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4 Local stability of the equilibrium point :) The nontrivial positive equilibrium point of (4) is Eo = (W, z) = (", 1) µ(e-1) e+3 such that e > 1. T (Wn + Wn-1) (žn-1+ žn-2) (Wn-1 + Wn-2) (2n + žn-1) Ln+1 + µ, Wn, Wn-1, + €, Zn, 2n-1 2n - € Wn - u 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 Ln+1 Q Ln, where Ln = (Wn, Zn, Wn-1, 2n-1, Wn-2, Zn-2)' and the Jacobian matrix determined at Eo is equal -(4u) (e2+2 €-3) €-1 -(e2+2e-3) 4 µ €-1 -(e+3) e+3 e+3 -(e+3) 2 µ 1-e 2 1-€ 1 (28) 1 1 0. 1 9.

4 Local stability of the equilibrium point :) The nontrivial positive equilibrium point of (4) is Eo = (W, z) = (", 1) µ(e-1) e+3 such that e > 1. T (Wn + Wn-1) (žn-1+ žn-2) (Wn-1 + Wn-2) (2n + žn-1) Ln+1 + µ, Wn, Wn-1, + €, Zn, 2n-1 2n - € Wn - u 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 Ln+1 Q Ln, where Ln = (Wn, Zn, Wn-1, 2n-1, Wn-2, Zn-2)' and the Jacobian matrix determined at Eo is equal -(4u) (e2+2 €-3) €-1 -(e2+2e-3) 4 µ €-1 -(e+3) e+3 e+3 -(e+3) 2 µ 1-e 2 1-€ 1 (28) 1 1 0. 1 9.

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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E Wn-p Wn-p > zn-h Zn-h p=1 h=0 + €, (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:E Wn-p Wn-p > zn-h Zn-h p=1 h=0 + €, (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.
4 Local stability of the equilibrium point ) The nontrivial positive equilibrium point of (4) is Eo = (w, z) = (", 1) such that e > 1. µ(E-1) e+3 > ((Wn + Wn-1) (Zn-1 + Zn-2) (Wn-1 + Wn-2) (žn + žn–1) Ln+1 + µ, Wn, Wn-1, + €, Zn, Zn–1 Zn - € Wn – µ 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 Ln+1 Q Ln, where Ln = (wn, Zn, Wn-1, Zn–1, Wn-2, 2n-2)" and the Jacobian matrix determined at Eo is equal - (4 μ) (e²+2 €-3) 2 µ e+3 e+3 -(e2+2€-3) 4 µ -(e+3) -(e+3) 1-€ 1-€ 2 µ 1 (28) 1 1 1 9.
Transcribed Image Text:4 Local stability of the equilibrium point ) The nontrivial positive equilibrium point of (4) is Eo = (w, z) = (", 1) such that e > 1. µ(E-1) e+3 > ((Wn + Wn-1) (Zn-1 + Zn-2) (Wn-1 + Wn-2) (žn + žn–1) Ln+1 + µ, Wn, Wn-1, + €, Zn, Zn–1 Zn - € Wn – µ 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 Ln+1 Q Ln, where Ln = (wn, Zn, Wn-1, Zn–1, Wn-2, 2n-2)" and the Jacobian matrix determined at Eo is equal - (4 μ) (e²+2 €-3) 2 µ e+3 e+3 -(e2+2€-3) 4 µ -(e+3) -(e+3) 1-€ 1-€ 2 µ 1 (28) 1 1 1 9.
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