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I have two curves and each one of them has two critical points which are saddles (i.e., I have four saddles). I try to draw them using Mathematica, but the saddles do not appear in the figure. (I need to draw them in the same plane). The first curve is given by

x y + (b/2) y^2 - (3/2) x^2 y^2 + (1/4) y^4 + (1/2 b) x^2 - 
  (-2 + 22 b^2 + 3 b^4 - 7 Sqrt[3] b^3 Sqrt[4 + 3 b^2])/(72 b^2) = 0

with 0 < b < (1/Sqrt[6]). The second one is given by

x y + (b/2) y^2 - (3/2) x^2 y^2 + (1/4) y^4 + (1/2 b) x^2 - 
  (-2 + 22 b^2 + 3 b^4 + 7 Sqrt[3] b^3 Sqrt[4 + 3 b^2])/(72 b^2) = 0

with 0 < b < (1/Sqrt[6]).

Please help me to get the correct figure.

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1 Answer 1

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It is not entirely clear what you are asking, but do you want something like this:

Manipulate[
 ContourPlot[{x y + (b/2) y^2 - (3/2) x^2 y^2 + (1/4) y^4 + (1/2 b) x^2 - 
         (-2 + 22 b^2 + 3 b^4 - 7 Sqrt[3] b^3 Sqrt[4 + 3 b^2])/(72 b^2) == 0, 
   x y + (b/2) y^2 - (3/2) x^2 y^2 + (1/4) y^4 + (1/2 b) x^2 - 
          (-2 +22 b^2 + 3 b^4 + 7 Sqrt[3] b^3 Sqrt[4 + 3 b^2])/(72 b^2) == 0},
         {x, -5, 5}, {y, -5, 5}], {b, 0.01, 1/Sqrt[6]}]

This gives you an interactive contour plot where you can change the value of $b$.

enter image description here

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  • $\begingroup$ Dear sir, Thank you so much for your help. But the two functions have four saddles, and in the figure, it does not appear anyone of them. This is the problem. $\endgroup$
    – Sara yaqob
    Commented Sep 7, 2018 at 9:28
  • $\begingroup$ @Sarayaqob, what do you mean? Where are they?> $\endgroup$
    – SPPearce
    Commented Sep 14, 2018 at 7:59

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