Author Topic: Q7 TUT 5102  (Read 1330 times)

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2511
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Q7 TUT 5102
« on: November 30, 2018, 04:12:23 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
$$\left\{\begin{aligned}
&\frac{dx}{dt} = (2 + x)( y - x),\\
&\frac{dy}{dt} = (4 - x)( y + x).
\end{aligned}\right.$$

Bonus: Computer generated picture

Chonghan Ma

  • Newbie
  • *
  • Posts: 4
  • Karma: 5
    • View Profile
Re: Q7 TUT 5102
« Reply #1 on: November 30, 2018, 04:19:46 PM »
(a)
Set (2+x)(y-x)=0 and (4-x)(y+x)=0
Then we have critical points (0,0), (4,4), (-2,2)
(b)
J = \begin{bmatrix}-2-2x+y & 2+x \\4-2y-2x & 4-x \end{bmatrix}
Linear systems are shown with each critical point:
J(0,0) =  \begin{bmatrix}-2 & 2 \\4 & 4 \end{bmatrix}
J(-2,2) =  \begin{bmatrix}4 & 0 \\6 & 6 \end{bmatrix}
J(4,4) =  \begin{bmatrix}-6 & 6 \\-8 & 0 \end{bmatrix}
(c)
Eigenvalues are computed by det(A - tI)= 0
So that
At (0,0): t=1±√17}
Critical point is a saddle point and it is unstable
At (-2,2): t= 4 and 6
Critical point is an unstable node
At ((4,4): t=-3±√9 i   
Critical point is a stable spiral point
« Last Edit: November 30, 2018, 04:23:26 PM by Chonghan Ma »

Xiaoyuan Wang

  • Jr. Member
  • **
  • Posts: 8
  • Karma: 9
    • View Profile
Re: Q7 TUT 5102
« Reply #2 on: November 30, 2018, 04:43:46 PM »
Here is my answer.

Jingze Wang

  • Full Member
  • ***
  • Posts: 30
  • Karma: 25
    • View Profile
Re: Q7 TUT 5102
« Reply #3 on: November 30, 2018, 05:17:58 PM »
This is computer generated picture

Mengfan Zhu

  • Jr. Member
  • **
  • Posts: 10
  • Karma: 5
    • View Profile
Re: Q7 TUT 5102
« Reply #4 on: December 01, 2018, 02:59:55 AM »
For this question, I draw the graph my hand.
If there is any problem, tell me as soon as possible.
Thanks for reading.

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2511
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Q7 TUT 5102
« Reply #5 on: December 01, 2018, 04:08:49 AM »
ok