Plot of a Vector Field and Nullclines

[maxbenczek]

Hello,
I’m writing my thesis and I need to put some pictures in it. I need to plot a vector field (which i found how to do) and in it the nullclines for the field.

The plot should look like the one in the attachment on the upper right. In that plot one can see the vector field for the system x’=y-x^2 ; y’=x-2 and the respective nullclines which are y=x^2 and x=2.
I would be very grateful if anyone could help me and tell me how I can plot that!

Best Regards

[shenghuiy (Shenghui Yang)]

You can use the VectorPlot function to plot the vector field. Use the ParametricPlot function to plot the line x == 2 and the parabola y == x^2

Show[
ParametricPlot[{{u,u^2},{2,3u^2}},{u,-3,3}],
VectorPlot[{y-x^2,x-2},{x,-3,3},{y,-1,5},
VectorScale->{Tiny,Small,None}],PlotRange->{-1,4}]

[mthiel (Marco Thiel)]

Hi,

what about this

Show[VectorPlot[{y - x^2, x - 2}, {x, -3, 3}, {y, -1, 9}, Epilog -> {PointSize[0.03], Point[{x, y} /. First[Solve[{y - x^2 == 0, x - 2 == 0}]]]}, 
  Frame -> False, Axes -> True], ContourPlot[{y - x^2 == 0, x - 2 == 0}, {x, -3, 3}, {y, -1, 9}, ContourStyle -> Red]]

where the length of the arrows represents the magnitude/speed. If you prefer the arrows of equal length like in your book this might work:

Show[VectorPlot[{y - x^2, x - 2}, {x, -3, 3}, {y, -1, 9}, Epilog -> {PointSize[0.03], Point[{x, y} /. First[Solve[{y - x^2 == 0, x - 2 == 0}]]]}, 
  Frame -> False, Axes -> True, VectorScale -> {Small, Small, None}, VectorPoints -> Coarse, ImageSize -> Medium], 
 ContourPlot[{y - x^2 == 0, x - 2 == 0}, {x, -3, 3}, {y, -1, 9}, ContourStyle -> Red]]

Cheers,

Marco