Consider the differential equation u' = F (u) where u = [ay] and the function on the right hand side is F (u) = [ -x+ 2xy y - xy (1) The Jacobian of F (u) has entries (3) Classify the behavior at the origin: Attractor Repellor Saddle Center J₁1 J21 (2) The origin is an equilibrium point. Evaluated at the origin, the Jacobian has entries J11 (5) At this equilibrium point, the Jacobian is J21 Spiral attractor Spiral repellor (4) The differential equation has another equilibrium point at = J₁1 J21 = = x= || || = ⒸJ12 = J22 J12 J22 y = || J12 || = J22 ||
Consider the differential equation u' = F (u) where u = [ay] and the function on the right hand side is F (u) = [ -x+ 2xy y - xy (1) The Jacobian of F (u) has entries (3) Classify the behavior at the origin: Attractor Repellor Saddle Center J₁1 J21 (2) The origin is an equilibrium point. Evaluated at the origin, the Jacobian has entries J11 (5) At this equilibrium point, the Jacobian is J21 Spiral attractor Spiral repellor (4) The differential equation has another equilibrium point at = J₁1 J21 = = x= || || = ⒸJ12 = J22 J12 J22 y = || J12 || = J22 ||
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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Question
![Consider the differential equation u' = F (u) where u = [ay] and the function on the right hand side is
F (u) = [
-x+ 2xy
y - xy
(1) The Jacobian of F (u) has entries
(3) Classify the behavior at the origin:
Attractor
Repellor
Saddle
Center
J₁1
J21
(2) The origin is an equilibrium point. Evaluated at the origin, the Jacobian has entries
J11
(5) At this equilibrium point, the Jacobian is
J21
Spiral attractor
Spiral repellor
(4) The differential equation has another equilibrium point at
=
J₁1
J21
=
=
X=
||
=
||
ⒸJ12
=
J22
J12
J22
y
=
||
J12
||
=
J22
||](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6b61458-362c-46a8-806a-8de7b1ef169c%2F4f794efe-d72a-4b09-b605-a9075a306a66%2Fw7wq51_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the differential equation u' = F (u) where u = [ay] and the function on the right hand side is
F (u) = [
-x+ 2xy
y - xy
(1) The Jacobian of F (u) has entries
(3) Classify the behavior at the origin:
Attractor
Repellor
Saddle
Center
J₁1
J21
(2) The origin is an equilibrium point. Evaluated at the origin, the Jacobian has entries
J11
(5) At this equilibrium point, the Jacobian is
J21
Spiral attractor
Spiral repellor
(4) The differential equation has another equilibrium point at
=
J₁1
J21
=
=
X=
||
=
||
ⒸJ12
=
J22
J12
J22
y
=
||
J12
||
=
J22
||
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