❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ▲×❯ ❱❿◆ ❙⑩❯ ❙Ü ❉❆❖ ✣❐◆● ❱⑨ ❍Ó❚ ❚❘❖◆● ▼➷ ❍➐◆❍ ▲❖●■❙❚■❈ ❈➶ ❚❘➍ ❘❮■ ❘❸❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ▲×❯ ❱❿◆ ❙⑩❯ ❙Ü ❉❆❖ ✣❐◆● ❱⑨ ❍Ó❚ ❚❘❖◆● ▼➷ ❍➐◆❍ ▲❖●■❙❚■❈ ❈➶ ❚❘➍ ❘❮■ ❘❸❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽ ✹✻ ✵✶ ữớ ữợ ❦❤♦❛ ❤å❝✿ ❚❙✳ ◆●❯❨➍◆ ❱❿◆ ❑❍❷■ ❍➔ ◆ë✐ ✲ ✷✵✶✽ ▲❮■ ❈❷▼ ❒◆ ▲í✐ ✤➛✉ t✐➯♥ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t tỵ✐ t ữợ ữớ ữợ t t t ữợ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ P❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❣✐↔♥❣ ❞↕② ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ❧✉ỉ♥ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❍➔ ◆ë✐✱ ♥❣➔② ✵✾ t❤→♥❣ ✵✾ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ▲÷✉ ❱➠♥ ổ ữợ sỹ ữợ t ❱➠♥ ❑❤↔✐✱ ❧✉➟♥ ✈➠♥ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐✿ ✧❙ü ❞❛♦ ✤ë♥❣ ✈➔ ❤ót tr♦♥❣ ♠ỉ ❤➻♥❤ ❧♦❣✐st✐❝ ❝â tr➵ rí✐ r↕❝✧ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜ð✐ sü ♥❤➟♥ t❤ù❝ ✈➔ t➻♠ ❤✐➸✉ ❝õ❛ ❜↔♥ t❤➙♥ t→❝ ❣✐↔✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ t→❝ ❣✐↔ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ ❦➳t q✉↔ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥ ❍➔ ◆ë✐✱ ♥❣➔② ✵✾ t❤→♥❣ ✵✾ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ▲÷✉ ❱➠♥ ❙→✉ Mục lục MỞ ĐẦU Chương Kiến thức chuẩn bị 1.1 Giới hạn 1.2 Tính ổn định tính hút điểm cân 1.2.1 Ôn định ổn định tuyến tính hóa 1.2.2 Tính hút tồn cục điểm cân dương 18 1.2.3 Dao động 24 Chương Sự dao động hút mơ hình logistic có trễ rời rạc 25 1.3 Mở đầu 25 1.4 Điều kiện cần đủ cho dao động 28 1.5 Tính bị chặn ổn định tiệm cận 33 1.6 Tính hút toàn cục điểm cân dương 34 Kết luận 42 Tài liệu tham khảo 43 é ỵ t ❤✐➺♥ t÷đ♥❣ ❝õ❛ s✐♥❤ ❤å❝ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥✳ ❈❤➼♥❤ ✈➻ ✈➟② ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ♥❣➔② ❝➔♥❣ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ αNt ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ▼ỉ ❤➻♥❤ ❧♦❣✐st✐❝ Nt+1 = ❧➔ ♠ët tr÷í♥❣ ❤đ♣ + βNt−k r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤ú✉ t✛✳ P✐❡❧♦✉ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ αNt ♠ỉ ❤➻♥❤ ❧♦❣✐st✐❝ Nt+1 = ✱ ♠ỉ t↔ ♠ët ❤✐➺♥ t÷đ♥❣ ❝õ❛ s✐♥❤ t❤→✐✳ + βNt−k ợ ố ữủ t tr t ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ αxn , n = 0, 1, ✱ ✈➔ ù♥❣ ❞ö♥❣ t❤ü❝ t✐➵♥ ❝õ❛ P❚ s❛✐ xn+1 = + xnk õ ữợ sỹ ủ ỵ t ữợ tổ t ❝ù✉ ✏❙ü ❞❛♦ ✤ë♥❣ ✈➔ ❤ót tr♦♥❣ ♠ỉ ❤➻♥❤ ❧♦❣✐st✐❝ ❝â tr➵ rí✐ r↕❝✑ ✳ ▲✉➟♥ ✈➠♥ ❞ü❛ ❝❤➼♥❤ ✈➔♦ ❬✹❪ ✈➔ ❬✺❪✱ ❝ư t❤➸✿ ❈❤÷ì♥❣ ✶ ❧✉➟♥ ✈➠♥ ❞ü❛ ✈➔♦ t➔✐ ❧✐➺✉ ❬✹❪✱ ❝❤÷ì♥❣ ✷ ❧✉➟♥ ✈➠♥ ❞ü❛ ✈➔♦ t➔✐ ❧✐➺✉ ❬✺❪✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ xn+1 = αxn , n = 0, 1, + βxn−k ✹ (1) ✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ▲➔♠ rã t➼♥❤ ❞❛♦ ✤ë♥❣✱ ❜à ❝❤➦♥✱ ❤ót ✈➔ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ✭✶✮ ✈➔ ✤÷❛ r❛ ♠ët sè ✈➼ ❞ư ❝ư t ố tữủ ự ã P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ αxn • P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ xn+1 = , n = 0, 1, + βxn−k ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❞ü ❦✐➳♥ ✤÷đ❝ ❞ị♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❣✐↔✐ t➼❝❤ t♦→♥ ❤å❝✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✳ ✻✳ ✣â♥❣ ❣â♣ ♠ỵ✐ ❈è ❣➢♥❣ ①➙② ❞ü♥❣ ❧✉➟♥ t ởt t ữợ t ✤ë♥❣✱ ❜à ❝❤➦♥✱ ❤ót ✈➔ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ✭✶✮✳ ✺ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ●✐ỵ✐ ❤↕♥ ●✐↔ sû (cn )∞ n=1 ❧➔ ♠ët ❞➣② ❜à ❝❤➦♥ ❝→❝ sè tỹ ỵ an = inf {ck : k n} , bn = sup {ck : k ≥ n} ❉➣② (an ) ❧➔ ❜à ❝❤➦♥ ✈➔ t➠♥❣✱ ❞♦ ✈➟②✱ ♥â ❝â ❣✐ỵ✐ ❤↕♥ ❧➔ a✱ ❣✐ỵ✐ ❤↕♥ ♥➔② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ lim inf ❝õ❛ ❞➣② (cn ) ữủ ỵ lim inf cn = lim inf {ck : k ≥ n} n n→∞ ❚÷ì♥❣ tü✱ ❞➣② (bn ) ❧➔ ❜à ❝❤➦♥ ✈➔ ❣✐↔♠✱ ❞♦ ✈➟②✱ ♥â ❝â ❣✐ỵ✐ ❤↕♥ ❧➔ b✱ ❣✐ỵ✐ ❤↕♥ ♥➔② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ lim sup ❝õ❛ ❞➣② (cn ) ữủ ỵ lim supcn = lim sup {ck : k ≥ n} n n→∞ ❱➼ ❞ö ✶✳✶✳ ✶✮ ❳➨t✿ (un ) = (−1)n ✻ ❑❤✐ ✤â✿ lim u2k = lim (−1)2k = 1; k→∞ k→∞ lim u2k+1 = lim (−1)2k+1 = −1 k→∞ k→∞ ❱➟②✿ lim sup un = 1; n lim inf un = −1 n ✷✮ ❳➨t✿ (un ) = (−1)n ❑❤✐ ✤â✿ n = 0; k→∞ 2k = lim (−1)2k+1 = k→∞ 2k + lim u2k = lim (−1)2k k→∞ lim u2k+1 k→∞ ❱➟②✿ lim sup un = 0; n lim inf un = n ✸✮ ❳➨t✿ (un ) = cos 2nπ ❑❤✐ ✤â✿ lim u3k = lim cos 2kπ = 1; k→∞ k→∞ 2π =− ; k→∞ 4π = lim cos =− k→∞ lim u3k+1 = lim cos k→∞ lim u3k+2 k→∞ ❱➟②✿ lim sup un = 1; n lim inf u = − n n ✼ ✶✳✷✳ ❚➼♥❤ ê♥ ✤à♥❤ ✈➔ t➼♥❤ ❤ót ❝õ❛ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❈→❝ ♥ë✐ ❞✉♥❣ ð ♠ư❝ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ tø ❬✹❪ ✈➔ ❬✺❪✳ ✶✳✷✳✶✳ ✃♥ ✤à♥❤ ✈➔ ê♥ ✤à♥❤ t✉②➳♥ t➼♥❤ ❤â❛ ●✐↔ sû I ❧➔ ♠ët ❦❤♦↔♥❣ ❝→❝ sè t❤ü❝ ✈➔ f : I × I → I ❧➔ ♠ët ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 , x−1 ∈ I, ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ xn+1 = f (xn , xn−1 ) , n = 0, 1, ✭✶✳✶✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t {xn }∞ n=−1 ▼ët ✤✐➸♠ x ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ✭❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t✮ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✶✳✶ ♥➳✉ ❝â✿ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ x = f (x, x) ♥❣❤➽❛ ❧➔ xn = x, n ≥ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣✱ x ❧➔ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ f ✳ ❱➼ ❞ư ✶✳✷✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ xn+1 = αxn + βxn−1 (∗) ❚r♦♥❣ ✤â✱ x−1 , x0 ∈ R ✈➔ x−1 x0 < 0, α ∈ (1; +∞) , β ∈ (0; +∞) ●å✐ x ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✯✮✳ ❑❤✐ ✤â✱ t❛ ❝â✿ x = f (x, x) = αx −1 , x= + βx β ✽ ❝â ♥❣❤✐➺♠ ❞÷ì♥❣ ❝✉è✐ ❝ị♥❣ {xn}✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❤â❛ ✭✷✳✶✹✮ ❝ơ♥❣ ❝â ♥❣❤✐➺♠ ❞÷ì♥❣ ❝✉è✐ ❝ị♥❣✳ ●✐↔ t❤✐➳t ✭✷✳✾✮ ✤ó♥❣ ✈➔ ❝❤♦ λ0 ❧➔ ♠ët ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ λ − + pλ−k = ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮✳ ❈❤♦ N1 ∈ N, N1 ≥ ✈➔ ϕ ∈ (0; ∞) ✈➔ ❣✐↔ sû {cn } ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❇ê ✤➲ ✷✳✷✳ cn+1 − cn + pcn−k ≥ 0, n = 0, 1, , N1 − ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ cn = ϕλn0 , n = −k, , t❤➻ t❛ ❝â cn ≥ ϕλn0 , n = 1, 2, , N1 ự ỵ ✭❇➡♥❣ ♣❤↔♥ ❝❤ù♥❣✮ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ ❝â ♥❣❤✐➺♠ {xn } ❦❤ỉ♥❣ ❞❛♦ ✤ë♥❣✱ ✈ỵ✐ {xn } ❧➔ ♠ët ♥❣❤✐➺♠ ❞÷ì♥❣ ❝✉è✐ ❝ị♥❣✳ ❚r÷í♥❣ ❤đ♣ {xn } ❧➔ ♥❣❤✐➺♠ ➙♠ ❝✉è✐ ❝ị♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳ ❉➵ t❤➜② limxn = n→∞ ✳ ❇➙② ❣✐í t❛ ❝â t❤➸ ✈✐➳t ❧↕✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ ♥❤÷ s❛✉✿ xn+1 − xn + p(n)xn−k = tr♦♥❣ ✤â p(n) = p f (xn−k ) xn−k ❚ø ✭✷✳✶✶✮ t❛ ❝â lim inf p(n) ≥ p ✈➔ ❣✐↔ t❤✐➳t ❝õ❛ ❇ê ✤➲ ✷✳✶ ✤÷đ❝ n→∞ t❤ä❛ ♠➣♥✳ ❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ ❝ơ♥❣ ❝â ữỡ ố ũ t ợ tt ỵ ữủ ự ự ỵ tt ú ợ f (u) u ợ u rữớ ❤đ♣ f (u) ≥ u ✈ỵ✐ −δ ≤ u ≤ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ ❝â ♠ët ♥❣❤✐➺♠ ❞÷ì♥❣ ❝✉è✐ ❝ị♥❣ {yn }✳ ❚❤➻ t❤❡♦ ❍➺ q✉↔ ✼✳✶✳✶ ✭t➔✐ ❧✐➺✉ ❬✷❪✮ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✳✼✮ λ − + pλ−k = ❝â ♠ët ♥❣❤✐➺♠ ❞÷ì♥❣ λ0 ✳ ❚÷ì♥❣ tü p > 0, λ0 ∈ (0, 1)✳ ❈❤♦ {xn } ởt ữỡ tr ợ ❜❛♥ ✤➛✉ xn = ϕλn0 ✈ỵ✐ n = −k, , −1, tr♦♥❣ ✤â ϕ = δλk00 ✣➸ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤ t❛ ❝❤➾ r❛ xn > ✈ỵ✐ n = 1, 2, ữủ tỗ t số N1 ≥ s❛♦ ❝❤♦ xn > ✈ỵ✐ −k ≤ n < N1 ✈➔ xN1 ≤ ❚ø ✭✷✳✽✮ t❛ t❤➜② r➡♥❣ xn+1 < xn ✈ỵ✐ ≤ n ≤ N1 − ❱➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝ư t❤➸ < xn < x0 = ϕ = δλk00 < δ ✈ỵ✐ n = 1, 2, , N1 − ❚❤➻ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ✭✷✳✶✸✮ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ t❛ ❝â ❜➜t ♣❤÷ì♥❣ tr➻♥❤ xn+1 − xn + pxn−k ≥ ✈ỵ✐ n = 0, 1, , N1 − 1 ❚❤❡♦ ❇ê ✤➲ ✷✳✷ t❛ ❝â xN1 ≥ ϕλN > ❱➔ ✈ỵ✐ ♠➙✉ t❤✉➝♥ ♥➔② ✤à♥❤ ỵ ữủ ự ữỡ tr ●✐↔ sû p ∈ (0; ∞) ✈➔ k ∈ {1, 2, }✳ ❚❤➻ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ yn+1 − yn + pyn−k = ❞❛♦ ✤ë♥❣ ①✉♥❣ q✉❛♥❤ ✤✐➸♠ ✵ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ p > kk (k + 1)k+1 ✳ ❚❤➟t ✈➟②✱ t❛ ❝â t❤➸ t❤➜② r➡♥❣ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ x(t) + px(t − τ ) = tr♦♥❣ ✤â p, τ ∈ R ❞❛♦ ✤ë♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ p, τ, e > ❇➙② ❣✐í q✉❛♥ s→t ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❇ê ✤➲ ✷✳✸ õ t ữủ t ữợ (k + 1)k >1 ✭✷✳✶✼✮ p(k + 1) kk (k + 1)k k tr♦♥❣ ✤â = (1 + ) ↑ e ❦❤✐ k → ∞✳ ❉♦ ✤â t❛ ❝â t❤➸ t❤➜② kk k tữỡ tỹ rớ r ợ tr (k + 1) ự ỵ ✷✳✶✿ α − yn e ❱➔ ❜✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮ t❛ ✤÷đ❝ β ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ s❛✉ ✣ê✐ ❜✐➳♥ xn = yn+1 − yn + tr♦♥❣ ✤â α−1 f (yn−k ) = 0, n = 0, 1, 2, α α (α − 1) eu + f (u) = ln α−1 α ✸✷ ✭✷✳✶✽✮ ❉♦ ✤â✱ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✵✮ ❞❛♦ ✤ë♥❣ ①✉♥❣ q✉❛♥❤ ✤✐➸♠ ✵ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮ ❞❛♦ ✤ë♥❣ α−1 ①✉♥❣ q✉❛♥❤ ✤✐➸♠ ✳ ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ t❤➜② r➡♥❣ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ β ❇ê ✤➲ ✷✳✽✳✷✱ ✷✳✽✳✸✱ ✷✳✽✳✺✱ ✷✳✽✳✻ ✤÷đ❝ tọ ố ợ ữỡ tr s t ữ ỵ r ữủ tọ f (u) u ợ u < Pữỡ tr t t t ợ ữỡ tr zn+1 zn + α−1 zn−k = α ✭✷✳✶✾✮ ❱➔ t❤❡♦ ❇ê ✤➲ ✷✳✸ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✶✮ ❞❛♦ ✤ë♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✭✷✳✼✮ ❝è ✤à♥❤✳ ❚❤➻ ❝❤ù♥❣ ♠✐♥❤ ❜➙② ❣✐í ❧➔ ♠ët ❤➺ q✉↔ ❝ì ❜↔♥ ❝õ❛ ❍➺ q✉↔ ✷✳✶ ✈➔ ❇ê ✤➲ ✷✳✸✳ ✷✳✸✳ ❚➼♥❤ ❜à ❝❤➦♥ ✈➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❜à ❝❤➦♥✳ ❱➔ t❤✐➳t ❧➟♣ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮✳ ●✐↔ t❤✐➳t α, β ∈ (0; ∞) ✈➔ k ∈ R✳ ❚❤➻ ♠é✐ ♥❣❤✐➺♠ {xn} ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❝â ❝→❝ ❣✐→ trà ❜❛♥ ✤➛✉ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✻✮ ❧➔ ❜à ❝❤➦♥ ✈➔ αk+1 xn ≤ ✈ỵ✐ n k + ỵ ự ♠✐♥❤✳ ❉♦ xn > ✈ỵ✐ n ≥ ✈➔ xn+1 ≤ αxn ✈ỵ✐ n ≥ 0✳ ❉♦ ✤â ✈ỵ✐ n≥k xn+1 αxn ≤ < βxn−k αxn β xn+1−k α ✸✸ αk+1 ≤ ≤ β ✈➔ ❝❤ù♥❣ ♠✐♥❤ ữủ t rữợ t t t ❝➟♥ ❝õ❛ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❝❤ó♥❣ t❛ ❝➛♥ ❦➳t q✉↔ s❛✉ ❝õ❛ ▲❡✈✐♥ ✈➔ ▼❛② ❬✺❪✳ ❇ê ✤➲ ✷✳✹✳ ●✐↔ t❤✐➳t p ∈ R ✈➔ k ∈ N✳ ❚❤➻ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ xn+1 − xn + pxn−k = 0, n = 0, 1, 2, ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♥➳✉ < p < cos kπ 2k + ✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❇ê ✤➲ ✷✳✹✱ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♥➳✉ α−1 kπ < p < cos 2k + ú ỵ r ữủ tọ ợ ∈ (0; ∞) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ k = ✈➔ k = 1✳ ❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦ ♠é✐ tr÷í♥❣ ❤đ♣ ♥❤÷ tr➯♥✱ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❧➔ ❤ót t♦➔♥ ❝ö❝✳ ❑❤✐ k = 2, 3, ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✈ỵ✐ 1 ❧➔ ❤ë✐ tử ỡ tợ ữỡ β ❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ①➨t ✸ tr÷í♥❣ ❤đ♣ t❤❡♦ tø♥❣ ❣✐→ trà ❜❛♥ ✤➛✉ x0✳ ❚r÷í♥❣ ❤đ♣ ✶✿ ◆➳✉ x0 = α−1 ✳ ❑❤✐ ✤â β α−1 α (α − 1) αx0 α−1 β β x1 = = = ⇒ x1 = α−1 + βx0 α β 1+β β α ●✐↔ sû xk = α−1 ✱ ❦❤✐ ✤â β α−1 α (α − 1) α−1 β β = = ⇒ xk+1 = α−1 α β 1+β β α xk+1 ❱➟② x0 = α−1 α−1 t❤➻ xn = , ∀n = 0, 1, β β ❚r÷í♥❣ ❤đ♣ ✷✿ α−1 t❤➻ βx0 > α − ✭✯✮ β αx0 ❳➨t x1 = ❦❤✐ ✤â + βx0 ◆➳✉ x0 > x1 − α−1 αx0 α−1 = − β + βx0 β = αβx0 − (α − 1) (1 + βx0 ) β (1 + βx0 ) ✸✺ = αβx0 − α + − αβx0 + βx0 β (1 + βx0 ) βx0 − (α − 1) >0 β (1 + βx0 ) α−1 ❱➟② x1 > ✳ β α−1 ●✐↔ sû xk > ⇒ βxk > α − ✭✯✯✮✳ ❳➨t β α−1 αxk α−1 xk+1 − = − β + βxk β = = αβxk − (α − 1) (1 + βxk ) β (1 + βxk ) = αβxk − α + − αβxk + βxk β (1 + βxk ) βxk − (α − 1) >0 β (1 + βxk ) α−1 ❱➟② xk+1 > ✳ β α−1 α−1 ❱➟② x0 > t❤➻ xn > , ∀n = 0, 1, β β = ❚✐➳♣ t❤❡♦✱ ①➨t xn+1 − xn = = αxn − xn + βxn αxn − xn − βxn + βxn (α − 1) xn − βxn = + βxn α−1 − xn β = βxn < 0, ∀n = 0, 1, + βxn ✸✻ ❱➟② xn+1 − xn < 0, ∀n = 0, 1, s✉② r❛ ❞➣② {xn } ỡ 1 ữợ õ tỗ t lim xn = x ổ ọ ❤ì♥ ✮✳ n→∞ β β αxn αx ◆❤÷ ✈➟② ❝❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ xn+1 = ❝â x = + βxn + βx α−1 ✳ ⇒ x = ❤♦➦❝ x = β α−1 α−1 ❱➻ xn > > 0, ∀n = 0, 1, ⇒ x = ✳ β β ❚r÷í♥❣ ❤đ♣ ✸✿ α−1 α−1 t❤➻ xn < , ∀n = β β α−1 0, 1, ✱ xn+1 > xn ✱ tø ✤â ❞➣② {xn } ❝â ợ x = ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚÷ì♥❣ tü ♥❤÷ ð tr÷í♥❣ ❤đ♣ ✷✳ x0 < ỵ t t tr÷í♥❣ ❤đ♣ k = ✈➔ ❝❤➾ r❛ r➡♥❣✱ ♥❤÷ tr÷í♥❣ ❤đ♣ k = 0✱ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❧➔ út t ợ trữớ ủ k = 0✱ sü ❤ë✐ tư ❝â t❤➸ ❦❤ỉ♥❣ ✤ì♥ ✤✐➺✉✳ ●✐↔ sû α ∈ (1; ∞) ✈➔ β ∈ (0; ∞)✳ ❑❤✐ ✤â ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ỵ xn+1 = ợ tử tợ ✤✐➸♠ ❝➙♥ ❜➡♥❣ αxn , n = 0, 1, 2, + βxn−1 x−1 ≥ ✈➔ x0 > ❞÷ì♥❣ α − ✳ β ✭✷✳✷✷✮ ✭✷✳✷✸✮ ❈❤ù♥❣ ♠✐♥❤✳ ✣ê✐ ❜✐➳♥ xn = α − yn + ✈ỵ✐ n ≥ −1 β β ✸✼ ✭✷✳✷✹✮ ❚❤❛② ✭✷✳✷✻✮ ✈➔♦ ✭✷✳✷✹✮ ❝â ⇒ (α − 1) + yn+1 β α − yn+1 + β β α − yn + β β = α − yn−1 1+β + β β α α [(α − 1) + yn ] β = α + yn−1 ⇒ (α − 1) + yn+1 = α (α − 1) + αyn α + yn−1 ⇒ yn+1 = α (α − 1) + αyn − (α − 1) α + yn−1 ⇒ yn+1 = α (α − 1) + αyn − α (α − 1) − (α − 1) yn−1 α + yn−1 ⇒ yn+1 = αyn − (α − 1) yn−1 α + yn−1 ❱➟②✱ yn+1 = αyn − (α − 1) yn−1 , n = 0, 1, 2, α + yn−1 ✭✷✳✷✺✮ ❍ì♥ ♥ú❛✱ ❞♦ ✭✷✳✷✺✮ ♥➯♥ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✹✮ ❧➔ ❞÷ì♥❣ ✈➔ ❞♦ ✭✷✳✷✻✮ α + yn > ợ n ữ ỵ r➡♥❣ ✈ỵ✐ n ≥ −1 α (yn − yn−1 ) + yn−1 , α + yn−1 (α − 1) + yn yn+1 − yn = − yn−1 α + yn−1 yn+1 = ✭✷✳✷✼✮ ✭✷✳✷✽✮ ❚❛ ❝â yn+2 = α (yn+1 − yn ) + yn = α + yn α ✸✽ α (yn − yn−1 ) + yn−1 − yn + yn α + yn−1 α + yn α ⇒ yn+2 = α (yn − yn−1 ) + yn−1 − yn (α + yn−1 ) + yn α + yn−1 α + yn ⇒ yn+2 = α [yn−1 (1 − α) − yn yn−1 ] + yn (α + yn−1 ) (α + yn ) (α + yn−1 ) ⇒ yn+2 = α (1 − α) yn−1 − αyn yn−1 + αyn + yn yn−1 (α + yn ) (α + yn−1 ) ⇒ yn+2 = αyn α (1 − α) yn−1 − αyn yn−1 + yn yn−1 + (α + yn ) (α + yn−1 ) (α + yn ) (α + yn−1 ) ⇒ yn+2 = αyn α (1 − α) yn−1 + (1 − α) yn yn−1 + (α + yn ) (α + yn−1 ) (α + yn ) (α + yn−1 ) ⇒ yn+2 = αyn (1 − α) yn−1 (α + yn ) + (α + yn ) (α + yn−1 ) (α + yn ) (α + yn−1 ) ⇒ yn+2 = αyn (α − 1) yn−1 − (α + yn ) (α + yn−1 ) α + yn−1 ❱➟②✱ yn+2 = αyn (α − 1) yn−1 − (α + yn−1 )(α + yn ) α + yn−1 ✭✷✳✷✾✮ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ♥➳✉ ❝❤ó♥❣ t❛ ❝❤➾ r❛ r➡♥❣ lim yn = t→0 ✭✷✳✸✵✮ ❚r♦♥❣ ❝ỉ♥❣ t❤ù❝ ✭✷✳✷✽✮✱ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ✭✷✳✸✵✮ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✼✮ ❧➔ rèt ❝✉ë❝ ❦❤ỉ♥❣ ➙♠ ❤♦➦❝ rèt ❝✉ë❝ ❦❤ỉ♥❣ ❞÷ì♥❣ ❤ë✐ tư ✤ì♥ ✤✐➺✉ ✈➲ ✵✳ ❉♦ ✤â✱ ✤➸ ❝â ❣✐ỵ✐ ợ ộ ữỡ tr ❞❛♦ ✤ë♥❣ ✏❝❤➦t✑✱ t❤❡♦ ♥❣❤➽❛ ♥â ✤↕t ✤÷đ❝ ❝↔ ❤❛✐ ❣✐→ trà ➙♠ ✈➔ ❞÷ì♥❣✳ ▼é✐ ♥❣❤✐➺♠ ❝❤ù❛ ♠ët ✏❝❤✉é✐✑ ❧✐➯♥ t✐➳♣ ❝→❝ ♣❤➛♥ tû ➙♠ ✤÷đ❝ s✐♥❤ ✸✾ ❜ð✐ ♠ët ❝❤✉é✐ ❧✐➯♥ t✐➳♣ ❝→❝ ♣❤➛♥ tû ❦❤æ♥❣ ➙♠✱✳ ✳ ✳ ❈❤ó♥❣ t❛ ❣å✐ ♥❤ú♥❣ ❝❤✉é✐ ♥➔② ❧➔ ❝❤✉é✐ ❜→♥ ❝❤✉ ❦ý ➙♠ ✈➔ ❜→♥ ❝❤✉ ❦ý ❞÷ì♥❣✳ ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ ✭✷✳✷✾✮ ✈➔ ✭✷✳✸✵✮ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ ♠é✐ ❜→♥ ❝❤✉ ❦ý ❝❤ù❛ ➼t ♥❤➜t ✸ ♣❤➛♥ tû ✈➔ ♣❤➛♥ tû t❤ù ✸ tr♦♥❣ ❜→♥ ❝❤✉ ❦ý ❞÷ì♥❣ ❧➔ ❞÷ì♥❣✳ ❚r➯♥ t❤ü❝ t➳ ♥➳✉ yn−1 < ✈➔ yn ≥ 0✱ t❤➻ tø ✭✷✳✷✽✮ ✈➔ ✭✷✳✸✵✮ t❛ t❤➜② yn+1 > yn ≥ 0✳ ❚÷ì♥❣ tü tø ✭✷✳✷✽✮ ✈➔ ✭✷✳✸✶✮ t❛ ❝â yn+2 > 0✳ ●✐→ trà ❧ỵ♥ ♥❤➜t tr♦♥❣ ❜→♥ ❝❤✉ ❦ý ❞÷ì♥❣ ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t tr♦♥❣ ❜→♥ ❝❤✉ ❦ý ➙♠ ❜➡♥❣ ✈ỵ✐ ❣✐→ trà ❝õ❛ ♣❤➛♥ tû t❤ù ✷ tr♦♥❣ ♠é✐ ❜→♥ ❝❤✉ ❦ý t÷ì♥❣ ù♥❣✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ♠é✐ ❜→♥ ❝❤✉ ❦ý ❞÷ì♥❣✳ ❱➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❜→♥ ❝❤✉ ❦ý ➙♠ ❧➔ t÷ì♥❣ tü ✈➔ ✤÷đ❝ ❜ä q✉❛✳ ❚❤ü❝ t➳✱ ♥❤÷ ❝❤ó♥❣ t❛ t❤➜② ð tr➯♥ ♣❤➛♥ tû t❤ù ❤❛✐ tr♦♥❣ ❜→♥ ❝❤✉ ý ữỡ ợ ỡ tỷ tự t ố ❝ị♥❣ ✭✷✳✸✵✮ ✤÷đ❝ ❜➢t ✤➛✉ tø ♣❤➛♥ tû t❤ù ✸ ✈➔ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❜→♥ ❝❤✉ ❦ý ❞÷ì♥❣ ❧➔ ❣✐↔♠✳ ✣➸ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤✱ ①➨t ✹ ❜→♥ ❝❤✉ ❦ý ❧✐➯♥ t✐➳♣ Cr−1 , Cr , Cr+1 , Cr+2 ✈ỵ✐ Cr−1 ❧➔ ♠ët ❜→♥ ❝❤✉ ❦ý ➙♠✱ ♥❤÷ s❛✉✿ Cr−1 = {yk+1 , yk+2 , , yl } ❜→♥ ❝❤✉ ❦ý ➙♠✱ Cr = {yl+1 , yl+2 , , ym } ❜→♥ ❝❤✉ ❦ý ❞÷ì♥❣✱ Cr+1 = {ym+1 , ym+2 , , yn } ❜→♥ ❝❤✉ ❦ý ➙♠✱ Cr+2 = {yn+1 , yn+2 , , yq } ý ữỡ ú t ữợ ữủ s❛✉✱ tø ✤â ❈▼ ❝õ❛ ✭✷✳✸✷✮ trð ♥➯♥ rã r➔♥❣ (α − 1)2 |ym+2 | ≤ |yk+2 | α −α+1 ✈➔ yn+2 ≤ α−1 α ✹✵ ✭✷✳✸✶✮ yl+2 ✭✷✳✸✷✮ ✣➸ ❦➳t t❤ó❝ ✤✐➲✉ ♥➔②✱ t❛ q✉❛♥ s→t tø ✭✷✳✸✶✮ (α − 1) yl−1 yl+2 ≤ − α + yl−1 ✈➔ tø ❤➔♠ x t➠♥❣ ♥❣➦t ✤è✐ ✈ỵ✐ x✱ α+x (α − 1) |yl−1 | yl+2 ≤ α − |yl−1 | ✭✷✳✸✸✮ ❑❤✐ ✤â✱ t❛ ❝â (α − 1) ym−1 α + ym−1 (α − 1) yl+2 ≤ α + yl+2 (α − 1) (α − 1) |yk+2 | α − |yk+2 | ≤ α + (α − 1) |yk+2 | α − |yk+1 | (α − 1)2 |yk+2 | = α2 − |yk+2 | |ym+2 | < ✭✷✳✸✹✮ (α − 1)2 ❙✉② r❛ |yk+2 | < α − 1✱ ❞♦ ✤â |ym+2 | ≤ |yk+2 | α −α+1 ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✭✷✳✸✸✮ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ❚÷ì♥❣ tü ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ (α − 1) |ym+2 | (α − 1) yn−1 ≤ , yn+2 ≤ − α + yn−1 α − |ym+2 | ❙û ❞ö♥❣ ✭✷✳✸✻✮ ✈➔ t➼♥❤ t➠♥❣ ❝õ❛ ❤➔♠ yn+2 x ✱ ❝❤ó♥❣ t❛ t❤➜② α+x (α − 1) (α − 1) yl+2 (α − 1)2 yl+2 (α − 1)2 α + yn+2 ≤ = < yk+2 α − (α − 1) yl+2 α2 + yl+2 α2 α + yl+2 ◆❤÷ ✈➟② ✭✷✳✸✹✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ỵ ữủ ự ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ✈➲ ❝→❝ tr↕♥❣ t❤→✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤ú✉ t✛ ❞↕♥❣ xn+1 = αxn , n = 0, 1, + βxn−k P❤➛♥ ✤â♥❣ ❣â♣ ❝õ❛ t→❝ ❣✐↔✿ • ◆❣❤✐➯♥ ❝ù✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët õ tố ữỡ tr ã t✐➳t ❤â❛ ð tr❛♥❣ ✷✻✳ • ❈❤✐ t✐➳t ❤â❛ ❝❤ù♥❣ ỵ ỵ ã ởt sè ✈➼ ❞ö✳ ✹✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▲✳ ❇r❛♥❞ ✭✶✾✺✺✮✱ ❆ s❡q✉❡♥❝❡ ❞❡❢✐♥❡❞ ❜② ❛ ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧② ✻✷✱ ✹✽✾✲✹✾✷✳ ❖s❝✐❧❧❛t✐♦♥ ❚❤❡♦r② ♦❢ ❉❡❧❛② ❉✐❢❢❡r✲ ❡♥❝❡ ❊q✉❛t✐♦♥s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❖①❢♦r❞ ❯♥✐✈✳ Pr❡ss✳ ☎ ❛♥❞ ●✳ ▲❛❞❛s ✭✶✾✾✶✮✱ ❬✷❪ ■✳ ●②❖r✐ ▲✐♥❡❛r✐③❡❞ ♦s❝✐❧❧❛t✐♦♥s ❢♦r ❡q✉❛t✐♦♥s ✇✐t❤ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❛r❣✉♠❡♥ts✱ ❉✐❢❢❡r❡♥t✐❛❧ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ☎ ❛♥❞ ●✳ ▲❛❞❛s ✭✶✾✽✾✮✱ ❬✸❪ ■✳ ●②❖r✐ ❏✳ ✷✱ ✶✷✸✲✶✸✶✳ ❬✹❪ ▼✳ ❘✳ ❙✳ ❑✉❧❡♥♦✈✐❝✱ ●✳ ▲❛❞❛s✱ ✭✷✵✵✶✮✱ ❉②♥❛♠✐❝s ♦❢ s❡❝♦♥❞ ♦r❞❡r r❛t✐♦♥❛❧ ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ ❈❤❛♣♠❛♥ ❛♥❞ ❍❛❧❧✴❈❘❈✳ ❬✺❪ ❙✳ ❙✳ ❑✉r✉❦❧✐s✱ ●✳ ▲❛❞❛s✱ ✭✶✾✾✷✮✱ ❖s❝✐❧❧❛t✐♦♥s ❛♥❞ ❣❧♦❜❛❧ ❛ttr❛❝t✐✈✐t② ✐♥ ❛ ❞✐s❝r❡t❡ ❞❡❧❛② ❧♦❣✐s✐❝ ♠♦❞❡❧✱ ◗✉❛rt❡r❧② ♦❢ ❛♣♣❧✐❡❞ ♠❛t❤❡♠❛t✐❝s✱ ❯❙❆✳ ❘❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥ts ✐♥ t❤❡ ♦s❝✐❧❧❛t✐♦♥ ♦❢ ❞❡❧❛② ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✿ ❙t❛❜✐❧✐t② ❛♥❞ ❈♦♥tr♦❧✱ ❬✻❪ ●✳ ▲❛❞❛s ✭✶✾✾✵✮✱ ▼❛r❝❡❧ ❉❡❦❦❡r✳ ❬✼❪ ❙✳ ▲❡✈✐♥ ❛♥❞ ❘✳ ▼❛② ✭✶✾✼✻✮✱ ❆ ♥♦t❡ ♦♥ ❞✐❢❢❡r❡♥❝❡✲❞❡❧❛② ❡q✉❛t✐♦♥s✱ ❚❤❡♦r❡t✳ P♦♣✉❧❛t✐♦♥ ❇✐♦❧✳ ✾✱ ✶✼✽✲✶✽✼✳ ✹✸ ❇✐♦❧♦❣✐❝❛❧ ♣♦♣✉❧❛t✐♦♥s ♦❜❡②✐♥❣ ❞✐❢❢❡r❡♥❝❡ ❡q✉❛✲ t✐♦♥s✿ st❛❜❧❡ ♣♦✐♥ts✱ st❛❜❧❡ ♣♦✐♥ts✱ st❛❜❧❡ ❝②❝❧❡s✱ ❛♥❞ ❝❤❛♦s✱ ❏✳ ❚❤❡♦✲ ❬✽❪ ❘✳ ▼✳ ▼❛② ✭✶✾✺✺✮✱ r❡t✱ ❇✐♦❧✳ ✺✶✱ ✺✶✶✲✺✷✹✳ ❈♦♠♣❡♥st❛t♦r② r❡❛❝t✐♦♥s ♦❢ ♣♦♣✉❧❛t✐♦♥s t♦ str❡ss❡s t♦ str❡ss❡s ❛♥❞ t❤❡✐r ❡✈♦❧✉t✐♦♥❛r② s✐❣♥✐❢✐❝❛♥❝❡✱ ❆✉str❛❧✳ ❏✳ ❬✾❪ ❆✳ ❏✳ ◆✐❝❤♦❧s♦♥ ✭✶✾✺✹✮✱ ❩♦♦❧✳ ✷✱ ✾✲✻✺✳ ❬✶✵❪ ❊✳ ❈✳ P✐❡❧♦✉ ✭✶✾✻✾✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❛t❤❡♠❛t✐❝❛❧ ❊❝♦❧♦❣②✱ ❲✐❧❡②✲■♥t❡rs❝✐❡♥❝❡✳ ❬✶✶❪ ❊✳ ❈✳ P✐❡❧♦✉ ✭✶✾✼✹✮✱ P♦♣✉❧❛t✐♦♥ ❛♥❞ ❈♦♠♠✉♥✐t② ❊❝♦❧♦❣②✱ ●♦r❞♦♥ ❛♥❞ ❇r❡❛❝❤✱ ◆❡✇ ❨♦r❦✳ ✹✹ ... định tính hút điểm cân 1.2.1 Ôn định ổn định tuyến tính hóa 1.2.2 Tính hút tồn cục điểm cân dương 18 1.2.3 Dao động 24 Chương Sự dao động hút mơ hình logistic có trễ rời rạc 25... rời rạc 25 1.3 Mở đầu 25 1.4 Điều kiện cần đủ cho dao động 28 1.5 Tính bị chặn ổn định tiệm cận 33 1.6 Tính hút toàn cục điểm cân dương 34 Kết luận 42 Tài liệu tham khảo