Q7 TUT 0401

[Victor Ivrii]

(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.
$$\left\{\begin{aligned}
&\frac{dx}{dt} = (2 + y)(2y - x), \\
&\frac{dy}{dt} = (2 - x)(2y + x).
\end{aligned}\right.$$

Bonus: Computer generated picture

[Xiaoyuan Wang]

Here is my answer.

[Jingze Wang]

Here is computer generated picture

[Victor Ivrii]

Xiaoyuan: correctly found points. However at point $(2,-2)$ you discovered two purely imaginary eigenvalues, and decided that this is a center. For linearized system it is a center, but we cannot make such conclusion for non-linear system, only "a center or a spiral point (with slow moving in/away)" and the pictures attached show that it is indeed such point