Find all equilibria of the given nonlinear system of differential equations, construct the linear approximating system near each equilibrium, identify the type of each equilibrium, and determine whether each equilibrium is stable, asymptotically stable, or unstable with respect to the nonlinear system. (4-1₁-I₂)F₁, −1²+1²+3, (a) (b) dx₁ dt dz₁ dt = dx₂ dt dx2 dt =(−2+x₁)x₂ = −11+ 2x₂ [45] (a) Three equilibria: (1₁, 12) = (0,0), (I₁, I₂) = (4,0), and (T₁, 1₂) = (2,2). The linear approximating system near (0,0): 0 |- [6 = ][₂] The equilibrium (0,0) is a saddle and is unstable. The linear approximating system near (4,0): 3-6 111 0 2 I2 I 0-2 The equilibrium (2, 1) is a saddle and is unstable. The linear -4 The equilibrium (4,0) is a saddle and is unstable. The linear approximating system near (2,2): [1] [3]-[2 = 3][ The equilibrium (2,2) is an attractive spiral focus and is asymptotically stable. (b) Two equilibria: (₁, ₂) = (2, 1) and (£₁, 1₂) = (-2,-1). -4 2 The linear approximating system near (2,1): [2] = [ 4-316-1 -1 2 -2 -2 IL- 1₂-2 -2 2][22+1]. I₂+1 4 approximating system near (-2,-1): [2] - [ = 2 The equilibrium (-2,-1) is a repulsive improper node and is unstable.
Find all equilibria of the given nonlinear system of differential equations, construct the linear approximating system near each equilibrium, identify the type of each equilibrium, and determine whether each equilibrium is stable, asymptotically stable, or unstable with respect to the nonlinear system. (4-1₁-I₂)F₁, −1²+1²+3, (a) (b) dx₁ dt dz₁ dt = dx₂ dt dx2 dt =(−2+x₁)x₂ = −11+ 2x₂ [45] (a) Three equilibria: (1₁, 12) = (0,0), (I₁, I₂) = (4,0), and (T₁, 1₂) = (2,2). The linear approximating system near (0,0): 0 |- [6 = ][₂] The equilibrium (0,0) is a saddle and is unstable. The linear approximating system near (4,0): 3-6 111 0 2 I2 I 0-2 The equilibrium (2, 1) is a saddle and is unstable. The linear -4 The equilibrium (4,0) is a saddle and is unstable. The linear approximating system near (2,2): [1] [3]-[2 = 3][ The equilibrium (2,2) is an attractive spiral focus and is asymptotically stable. (b) Two equilibria: (₁, ₂) = (2, 1) and (£₁, 1₂) = (-2,-1). -4 2 The linear approximating system near (2,1): [2] = [ 4-316-1 -1 2 -2 -2 IL- 1₂-2 -2 2][22+1]. I₂+1 4 approximating system near (-2,-1): [2] - [ = 2 The equilibrium (-2,-1) is a repulsive improper node and is unstable.
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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![[45] Find all equilibria of the given nonlinear system of differential equations, construct the
linear approximating system near each equilibrium, identify the type of each equilibrium,
and determine whether each equilibrium is stable, asymptotically stable, or unstable with
respect to the nonlinear system.
(a)
(b)
dx₁
dt
dx₁
dt
-
=
(4- x₁-x₂)x₁,
-x²+x² +3,
dx2
dt
dx2
dt
=(−2+x₁)x₂
=−x1 + 2x₂
[45] (a) Three equilibria: (1, 2) = (0,0), (1, 2) = (4,0), and (₁,₂)=(2, 2).
4
I
22][2]
The linear approximating system near (0,0): [4]
The equilibrium (0,0) is a saddle and is unstable.
The linear approximating system near (4,0): [1]
=
=
-4
=
0
-4
I[¹].
2
The equilibrium (4,0) is a saddle and is unstable.
The linear approximating system near (2, 2): [1]
-2
-2
2-2
The equilibrium (2, 2) is an attractive spiral focus and is asymptotically stable.
(b) Two equilibria: (₁, ₂) = (2, 1) and (₁, ₂) = (-2,-1).
anattractive para (233)
[2] = []
The linear approximating system near (2,1):
The equilibrium (2, 1) is a saddle and is unstable.
The linear approximating system near (-2,-1):
-4 2]
I1
2 I₂
2
-2
+2
[14]-[4 363
=
12
2
The equilibrium (-2, -1) is a repulsive improper node and is unstable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c89c353-252e-4eca-b673-064be33ffdb0%2F89c2967e-7fab-4d5b-b717-4c3e49b58e94%2Flhcldjp_processed.png&w=3840&q=75)
Transcribed Image Text:[45] Find all equilibria of the given nonlinear system of differential equations, construct the
linear approximating system near each equilibrium, identify the type of each equilibrium,
and determine whether each equilibrium is stable, asymptotically stable, or unstable with
respect to the nonlinear system.
(a)
(b)
dx₁
dt
dx₁
dt
-
=
(4- x₁-x₂)x₁,
-x²+x² +3,
dx2
dt
dx2
dt
=(−2+x₁)x₂
=−x1 + 2x₂
[45] (a) Three equilibria: (1, 2) = (0,0), (1, 2) = (4,0), and (₁,₂)=(2, 2).
4
I
22][2]
The linear approximating system near (0,0): [4]
The equilibrium (0,0) is a saddle and is unstable.
The linear approximating system near (4,0): [1]
=
=
-4
=
0
-4
I[¹].
2
The equilibrium (4,0) is a saddle and is unstable.
The linear approximating system near (2, 2): [1]
-2
-2
2-2
The equilibrium (2, 2) is an attractive spiral focus and is asymptotically stable.
(b) Two equilibria: (₁, ₂) = (2, 1) and (₁, ₂) = (-2,-1).
anattractive para (233)
[2] = []
The linear approximating system near (2,1):
The equilibrium (2, 1) is a saddle and is unstable.
The linear approximating system near (-2,-1):
-4 2]
I1
2 I₂
2
-2
+2
[14]-[4 363
=
12
2
The equilibrium (-2, -1) is a repulsive improper node and is unstable.
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