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Fixed points for system using PlotVectorField

Math folks,

I am trying to determine the fixed points of the following system:

dx/dt = x(1-x^2-y^2)

dy/dt = y(2-x^4-y^4)

I understand that you study

dy/dx = y(2-x^4-y^4)/x(1-x^2-y^2) and see what happens to the field when 
dx/dt and dy/dt = 0.

I would like to know if I am using the PlotVectorField function in the 
correct way.

What I coded was the following:

PlotVectorField[{1, y(2-x^4-y^4)/x(1-x^2-y^2)},{x,-2,2},{y,-2,2},Axes -> 

My plot shows a fixed point at (1.18921, 0), which is the (fourth root of 2, 

This makes sense, as one of the solutions for x...

Solve[y(2-x^4-y^4)==0, x] is the fourth root of 2.

What I don't understand is the following:

There seem to be other points of interest on the plot at the following 
points: (+/- 1.16, +/- 0.7)

When I search for x's and y's close to one another computationally which 
make y(2-x^4-y^4)/x(1-x^2-y^2 -> 0, I get (+/- 0.6, +/- 0.8).

Can someone tell me if I am applying the PlotVectorField appropriately, and 
if so, how I should interpret the points of interest on the plot?


"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

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