# I have a potential but I can not obtained the equilibrium points from this potential. How can I plot the basin of attraction at equilibrium points

My potential is the form as:

$$U = 0.515755 (x^2 + y^2) - 1.0156333 x(0.0200/((-0.1 + x)^2 + y^2)^{(3/2)} + 0.006750/((0.9 + x)^2 + y^2)^{(7/2)} - 0.0180000/((0.9 + x)^2 + y^2)^{(5/2)} - 3.6/((0.9 + x)^2 + y^2)^{(3/2)} - 0.3600000/(((0.9 + x)^2 + y^2) \sqrt{0.0001 + (x^2 + y^2)^2})) + 1.0156333 (x^2 + y^2) (0.2/((-0.1 + x)^2 + y^2)^{(3/2)} - 0.0075/((0.9 + x)^2 + y^2)^{(7/2)} + 0.02/((0.9 + x)^2 + y^2)^{(5/2)} + 4/((0.9 + x)^2 + y^2)^{(3/2)} + 0.4/(((0.9 + x)^2 + y^2) \sqrt{0.0001 + (x^2 + y^2)^2}))$$

In Mathematica code:

U[x_, y_]:=
0.515755 (x^2 + y^2) - 1.0156333 x (0.0200/((-0.1 + x)^2 + y^2)^( 3/2) +
0.006750/((0.9 + x)^2 + y^2)^(7/2) - 0.0180000/((0.9 + x)^2 + y^2)^(5/2) -
3.6/((0.9 + x)^2 + y^2)^(3/2) - 0.3600000/(((0.9 + x)^2 + y^2)*Sqrt[0.0001 + (x^2 + y^2)^2])) +
1.0156333*(x^2 + y^2)*(0.2/((-0.1 + x)^2 + y^2)^(3/2) - 0.0075/((0.9 + x)^2 + y^2)^(7/2) +
0.02/((0.9 + x)^2 + y^2)^( 5/2) + 4/((0.9 + x)^2 + y^2)^(3/2) +
0.4/(((0.9 + x)^2 + y^2)*Sqrt[0.0001 + (x^2 + y^2)^2])

• What have you tried? Mar 15, 2020 at 14:20
• Welcome to Mathematica.SE! I hope you will become a regular contributor. To get started, 1) take the introductory tour now, 2) when you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge, 3) remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign, and 4) give help too, by answering questions in your areas of expertise. Mar 15, 2020 at 14:22

## 1 Answer

your Mathematica code has some syntax error I corrected it. This is what I have done:

1) eqn

 U[x_, y_] :=  0.515755 (x^2 + y^2) - 1.0156333 x (0.0200/((-0.1 + x)^2 + y^2)^(3/2) +
0.006750/((0.9 + x)^2 + y^2)^(7/2) -
0.0180000/((0.9 + x)^2 + y^2)^(5/2) -
3.6/((0.9 + x)^2 + y^2)^(3/2) -
0.3600000/(((0.9 + x)^2 + y^2)*Sqrt[0.0001 + (x^2 + y^2)^2])) + +1.0156333*(x^2 + y^2) (0.2/((-0.1 + x)^2 + y^2)^(3/2) -
0.0075/((0.9 + x)^2 + y^2)^(7/2) +
0.02/((0.9 + x)^2 + y^2)^(5/2) + 4/((0.9 + x)^2 + y^2)^(3/2) +
0.4/(((0.9 + x)^2 + y^2)*Sqrt[0.0001 + (x^2 + y^2)^2]));


2) Take partial Derivatives to find functions to minimize

   funs = D[U[x, y], #] & /@ {x, y};


3) Generally, you equate these to zero and solve it for variables but for a complex equation often we resort to numerical methods. You can use FindAllCrossings2D for this purpose but you need to know ranges of x and y's:

   pts = FindAllCrossings2D[funs, {x, -1, 1}, {y, -1, 1}, Method ->
{"Newton", "StepControl" -> "LineSearch"},  PlotPoints -> 85]//Chop
`

Just do a ListPlot of these to find the plot of x and y values.