Equation Solving, intersection of graphs [duplicate]

FindRoot is the first thing that comes to mind with this type of problems. First, create a list of initial conditions for the root search, based on the plot (rough estimates are sufficient):

init = {{0.5, 0}, {0.75, 0.75}, {0.75, -0.75}, {1, 1}, {1, -1}}; 

Make a pure function that will find the roots:

f = FindRoot[{x == 1/2 (1 + Tanh[(x y^2)/0.5]), y == Tanh[(x^2 y)/0.5]}, {{x, #1}, {y, #2}}] & 

and apply it to the initial conditions:

roots = f @@@ init 

{{x -> 0.5, y -> 0.}, {x -> 0.764865, y -> 0.620918}, {x -> 0.764865, y -> -0.620918}, {x -> 0.969417, y -> 0.944097}, {x -> 0.969417, y -> -0.944097}}

or just

{x, y} /. roots 

{{0.5, 0.}, {0.764865, 0.620918}, {0.764865, -0.620918}, {0.969417, 0.944097}, {0.969417, -0.944097}}

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EDIT: I developed a different method, suitable for this problem also, while answering another question.

Let's take the plot in the form

curve = ContourPlot[{x == 1/2 (1 + Tanh[(x y^2)/0.5]), y == Tanh[(x^2 y)/0.5]}, {x, 0, 2}, {y, -1, 1}, Frame -> None, PlotRangePadding -> None] 

and extract the pixel positions of intersection with

px = PixelValuePositions[#, White] & @ MorphologicalBranchPoints @ Thinning @ Binarize @ ColorNegate @ curve 

We need to connect them to the actual coordinates on the plot:

pl = PlotRange@curve 

{{0., 2.}, {-1., 1.}}

id = ImageDimensions@curve 

{360, 360}

The relation between pl and id is linear, $y=ax+b$ and $y=cx+d$ in the horizontal and vertical directions, respecively:

{a, b} = {a, b} /. First@Solve[{pl[[1, 1]] == b, pl[[1, 2]] == a id[[1]] + b}, {a, b}] {c, d} = {c, d} /. First@Solve[{pl[[2, 1]] == d, pl[[2, 2]] == c id[[2]] + d}, {c, d}] 

{0.00555556, 0.}

{0.00555556, -1.}

The pixel positions transformed to plot coordinates:

ic = {a #1 + b, c #2 + d} & @@@ px 

{{0.983333, 0.961111}, {0.988889, 0.955556}, {0.977778, 0.95}, {0.961111, 0.933333}, {0.783333, 0.638889}, {0.761111, 0.605556}, {0.755556, 0.6}, {0.761111, 0.594444}, {0.505556, 0.}, {0.711111, 0.}, {0.716667, 0.}, {0.761111, -0.594444}, {0.755556, -0.6}, {0.761111, \ -0.605556}, {0.777778, -0.638889}, {0.961111, -0.933333}, {0.977778, \ -0.95}}

look like this:

Not exactly at the intersections, but this is good enough. We can get rid of the ambiguity with clustering:

clu = ClusterClassify[ic, Method -> "DBSCAN"] 

Number of clusters: 6

g = GatherBy[ic, clu] icmean = Reverse[Mean /@ g] 

{{0.969444, -0.941667}, {0.763889, -0.609722}, {0.713889, 0.}, {0.505556, 0.}, {0.765278, 0.609722}, {0.977778, 0.95}}

Show[curve, Graphics[{PointSize[Large], Point[#]}] &@icmean] 

which looks very good. The additional intersection of the orange curve with itself was found additinally.