɚɟɜ ɇ.Ƚ. ɍɫɬɨɣɱɢɜɨɫɬɶ ɞɜɢɠɟɧɢɹ. ɂɡɞ. 2, ɢɫɩɪ. Ɇ., Ƚɨɫɬɟɯɬɟɨɪɢɡɞɚɬ. 1955. 175 ɫ.
8. Ȼɟɪɟɡɢɧɫɤɚɹ ɋ.ɇ., Ʉɭɝɭɲɟɜ ȿ.ɂ., ɋɨɪɨɤɢɧɚ Ɉ.ȼ. Ɉ ɞɜɢɠɟɧɢɢ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ ɨɞɧɨɫɬɨɪɨɧɧɢɦɢ ɫɜɹɡɹɦɢ // ȼɟɫɬɧ. ɆȽɍ, ɋɟɪ. 1. Ɇɚɬɟɦɚɬɢɤɚ, ɦɟɯɚɧɢɤɚ. 2005. ȼɵɩ. 3. ɋ. 18-24.
9. Ȼɨɪɢɫɨɜ Ⱥ.ȼ., Ɇɚɦɚɟɜ ɂ.ɋ. Ƚɚɦɢɥɶɬɨɧɢɡɚɰɢɹ ɧɟɝɨɥɨɧɨɦɧɵɯ ɫɢɫɬɟɦ, ArXiv:
nlin.SI/0509036, 2005.
10. ȼɚɝɧɟɪ Ƚ.ȼ. Ƚɟɨɦɟɬɪɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɞɜɢɠɟɧɢɹ ɧɟɝɨɥɨɧɨɦɧɵɯ ɞɢɧɚɦɢɱɟɫɤɢɯ
ɫɢɫɬɟɦ. Ɍɪɭɞɵ ɫɟɦɢɧɚɪɚ ɩɨ ɜɟɤɬɨɪɧɨɦɭ ɢ ɬɟɧɡɨɪɧɨɦɭ ɚɧɚɥɢɡɭ, 1941 ȼɵɩ. 5, ɫ.301-327.
11. ɋɭɫɥɨɜ Ƚ.Ʉ. Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ. Ɇ., Ƚɨɫɬɟɯɢɡɞɚɬ, ɆɅ, 1946.
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ВЕСТНИК
МГСУ
Literature
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Sbornik. T.74 vyp 2., 1967, str. 202-208.
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vozmuscheniyah.//Avtomatika i telemehanika. 2007, ʋ10, str. 125-133.
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11. Suslov G.K. Teoreticheskaya mehanika. M., Gostehizdat, ML, 1946.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɧɟɭɞɟɪɠɢɜɚɸɳɚɹ ɫɜɹɡɶ,ɧɟɝɨɥɨɧɨɦɧɚɹ ɫɜɹɡɶ, ɭɞɚɪ, ɪɟɚɤɰɢɹ ɫɜɹɡɢ, ɭɫɬɨɣɱɢɜɨɫɬɶ ɩɨ Ʌɹɩɭɧɨɜɭ, ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ ɫɜɹɡɶ, ɡɚɞɚɱɚ ɋɭɫɥɨɜɚ
Key words: one sided restriction, nonholonomic restricition,impact, reaction, Lyapunov stability,
differential restriction, Suslov's problem
e - m a i l : e o t s t a v n o v @y a . r u
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