The nontrivial positive equilibrium point of (4) is Eo = (w, z) = T (Wn+Wn-1) (2n-1 + Zn-2) Zn € (Wn-1 + Wn-2) (Zn + Zn-1) Wn-fl 1²2-1) ². Intl = Theorem 4.1. The positive equilibrium point Eo 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 where Ln = Q = +H, Wn, Wn-1; 3 -(e²+2e-3) 4μ 1 0 0 0 Ln+1 = (Wn, Zn, Wn-1, Zn-1, Wn-2, Zn-2) and the Jacobian matrix determined at Eo is equal 3 -(e+3) 2μ 0 0 1 0 Q Ln, 0 -(€+3) 2μ 0 0 0 0 9 -(4μ) (€²+2€-3) 1-e 2 0 1 0 0 such that € > 1. €+37 2μ €+3 2 0 0 0 1 2μ €+3 +E, Zn, 2n-1 0 0 0 0 (28)
The nontrivial positive equilibrium point of (4) is Eo = (w, z) = T (Wn+Wn-1) (2n-1 + Zn-2) Zn € (Wn-1 + Wn-2) (Zn + Zn-1) Wn-fl 1²2-1) ². Intl = Theorem 4.1. The positive equilibrium point Eo 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 where Ln = Q = +H, Wn, Wn-1; 3 -(e²+2e-3) 4μ 1 0 0 0 Ln+1 = (Wn, Zn, Wn-1, Zn-1, Wn-2, Zn-2) and the Jacobian matrix determined at Eo is equal 3 -(e+3) 2μ 0 0 1 0 Q Ln, 0 -(€+3) 2μ 0 0 0 0 9 -(4μ) (€²+2€-3) 1-e 2 0 1 0 0 such that € > 1. €+37 2μ €+3 2 0 0 0 1 2μ €+3 +E, Zn, 2n-1 0 0 0 0 (28)
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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show me steps of the determine yellow and all information is here explain to me step by step i need every details
Transcribed Image Text:In this paper, we solve and study the properties of the following system
E Wn-p
Wn-P
> zn-h
Zn-h
h=0
p=1
+ €,
(4)
p=0
h=1
+µ and zn+1
Wn+1
Wn - H
Zn - €
where u and e are arbitrary positive real numbers with initial conditions w; and z; for i =
-2, –1,0.
Transcribed Image Text:The nontrivial positive equilibrium point of (4) is Eo = (w, z) = ( ,1) such that e > 1.
e+3
T
(wn + Wn-1) (žn-1+ žn-2)
(Wn-1 + Wn-2) (2, + 2m–1)
Ln+1
+ l, Wn, Wn-1,
+ €, Zn, Zn-1
Zn - €
Wn - l
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, Zn, Wn-1, Zn–1, Wn-2, žn–2)' and the Jacobian matrix determined at Eo is equal
-2
E-1
-(4µ)
(e²+2 e-3)
2 µ
e+3
e+3
-(²+2e-3)
4 µ
-(e+3)
2 u
-(c+3)
2 u
1-e
2
(28)
1
1
1
9.
- O O o
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