12-Mavzu: Turg’unlik nazariyasi asoslari

Lyapunovning asimptotik turg’unlik haqidagi teoremasi.

Agar (1) differensial tenglamalar sistemasi uchun shunday musbat aniqlangan

funksiya topilib, bu funksiyaning (1) sistema yordamida hisoblangan ^𝑑𝑉
𝑑𝑡
hosilasi manfiy

aniqlangan bo’lsa, (1) sistemaning trivial yechimi asiptotik turg’un bo’ladi.

1. misol

𝑑𝑥_{1 = 𝑥}

− 𝑥³ , ^𝑑𝑥2 = −𝑥

− 𝑥³

𝑑𝑡
3 1 _𝑑𝑡 1 2

Sistemani 𝑥₁ = 0, 𝑥₂ = 0 yechimni tekshiramiz.
𝑉⁽𝑥₁, 𝑥₂⁾ = 𝑥²+𝑥² - musbat aniqlangan, sistemaga ko’ra hisoblangan hosilasi
1 2

𝑑𝑉 𝜕𝑉
=

𝑑𝑥_{1 +} 𝜕𝑉
^𝑑𝑥2 = −2(𝑥⁴ + 𝑥⁴) < 0

𝑑𝑡 𝜕𝑥₁
𝑑𝑡 𝜕𝑥₂
𝑑𝑡 ^{1 2}

𝑥₁ = 0, 𝑥₂ = 0 yechim asimptotik turg’un bo’ladi.

2. misol

^𝑑𝑥1 = −𝑥

+ 𝑥

− 𝑥³,

𝑑𝑡
1 2 1

^𝑑𝑥2 = −𝑥

− 𝑥

+ 𝑥³

𝑑𝑡
1 2 2

Sistemani 𝑥₁ = 0, 𝑥₂ = 0 yechimni tekshiramiz.
Lyapunov funksiyasi sifatida 𝑉 = 𝑥² + 𝑥² musbat aniqlangan funksiyani olamiz,
1 2
u holda sistemaga ko’ra hisoblangan hosilasi
^𝑑𝑉 = −2⁽𝑥² + 𝑥² − 𝑥⁴ − 𝑥⁴⁾,

𝑑𝑡
1 2 1 2

𝑥² + 𝑥² < 1 doirada manfiy aniqlanganligi sababli sistemaning trivial yechimi asiptotik
1 2
turg’un bo’ladi.

Takrorlash uchun savollar:

1. 
   Differensial tenglamalarning normal sistemasini yozing?
   
2. 
   Lyapunovning turg’unlik haqidagi teoremasini ayting.
   
3. 
   Lyapunovning asimptotik turg’unlik haqidagi teoremasini ayting.
   

4.
kerak?
y  (t)
harakat asimptotik turg’un bo’lishi uchun, qanday shartlar bajarilishi

5. 
   Nol yechim asimptotik turg’un bo’lishi uchun qanday shartlar bajarilishi kerak?
   
6. 
   Normal sistemaga mos bo’lgan birinchi yaqinlashish tenglamasi qanday tuziladi.
   

Mustaqil ishlash uchun misollar

Quyidagi misollarda berilgan yechimlarning turg’unligini, turg’unlik ta’rifidan foydalanib, tekshiring:

1. ^𝑑𝑦 + 𝑦 = 0, 𝑦⁽𝑡

⁾ = 𝑦⁰.

𝑑𝑥 ⁰

2. ^𝑑𝑦 + 𝑦 = 1, 𝑦⁽0⁾ = 1.
𝑑𝑥

3. ^𝑑𝑦 = 𝑎²𝑦, 𝑦⁽𝑡

⁾ = 𝑦⁰, 𝑎 ≠ 0.

𝑑𝑥 ⁰
Quyidagi tenglamalar sistemasining trivial yechimini turg’unligini tekshiring:

1. ^𝑑𝑥1 = 𝑥
− 𝑥³, ^𝑑𝑥2 = −𝑥
− 7𝑥³.

𝑑𝑡
2 1 _𝑑𝑡 1 2

2. ^𝑑𝑥1 = −2𝑥
− 𝑥
+ 2𝑥
𝑥³ − 3𝑥⁴,

𝑑𝑡
1 2 1 2 1

Berilgan Lyapunov funksiyadan foydalanib sistemani nol yechimini turg’unligini ko’rsating

1. ^𝑑𝑥1 = −𝑥
𝑒^𝑥1 − 𝑥 , ^𝑑𝑥2 = 𝑥³, 𝑉 = 𝑥⁴ + 𝑥⁴.

𝑑𝑡 ¹
2 _𝑑𝑡 1 1 2

2. ^𝑑𝑥1 = −𝑓⁽𝑥 ⁾ − 𝑥³, ^𝑑𝑥2 = 𝑥³ − 𝜑⁽𝑥 ⁾,

𝑑𝑡
1 2 _𝑑𝑡 1 2

Birinchi yaqinlashishlar yordamida sistemaning 𝑥₁ = 0, 𝑥₂ = 0
muvozanat nuqtasining turg’unligini tekshiring:

1.^𝑑𝑥1 = 2𝑒^𝑥1 + 5𝑥
− 2 + 𝑥⁴ , ^𝑑𝑥2 = 𝑥
+ 6 cos 𝑥
− 6 − 𝑥².

𝑑𝑡
2 1 _𝑑𝑡 1 2 2

2.^𝑑𝑥1 = 10 sin 𝑥
− 29𝑥
+ 3𝑥³ , ^𝑑𝑥2 = 5𝑥
− 14 sin 𝑥
+ 𝑥².

𝑑𝑡
1 2 2
_𝑑𝑡 1 2 2

3.^𝑑𝑥1 = −4𝑥
− 𝑥³ , ^𝑑𝑥2 = 3𝑥
− 𝑥³.

𝑑𝑡
2 1 _𝑑𝑡 1 2