$ = g(ø, v, ø) = aø + - b = g(4, 4, 4) = arb + which can be rewritten as ayoy – ado? + Bo², ayoy – ady² + Bb?. (9) Subtracting Eq. (9) from Eq. (8) leads to [8(1 – a) + B](4² – o?) = 0. Hence, if d(a – 1) + B, then o = v. Therefore, Theorem B in [21] ensures that the 1:1
$ = g(ø, v, ø) = aø + - b = g(4, 4, 4) = arb + which can be rewritten as ayoy – ado? + Bo², ayoy – ady² + Bb?. (9) Subtracting Eq. (9) from Eq. (8) leads to [8(1 – a) + B](4² – o?) = 0. Hence, if d(a – 1) + B, then o = v. Therefore, Theorem B in [21] ensures that the 1:1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
solving the determine yellow in the same way of the determine green
Transcribed Image Text:Bun-1un-5
Un+1 = aun-1+
n = 0, 1, ..,
(1)
yun-3 – Sun-5
Bun-1un-5
Un+1 = QUn-1 –
n = 0, 1, ...,
(2)
Yun-3 + dun-5
Eq. (1). In order to discuss the stability of Eq. (1), we first define the function
g : (0, 00)³ -
→ (0, o0) by
Brz
g(x, y, z) =
= ax +
(3)
YY – 8z
Then,
ag(x, y, z)
Bz
a +
YY – 8z'
Byrz
(4)
ag(x, y, z)
ду
ag(x, y, z)
(5)
(ry – ôz)2
Byry
(6)
dz
(ry – 8z)2 °
![Theorem 2 ASsume that yy > dz, then the equilibrium point of Eq. (1) is a global
attractor if 8(a – 1) + B.
Proof.
3
Assume that p, q E R and let g : p, q]°
p, q] is a function defined by
Eq. (3). Since yy > 8z, the function g is increasing in x and z and decreasing in y.
Let (ø, v) be a solution for the following system:
$ = g(ø, V, ¢), v = g(y, 4, v).
Plugging this into Eq. (1) leads to
Ø = g($, Þ, ø) = aø +
V = g(4, Ø, v) = ar) +
which can be rewritten as
ayop – adø? + B¢²,
ayop – adıy? + By?.
(9)
-
Subtracting Eq. (9) from Eq. (8) leads to
[8(1 – a) + B](4² – 4²) = 0.
Hence, if d (a - 1) + B, then $ = p. Therefore, Theorem B in [21] ensures that the
equilibrium point is a global attractor.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d564931-7cab-47e3-baee-f646b4270efc%2F53a12363-d7f8-418f-bb33-8db1cb1a160f%2Fnevg4i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 2 ASsume that yy > dz, then the equilibrium point of Eq. (1) is a global
attractor if 8(a – 1) + B.
Proof.
3
Assume that p, q E R and let g : p, q]°
p, q] is a function defined by
Eq. (3). Since yy > 8z, the function g is increasing in x and z and decreasing in y.
Let (ø, v) be a solution for the following system:
$ = g(ø, V, ¢), v = g(y, 4, v).
Plugging this into Eq. (1) leads to
Ø = g($, Þ, ø) = aø +
V = g(4, Ø, v) = ar) +
which can be rewritten as
ayop – adø? + B¢²,
ayop – adıy? + By?.
(9)
-
Subtracting Eq. (9) from Eq. (8) leads to
[8(1 – a) + B](4² – 4²) = 0.
Hence, if d (a - 1) + B, then $ = p. Therefore, Theorem B in [21] ensures that the
equilibrium point is a global attractor.
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