Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz7 => Topic started by: Victor Ivrii on November 30, 2018, 04:02:45 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.
$$\left\{\begin{aligned}
&\frac{dx}{dt} = x + x^2 + y^2, \\
&\frac{dy}{dt} = y  xy.
\end{aligned}\right.$$
Bonus: Computer generated picture

(a)
Set x’ = 0 and y’=0
Then we have critical points (0,0), (1,0)
(b)
J = \begin{bmatrix}2x+1 & 2y \\y & 1x \end{bmatrix}
Linear systems are shown with each critical point:
J(0,0) = \begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}
J(1,0) = \begin{bmatrix}1 & 0 \\0 & 2 \end{bmatrix}
(c)
Eigenvalues are computed by det(A  tI)= 0
So that
At (0,0): t=1
Critical point is an unstable proper node or spiral point
At (1,0): t=1 and 2
Critical point is an unstable saddle point

This is computer generated picture :)

Hello, I solve this question step by step.
Please see the picture below.
If there are any mistakes, reply to me anytime.
Thank you very much.

Jingze,
You took too large range by $y$ and ...
When we have nonzero repeated eigenvalues, there is always a node (stable or unstable). Spiral may appear in this situation only when the righthand is not smooth, which is not the case. On the picture attached you can see...