To solve self consistent equations by simultaneously plotting them

Question

How can we solve a pair of equations that have to be solved self-consistently (even by plotting them simultaneously) .

y = 1/2 + 1/\[Pi] ArcTan[c (0.5 - x)];
x = 1/2 + 1/\[Pi] ArcTan[c (0.5 - y)];

For c=0.5 we must have a plot as below one in which the vertical axis is y and horizontal one is devoted to x.

For c=Pi we must have

I wanted to plot the first function y vs x ordinary and by InverseFunction[f][y] I wanted to plot the second one (x vs y). For the second one the output plot is not symmetric relative to bisectrix of the first quarter of the coordination plate. — Unbelievable, Mar 27, 2019 at 10:38

Hugh · Accepted Answer · 2019-03-27 10:53:00Z

2

With[{c = 0.5},
 ContourPlot[{
   y == 1/2 + 1/π ArcTan[c (0.5 - x)],
   x == 1/2 + 1/π ArcTan[c (0.5 - y)]
   }, 
  {x, 0, 1}, {y, 0, 1}]
 ]

Mathematica graphics

With[{c = π},
 ContourPlot[{
   y == 1/2 + 1/π ArcTan[c (0.5 - x)],
   x == 1/2 + 1/π ArcTan[c (0.5 - y)]
   }, 
  {x, 0, 1}, {y, 0, 1}]
 ]

Mathematica graphics

answered Mar 27, 2019 at 10:53

Hugh

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1

$\begingroup$ One can even use MeshFunctions to mark the intersection point. $\endgroup$
– J. M.'s eventual burnout ♦
Mar 27, 2019 at 13:36

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