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# How to plot multiple equilibria?

Posted 9 years ago
 I have a function In[186]:= v[k_, x_] = u1[x] - u2[x] Out[186]= 1 - 10 (2 - k)^x + (1 - k)/(2 k) - 2 k - k/(2 (1 - k)) + 10 (1 + k)^x + 1/2 (-(1 - k)^2 - k^2) + 1/2 ((1 - k)^2 + k^2) - Log[2/(Sqrt[15] Sqrt[(1 - k)/k])] + Log[(2 Sqrt[3/5])/Sqrt[k/(1 - k)]]  k is the variable of interest and x is a parameter. A solution is found when v[k,x]==0. As x increases multiple equilibria arise. What I want to do is have a plot that shows the multiple equilibria, i.e. a function k[x], to see the point where multiple equilibria arise. I tried to write the function as the following: k1[x_] := FindRoot[v[k, x] == 0, {k, 1}] k1[x_?NumericQ] := k /. FindRoot[v[k, x] == 0, {k, .5}]  This provides a solution, but only provides a local solution around the starting point and if I change the starting point I get a different solution. How can I plot a figure that will show all the equilibria over a certain range? Any help would be appreciated.
 This is a way: v[k_, x_] = 1 - 10 (2 - k)^x + (1 - k)/(2 k) - 2 k - k/(2 (1 - k)) + 10 (1 + k)^x + 1/2 (-(1 - k)^2 - k^2) + 1/2 ((1 - k)^2 + k^2) - Log[2/(Sqrt[15] Sqrt[(1 - k)/k])] + Log[(2 Sqrt[3/5])/Sqrt[k/(1 - k)]]; Manipulate[ Plot[v[k, x], {k, 0, 1}, Epilog -> {PointSize[Large], Point[{k, 0} /. Solve[Rationalize[v[k, x]] == 0 && 0 < k < 1, k]]}], {x, 0, 4}]