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Hence, the characteristic equation of (28) is P(A) = X® – nz X³ – na X* – n3 X³ – n2 X² – nd – no, (29) - nA where, (2ε+6μ-2εμ+ &-3) (2 µ (1 – e)) (2ε+4μ-6εμ+2 έμ-+15) (2μ (ε- 1)) (2ε+3μ +2 εμ- μ+2@-12 (2 µ (e – 1)) n5 n4 = (E+2μ-2εμ+3) n2 = n3 (µ (1 – e)) n1 no = -1. Now, let T(A) = 1® and V(A) = -ng 15 – n4 X4 – ng 13 – n2 X² – n1 A – no. Consider |A|=1, then one has (2ε+6μ-2εμ + ε23) (2μ (1 - 0) ) (2e+4μ-6εμ+2ε μ-+15) (2μ (ε - 1) (2ε+3μ +2εμ- μ+2 22-12) (2μ (ε -1) ) (ε+2μ-2εμ+3) (µ (1 – €)) %3D (1– +1, 2 + (2ε46μ-2εμ + 2-3) (4ε+7μ-4εμ+ θμ + +3) (2 µ (1 – €)) (e + 2µ – 2 e µ +3) (1– e) (и (1 — е)) (2 e + µ – 2 €u + e² µ+ 6) , (e +2µ – 2 €µ+3) , (1– e) (2μ (ε - 1) ) (-3μ+2εμ + μ) μ(ε-1) (1-) (2 µ (e – 1)) –3 µ +2 € µ+ e² µ –µ(e – 1)² 2 и (е — 1) (2 µ (e – 1)) +1, + 1, (H (1 – €)) +1, %3D µ(E – 1) -4μ+4με + 1, +1 = 2µ (e – 1) = 3 > 1. Then, from Rouché's Theorem 1.2, V(A) and Y(A) + ¥(A) = P(^) have the same number of zeros in an open unit disk |A| < 1, that means there are only five roots lie inside unit disk. Thus, from Theorem 1.1 the equilibrium point Eo is not asymptotically stable. By solving (29), we find that 1, implies Eo is nonhyperbolic point.

Hence, the characteristic equation of (28) is P(A) = X® – nz X³ – na X* – n3 X³ – n2 X² – nd – no, (29) - nA where, (2ε+6μ-2εμ+ &-3) (2 µ (1 – e)) (2ε+4μ-6εμ+2 έμ-+15) (2μ (ε- 1)) (2ε+3μ +2 εμ- μ+2@-12 (2 µ (e – 1)) n5 n4 = (E+2μ-2εμ+3) n2 = n3 (µ (1 – e)) n1 no = -1. Now, let T(A) = 1® and V(A) = -ng 15 – n4 X4 – ng 13 – n2 X² – n1 A – no. Consider |A|=1, then one has (2ε+6μ-2εμ + ε23) (2μ (1 - 0) ) (2e+4μ-6εμ+2ε μ-+15) (2μ (ε - 1) (2ε+3μ +2εμ- μ+2 22-12) (2μ (ε -1) ) (ε+2μ-2εμ+3) (µ (1 – €)) %3D (1– +1, 2 + (2ε46μ-2εμ + 2-3) (4ε+7μ-4εμ+ θμ + +3) (2 µ (1 – €)) (e + 2µ – 2 e µ +3) (1– e) (и (1 — е)) (2 e + µ – 2 €u + e² µ+ 6) , (e +2µ – 2 €µ+3) , (1– e) (2μ (ε - 1) ) (-3μ+2εμ + μ) μ(ε-1) (1-) (2 µ (e – 1)) –3 µ +2 € µ+ e² µ –µ(e – 1)² 2 и (е — 1) (2 µ (e – 1)) +1, + 1, (H (1 – €)) +1, %3D µ(E – 1) -4μ+4με + 1, +1 = 2µ (e – 1) = 3 > 1. Then, from Rouché's Theorem 1.2, V(A) and Y(A) + ¥(A) = P(^) have the same number of zeros in an open unit disk |A| < 1, that means there are only five roots lie inside unit disk. Thus, from Theorem 1.1 the equilibrium point Eo is not asymptotically stable. By solving (29), we find that 1, implies Eo is nonhyperbolic point.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hence, the characteristic equation of (28) is P(A) = X° – ng X* – na X* – n3 X³ – n2 1² – n1 1 – no, (29) - where, (2ε+6μ-2εμ + -3) (2μ (1 - e) (2ε+4μ =6εμ# 2 μ- 2+ 15 ) (2μ (ε-1) ) (2ε+3μ+2 εμ- έμ+ 222-12) n5 n4 = (2μ (ε - 1) ) (ε+2μ-2εμ + 3) n2 = n3 (µ (1 – ) (1 – e) n1 no = -1. Now, let T(A) = 16 and V(A) = -nz 15 – n4 X4 – n3 13 – n2 X? – nị 1 – no. Consider |A|=1, then one has (2ε+6μ-2 εμ # ε- 3) (2 μ (1-) (2ε+ 3μ + 2 εμ - ε?μ + 22-12 ) (2μ (ε - 1) ( E+2μ-2εμ# 3) (µ (1 – e)) (4ε+7μ-4εμ + ε'μ + 243) (2μ (ε-1)) (2ε+4μ = 6εμ+ 2 μ - ?+ 15) + (1 – €) + + 1, (2 μ (ε - 1) ) (2ε + 6μ = 2εμ + ε? - 3) (2μ (1 - ε) ) ( ε+2μ-2 εμ+3) (и (1 — е)) (2ε+μ-2εμ+ μ+6) (2μ (ε -1) ) 2 (1 €) + 1, (e + 2 µ – 2 € µ + 3) (1 (и (1 — е)) +1, 2 (-3μ +2εμ+' μ) μ (ε-1) (1-e) + 1, (2 μ (ε- 1) ) н (€ — 1) -3 u +2 € u + e? µ - µ (e – 1)2 -4μ+4με + 1, 2 µ (e – 1) +1= 2μ (ε- 1) = 3 > 1. Then, from Rouché's Theorem 1.2, V(A) and Y(A) + ¥(A) = P(X) have the same number of zeros in an open unit disk |A| < 1, that means there are only five roots lie inside unit disk. Thus, from Theorem 1.1 the equilibrium point Eo is not asymptotically stable. By solving (29), we find that A1 = 1, implies Eo is nonhyperbolic point.
Transcribed Image Text:Hence, the characteristic equation of (28) is P(A) = X° – ng X* – na X* – n3 X³ – n2 1² – n1 1 – no, (29) - where, (2ε+6μ-2εμ + -3) (2μ (1 - e) (2ε+4μ =6εμ# 2 μ- 2+ 15 ) (2μ (ε-1) ) (2ε+3μ+2 εμ- έμ+ 222-12) n5 n4 = (2μ (ε - 1) ) (ε+2μ-2εμ + 3) n2 = n3 (µ (1 – ) (1 – e) n1 no = -1. Now, let T(A) = 16 and V(A) = -nz 15 – n4 X4 – n3 13 – n2 X? – nị 1 – no. Consider |A|=1, then one has (2ε+6μ-2 εμ # ε- 3) (2 μ (1-) (2ε+ 3μ + 2 εμ - ε?μ + 22-12 ) (2μ (ε - 1) ( E+2μ-2εμ# 3) (µ (1 – e)) (4ε+7μ-4εμ + ε'μ + 243) (2μ (ε-1)) (2ε+4μ = 6εμ+ 2 μ - ?+ 15) + (1 – €) + + 1, (2 μ (ε - 1) ) (2ε + 6μ = 2εμ + ε? - 3) (2μ (1 - ε) ) ( ε+2μ-2 εμ+3) (и (1 — е)) (2ε+μ-2εμ+ μ+6) (2μ (ε -1) ) 2 (1 €) + 1, (e + 2 µ – 2 € µ + 3) (1 (и (1 — е)) +1, 2 (-3μ +2εμ+' μ) μ (ε-1) (1-e) + 1, (2 μ (ε- 1) ) н (€ — 1) -3 u +2 € u + e? µ - µ (e – 1)2 -4μ+4με + 1, 2 µ (e – 1) +1= 2μ (ε- 1) = 3 > 1. Then, from Rouché's Theorem 1.2, V(A) and Y(A) + ¥(A) = P(X) have the same number of zeros in an open unit disk |A| < 1, that means there are only five roots lie inside unit disk. Thus, from Theorem 1.1 the equilibrium point Eo is not asymptotically stable. By solving (29), we find that A1 = 1, implies Eo is nonhyperbolic point.
Σ 1 E zn-h Wn-p h=0 Wn-p Zn-h h=1 p=0 p=1 Wn+1 +µ and zn+1 + €, (4) Zn - € Wn - H where u and e are arbitrary positive real numbers with initial conditions w; and z; for i = -2, -1,0. The nontrivial positive equilibrium point of (4) is Eo = (w, z) = ( ,1) such that e > 1. µ(e-1) e+3 T (Wn + Wn-1) (zn-1 + Zn-2) (Wn-1 + Wn-2) (žn + žn-1) Ln+1 = + l, Wn, Wn-1, + €, Zn, Zn-1 Zn - € Wn - µ Theorem 4.1. The positive equilibrium point E, of system (4) is not asymptotically stable and nonhyperbolic point . Proof. The linearized system of (4) evaluated at Eo is written in the matrix form as Ln+1 Q Ln, where Ln (Wn, 2n, Wn-1, žn-1, Wn-2, zn-2) and the Jacobian matrix determined at Eo is equal 금 -(44) (e+2e-3) e+3 e+3 -(e2+2e-3) 4 µ -(e+3) 2 u -(c+3) 1-e 2 (28) 1 1 1 9. O O
Transcribed Image Text:Σ 1 E zn-h Wn-p h=0 Wn-p Zn-h h=1 p=0 p=1 Wn+1 +µ and zn+1 + €, (4) Zn - € Wn - H where u and e are arbitrary positive real numbers with initial conditions w; and z; for i = -2, -1,0. The nontrivial positive equilibrium point of (4) is Eo = (w, z) = ( ,1) such that e > 1. µ(e-1) e+3 T (Wn + Wn-1) (zn-1 + Zn-2) (Wn-1 + Wn-2) (žn + žn-1) Ln+1 = + l, Wn, Wn-1, + €, Zn, Zn-1 Zn - € Wn - µ Theorem 4.1. The positive equilibrium point E, of system (4) is not asymptotically stable and nonhyperbolic point . Proof. The linearized system of (4) evaluated at Eo is written in the matrix form as Ln+1 Q Ln, where Ln (Wn, 2n, Wn-1, žn-1, Wn-2, zn-2) and the Jacobian matrix determined at Eo is equal 금 -(44) (e+2e-3) e+3 e+3 -(e2+2e-3) 4 µ -(e+3) 2 u -(c+3) 1-e 2 (28) 1 1 1 9. O O
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