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:

Question

Want to see the full answer?

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:

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

== 1. A separable differential equation can be written in the form hy) = g(a) where h(y) is a function of y only, and g(x) is a function of r only. All of the equations below are separable. Rewrite each of these in the form h(y) = g(x), then find a general solution by integrating both sides. Determine whether the solutions you found are explicit (functions) or implicit (curves but not functions) (a) 1' = — 1/3 (b) y' = = --- Y (c) y = x(1+ y²)

Ja дх dx dx Q3: Define the linear functional J: H()-R by تاریخ (v) = ½a(v, v) - (v) == Let u be the unique weak solution to a(u,v) = L(v) in H₁(2) and suppose that a(...) is a symmetric bilinear form on H() prove that a Buy v) = 1- u is minimizer. 2- u is unique. 3- The minimizer J(u,) can be rewritten under J(u)=u' Au-ub, algebraic form Where A, b are repictively the stiffence matrix and the load vector Q4: A) Answer only 1-show that thelation to -Auf in N, u = 0 on a satisfies the stability Vulf and show that V(u-u,)||² = ||vu||2 - ||vu||2 lu-ulls Chu||2 2- Prove that Where =1 ||ul|= a(u, u) = Vu. Vu dx + fu. uds B) Consider the bilinear form a(u, v) = (Au, Av) + (Vu, Vv) + (Vu, v) + (u, v) Show that a(u, v) continues and V- elliptic on H(2) (3) (0.0), (3.0)

Q1: A) fill the following: 1- The number of triangular in a triangular region with 5 nodes is quadrilateral with n=5 and m=6 nodés is 2- The complex shape function in 1-D 3- dim(P4(K))=- (7M --- and in the and multiplex shape function in 2-D is 4- The trial space and test space for problem -Auf, u = go on and B) Define the energy norm and prove that the solution u, defined by Galerkin orthogonal satisfies the best approximation. Q2: A) Find the varitional form for the problem 1330 (b(x)) - x²=0, 0

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:

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

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

Want to see the full answer?

Check out a sample textbook solution

See solution

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

Want to see the full answer?

Check out a sample textbook solution

See solution

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

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 13

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

Want to see the full answer?

Chapter 3 Solutions