The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x ' = f ( x ) with critical points at A ( − 1 , 0 ) , B ( 0 , 0 ) , C ( 0 , 1 ) are shown in the figure below:

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Answer 1

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

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Answer 2

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

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Answer 3

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Expert Solution & Answer

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Answer 4

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Expert Solution & Answer

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Answer 5

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Answer 6

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

Answer 7

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

Answer 8

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Answer 9

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Answer 10

Expert Solution & Answer

Answer 11

Expert Solution & Answer

Answer 12

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Answer 13

Solution Summary: The author explains that the critical point B(0,0) is an unstable saddle point.

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Chapter 3 Solutions