# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Quiz-7 => Topic started by: Victor Ivrii on November 30, 2018, 04:07:36 PM

Title: Q7 TUT 0401
Post by: Victor Ivrii on November 30, 2018, 04:07:36 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 + y)(2y - x), \\ &\frac{dy}{dt} = (2 - x)(2y + x). \end{aligned}\right.

Bonus: Computer generated picture
Title: Re: Q7 TUT 0401
Post by: Xiaoyuan Wang on November 30, 2018, 05:23:14 PM
Here is my answer.
Title: Re: Q7 TUT 0401
Post by: Jingze Wang on November 30, 2018, 08:33:46 PM
Here is computer generated picture
Title: Re: Q7 TUT 0401
Post by: Victor Ivrii on December 01, 2018, 03:49:13 AM
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