Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz7 => Topic started by: Victor Ivrii on November 30, 2018, 04:08:22 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.
$$\left\{\begin{aligned}
&\frac{dx}{dt} = 2x  y x(x^2 + y^2),\\
&\frac{dy}{dt} = x  y + y(x^2 + y^2).
\end{aligned}\right.$$
Bonus: Computer generated picture

To find critical points, let $x'=0, y'=0$
Thus, $2xyx(x^2+y^2)=0, xy+y(x^2+y^2)=0$
$ (0,0)$ is one crtical point
Also, $(0.330757,1.09242), (0.330757,1.09242)$ are also critical points
Let $f(x)=2xyx(x^2+y^2), g(x)=xy+y(x^2+y^2)$
$f_x=23x^2y^2, f_y=12xy$
Also, $ g_x=1+2xy, g_y=1+x^2+3y^2$
$J=\begin{bmatrix}
23x^2y^2&12xy\\
1+2xy&1+x^2+3y^2\\
\end{bmatrix}$
Plug in the critical points to find eigenvalues of each linear system
For $(0,0)$,
We get $\begin{bmatrix}
2&1\\
1&1\\
\end{bmatrix}$,
$r=\frac{3\pm\sqrt{3}i}{2}$
Therefore, it is stable at $(0,0)$
Similarly, for others critical points,
Plug in and eigenvalues are $3.5092, 2.6771$
Thus, it is a saddle point therefore unstable

This is my answer

I am not sure why in the textbook was the problem with difficult to calculate stationary points. The easiest way:
\begin{align*}
&2xyx(x^2+y^2)=0,\\
&xy+y(x^2+y^2)=0.
\end{align*}
Multiplying these equations by $y$ and $x$ correspondingly and adding we get $y^23xy +x^2=0$ and we can find $x = (3\pm \sqrt{13})y/2$. Then plugging to one of the equations in addition to $(0,0)$ we get two other points.
I will instruct TAs to consider only $(0,0)$.
Guanyao: calculation of eigenvalues missing; picture wrong
Jingze: again, wrong range for variables in the picture