Theorem 10 Every solution of Eq. (8) satisfies both of the following asymptotic relations Yn+1 lim n 00 Yn lim sup (lyn - y|)/" n 00 where je {1,... , k} and X; are the roots of characteristic equation (16). Proof. We get from Eq.(8): -- - (1+)-(1+) Yn Yn+1 Yn-m p(y + Yn-m) ý. Yn-m (Yn – 9) (Yn-m – 9) . | yn-m Set en = Yn - g. Therefore we have en+1 + Pnen + Anen-m 0, where p (y + Yn-m) , In Pn 2. Yn-m Due to the equilibrium point y of Eq.(8) is globally asymptotically stable, we get 2p lim Pn lim qn n 00 Hence, the limiting equation of Eq.(8) is the linearized equation (15). I
Theorem 10 Every solution of Eq. (8) satisfies both of the following asymptotic relations Yn+1 lim n 00 Yn lim sup (lyn - y|)/" n 00 where je {1,... , k} and X; are the roots of characteristic equation (16). Proof. We get from Eq.(8): -- - (1+)-(1+) Yn Yn+1 Yn-m p(y + Yn-m) ý. Yn-m (Yn – 9) (Yn-m – 9) . | yn-m Set en = Yn - g. Therefore we have en+1 + Pnen + Anen-m 0, where p (y + Yn-m) , In Pn 2. Yn-m Due to the equilibrium point y of Eq.(8) is globally asymptotically stable, we get 2p lim Pn lim qn n 00 Hence, the limiting equation of Eq.(8) is the linearized equation (15). I
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 the steps of determine blue and inf is here
Transcribed Image Text:Hence we obtain the linearized equation of Eq.(8) about its unique positive
equilibrium point y as follow:
2p
Zn-m
= 0.
(15)
Zn+1
Therefore, the characteristic equation of Eq.(8) is
2p
0.
т
Am+1
(16)
Yn
Yn+1 = 1++p
(8)
Yn-m
..bone
B
.u. hendle the
inen og uetion (0)
The unigue
Transcribed Image Text:Theorem 10 Every solution of Eq. (8) satisfies both of the following asymptotic
relations
Yn+1
lim
Ajl,
Yn – ỹ
n 00
lim sup (lyn – y|)/"
where j e {1,.. , k} and X; are the roots of characteristic equation (16).
Proof. We get from Eq.(8):
-9 - (1+)-(*)
Yn
1+P.2
Yn+1
n-m
p(y + Yn-m)
ý · Yn-m
(Yn–m – 9) .
(Yn – 9)
.2
yn-m
Set en = Yn – ỹ. Therefore we have
en+1 + Pnen + Inen-m = 0,
where
p(y + Yn-m)
In
Yn-m
,2
y· Yn-m
Pn
,2
Due to the equilibrium point j of Eq.(8) is globally asymptotically stable, we
get
2p
lim Pn
lim qn
Hence, the limiting equation of Eq.(8) is the linearized equation (15). -
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