Analyzing Nonlinear PDE Systems with Polar Coordinates

[menphis]

Homework Statement

Hi, i have the following system of equation. In the task is that system have periodic solution and have to be used polar coordinates.

Homework Equations

x'=1+y-x^2-y^2
y'=1-x-x^2-y^2

The Attempt at a Solution

After transfer to polar system i tried to use the method of variation of parameters, but without success.

 

[HallsofIvy]

1) Why is this posted in the "preCalculus" section?
I assume that was a mistake and I will move it to "Calculus and Beyond" homework.

2) Why is this tltled "PDE"? I see no partial differential equations. I see a system of two ordinary differential equations.

Changing to polar coordinates looks like a very good idea but I don't know what you mean by "variation of parameters" for a non-linear equation. What equations did you get after changing to polar coordinates?

 

[menphis]

I'm sorry for PDE and wrong section
In the polar coordinates have equations this shape:

rho' cos(phi)=rho sin(phi)(1+phi')
rho' sin(phi)=- rho cos(phi)(1+phi')

 

[epenguin]

Did you get anywhere yet?
One thing you might notice straightaway is that any point on the circle x² + y² - 1 = 0 you find the equations become those of SHM whose solution is that same circle, so that circle is a solution.
However it is not SHM in general, for no other points have that property and (0, 0) is not a stationary point.

Do you know how to analyse such systems qualitatively? This one appears quite complex and surprising.
Main thing, you have to find the stationary points (i.e. where x' = y' = 0) and analyse the stability of the linear approximation around them.

Perhaps the d.e. s can be solved too, I don't know yet.